Along the lines (of force). |

Conversion Between Tesla and Gauss

The Gauss (G) and Tesla (T) units are related as:

[math]1 \text{ Tesla} = 10^4 \text{ Gauss}[/math]

Thus, for a given magnetic flux density B:

[math]B \text{ (in Gauss)} = B \text{ (in Tesla)} \times 10^4[/math]

Understanding Steinmetz’s Definition

Steinmetz used Magnetic Lines of Force in his electromagnetic theory. His system defined:

One line per square centimeter as one Gauss.

Since one Maxwell is one line of magnetic flux, this means:

[math]1 \text{ Gauss} = \frac{1 \text{ Maxwell}}{1 \text{ cm}^2}[/math]

This was a part of the CGS (centimeter-gram-second) unit system.

Comparing to SI Units

In SI units:

One Tesla is one Weber per square meter:

[math]1 \text{ Tesla} = \frac{1 \text{ Weber}}{1 \text{ m}^2}[/math]

Since one Weber is equal to [math]10^8[/math] Maxwells, we can rewrite:

[math]1 \text{ Tesla} = \frac{10^8 \text{ Maxwells}}{10^4 \text{ cm}^2} = 10^4 \text{ Gauss}[/math]

which confirms our earlier Tesla-to-Gauss conversion.

Conclusion: One Line per cm² = One Gauss

Steinmetz’s definition aligns with the CGS system, where:

[math]1 \text{ Gauss} = 1 \text{ Maxwell per cm}^2[/math]

And using modern SI units:

[math]1 \text{ Tesla} = 10^4 \text{ Gauss}[/math]

So, in Steinmetz’s scheme, a field of one line per square centimeter directly translates to one Gauss. This is consistent with the classical CGS electromagnetic system.

Below are quotes from Charles Steinmetz

Transients in Time

Chapter One

The Constants of the Electric Circuit

One. To transmit electric energy from one place where it is generated to another place where it is used, an electric circuit is required, consisting of conductors which connect the point of generation with the point of utilization.

When electric energy flows through a circuit, phenomena take place inside of the conductor as well as in the space outside of the conductor.

In the conductor, during the flow of electric energy through the circuit, electric energy is consumed continuously by being converted into heat. Along the circuit, from the generator to the receiver circuit, the flow of energy steadily decreases by the amount consumed in the conductor, and a power gradient exists in the circuit along or parallel with the conductor.

Thus, while the voltage may decrease from generator to receiver circuit, as is usually the case, or may increase, as in an alternating current circuit with leading current, and while the current may remain constant throughout the circuit, or decrease, as in a transmission line of considerable capacity with a leading or non-inductive receiver circuit, the flow of energy always decreases from generating to receiving circuit, and the power gradient therefore is characteristic of the direction of the flow of energy.

In the space outside of the conductor, during the flow of energy through the circuit, a condition of stress exists which is called the electric field of the conductor. That is, the surrounding space is not uniform, but has different electric and magnetic properties in different directions.

No power is required to maintain the electric field, but energy is required to produce the electric field, and this energy is returned, more or less completely, when the electric field disappears by the stoppage of the flow of energy.

Thus, in starting the flow of electric energy, before a permanent condition is reached, a finite time must elapse during which the energy of the electric field is stored, and the generator therefore gives more power than consumed in the conductor and delivered at the receiving end; again, the flow of electric energy cannot be stopped instantly, but first the energy stored in the electric field has to be expended. As a result hereof, where the flow of electric energy pulsates, as in an alternating current circuit, continuously electric energy is stored in the field during a rise of the power, and returned to the circuit again during a decrease of the power.

The electric field of the conductor exerts magnetic and electrostatic actions.

The magnetic action is a maximum in the direction concentric, or approximately so, to the conductor. That is, a needle-shaped magnetizable body, as an iron needle, tends to set itself in a direction concentric to the conductor.

The electrostatic action has a maximum in a direction radial, or approximately so, to the conductor. That is, a light needle-shaped conducting body, if the electrostatic component of the field is powerful enough, tends to set itself in a direction radial to the conductor, and light bodies are attracted or repelled radially to the conductor.

Thus, the electric field of a circuit over which energy flows has three main axes which are at right angles with each other:

The electromagnetic axis, concentric with the conductor.
The electrostatic axis, radial to the conductor.
The power gradient, parallel to the conductor.

This is frequently expressed pictorially by saying that the lines of magnetic force of the circuit are concentric, the lines of electrostatic force radial to the conductor.

Where, as is usually the case, the electric circuit consists of several conductors, the electric fields of the conductors superimpose upon each other, and the resultant lines of magnetic and of electrostatic forces are not concentric and radial respectively, except approximately in the immediate neighborhood of the conductor.

In the electric field between parallel conductors, the magnetic and the electrostatic lines of force are conjugate pencils of circles.

Two. Neither the power consumption in the conductor, nor the electromagnetic field, nor the electrostatic field, are proportional to the flow of energy through the circuit.

The product, however, of the intensity of the magnetic field, represented as [math]\Phi[/math], and the intensity of the electrostatic field, represented as [math]\Psi[/math], is proportional to the flow of energy or the power, denoted as [math]P[/math]. The power [math]P[/math] is therefore resolved into a product of two components, [math]i[/math] and [math]e[/math], which are chosen proportional respectively to the intensity of the magnetic field [math]\Phi[/math] and of the electrostatic field [math]\Psi[/math].

That is, putting the power [math]P[/math] as equal to the product of [math]i[/math] and [math]e[/math], we have the following relationships:

[math]\Phi = L i[/math] and represents the intensity of the electromagnetic field.

[math]\Psi = C e[/math] and represents the intensity of the electrostatic field.

The component [math]i[/math], called the current, is defined as that factor of the electric power [math]P[/math] which is proportional to the magnetic field, and the other component [math]e[/math], called the voltage, is defined as that factor of the electric power [math]P[/math] which is proportional to the electrostatic field.

Current [math]i[/math] and voltage [math]e[/math], therefore, are mathematical fictions, factors of the power [math]P[/math], introduced to represent respectively the magnetic and the electrostatic or dielectric phenomena.

The current [math]i[/math] is measured by the magnetic action of a circuit, as in the ammeter; the voltage [math]e[/math], by the electrostatic action of a circuit, as in the electrostatic voltmeter, or by producing a current [math]i[/math] by the voltage [math]e[/math] and measuring this current [math]i[/math] by its magnetic action, in the usual voltmeter.

The coefficients [math]L[/math] and [math]C[/math], which are the proportionality factors of the magnetic and of the dielectric component of the electric field, are called the inductance and the capacity of the circuit, respectively.

As electric power [math]P[/math] is resolved into the product of current [math]i[/math] and voltage [math]e[/math], the power loss in the conductor, represented as [math]P_l[/math], therefore can also be resolved into a product of current [math]i[/math] and voltage [math]e_l[/math], which is consumed in the conductor. That is,

[math]P_l = i e_l[/math].

It is found that the voltage consumed in the conductor, represented as [math]e_l[/math], is proportional to the factor [math]i[/math] of the power [math]P[/math]. That is,

[math]e_l = r i[/math],

where [math]r[/math] is the proportionality factor of the voltage consumed by the loss of power in the conductor, or by the power gradient, and is called the resistance of the circuit.

Any electric circuit, therefore, must have three constants: [math]r[/math], [math]L[/math], and [math]C[/math], where

[math]r[/math] is the circuit constant representing the power gradient, or the loss of power in the conductor, called resistance.
[math]L[/math] is the circuit constant representing the intensity of the electromagnetic component of the electric field of the circuit, called inductance.
[math]C[/math] is the circuit constant representing the intensity of the electrostatic component of the electric field of the circuit, called capacity.

In most circuits, there is no current consumed in the conductor, represented as [math]i_l[/math], and proportional to the voltage factor [math]e[/math] of the power [math]P[/math]. That is,

[math]i_l = g e[/math],

where [math]g[/math] is the proportionality factor of the current consumed by the loss of power in the conductor, which depends on the voltage, such as dielectric losses. Where such losses exist, a fourth circuit constant appears, called the conductance [math]g[/math]. For further details, refer to sections three and four.

Three. A change of the magnetic field of the conductor, that is, if the number of lines of magnetic force [math]\Phi[/math] surrounding the conductor changes, generates an electromotive force, denoted as [math]e'[/math], which is equal to the derivative of [math]\Phi[/math] with respect to time.

Thus,

[math]e’ = \frac{d\Phi}{dt}[/math],

and the conductor absorbs a power [math]P'[/math], which is equal to

[math]P’ = i e'[/math].

Substituting, this becomes

[math]P’ = i \frac{d\Phi}{dt}[/math].

Using the relationship where

[math]\Phi = L i[/math],

we find that

[math]\frac{d\Phi}{dt} = L \frac{di}{dt}[/math].

Substituting back, we find that the power [math]P'[/math] is

[math]P’ = L i \frac{di}{dt}[/math].

The total energy absorbed by the magnetic field during the rise of current from zero to [math]i[/math] is represented as [math]W_M[/math] and is calculated as the integral of [math]P'[/math] over time.

This simplifies to

[math]W_M = \int P’ dt = L \int i \frac{di}{dt} dt[/math].

Since [math]\frac{di}{dt} dt = di[/math], we obtain

[math]W_M = L \int i di[/math].

That is, the energy stored in the magnetic field of the circuit, represented as [math]W_M[/math], is equal to

[math]W_M = \frac{1}{2} L i^2[/math].

A change of the dielectric field of the conductor, represented as [math]\Psi[/math], absorbs a current proportional to the change of the dielectric field. This current is denoted as [math]i'[/math] and is equal to

[math]i’ = \frac{d\Psi}{dt}[/math].

The power absorbed by this process, represented as [math]P”[/math], is equal to

[math]P” = e i’ = e \frac{d\Psi}{dt}[/math].

Using the relationship where

[math]\Psi = C e[/math],

we substitute to find that the power [math]P”[/math] is equal to

[math]P” = C e \frac{de}{dt}[/math].

The total energy absorbed by the dielectric field during a rise of voltage from zero to [math]e[/math] is represented as [math]W_K[/math] and is calculated as the integral of [math]P”[/math] over time. This simplifies to

[math]W_K = \int P” dt = C \int e \frac{de}{dt} dt[/math].

Since [math]\frac{de}{dt} dt = de[/math], we obtain

[math]W_K = C \int e de[/math].

That is, the energy stored in the dielectric field, represented as [math]W_K[/math], is equal to

[math]W_K = \frac{1}{2} C e^2[/math].

The power consumed in the conductor by its resistance, represented as [math]r[/math], is denoted as [math]P_r[/math] and is equal to

[math]P_r = i e_l[/math].

Substituting, we find that

[math]P_r = i^2 r[/math].

Thus, when the electric power, represented as [math]P[/math], is equal to

[math]P = e i[/math],

the following components exist in a circuit:

The power lost in the conductor is represented as [math]P_r[/math] and is equal to [math]i^2 r[/math].
The energy stored in the magnetic field of the circuit is represented as [math]W_M[/math] and is equal to [math]\frac{1}{2} L i^2[/math].
The energy stored in the dielectric field of the circuit is represented as [math]W_K[/math] and is equal to [math]\frac{1}{2} C e^2[/math].

…and the three circuit constants, resistance denoted as [math]r[/math], inductance denoted as [math]L[/math], and capacity denoted as [math]C[/math], therefore appear as the components of energy conversion into heat, magnetism, and electric stress, respectively, in the circuit.

Circuit Constants and Their Dependence

Four. The circuit constant resistance, denoted as [math]r[/math], depends only on the size and material of the conductor, but not on the position of the conductor in space, nor on the material filling the space surrounding the conductor, nor on the shape of the conductor section.

The circuit constants inductance, denoted as [math]L[/math], and capacity, denoted as [math]C[/math], almost entirely depend on the position of the conductor in space, on the material filling the space surrounding the conductor, and on the shape of the conductor section, but do not depend on the material of the conductor, except to a small extent as represented by the electric field inside the conductor section.

Five. The resistance, denoted as [math]r[/math], is proportional to the length and inversely proportional to the section of the conductor. This relationship is expressed as

[math]r = \rho \frac{l}{A}[/math],

where [math]\rho[/math] is a constant of the material, called the resistivity or specific resistance.

For different materials, the value of [math]\rho[/math] varies probably over a far greater range than almost any other physical quantity. Expressed in ohms per cubic centimeter, it is approximately as follows at ordinary temperatures:

Metals:

Copper: [math]1.6 \times 10^{-6}[/math]
Aluminum: [math]2.8 \times 10^{-6}[/math]
Iron: [math]10 \times 10^{-6}[/math]
Mercury: [math]94 \times 10^{-6}[/math]
Gray cast iron: up to [math]100 \times 10^{-6}[/math]
High-resistance alloys: up to [math]150 \times 10^{-6}[/math]

Electrolytes:

Nitric acid (30% concentration): as low as [math]1.3[/math]
Potassium hydroxide (25% concentration): as low as [math]1.9[/math]
Sodium chloride (25% concentration): as low as [math]4.7[/math]

Values increase from pure river water and extend over alcohols, oils, and similar materials to practically infinity.

Note: These values assume a conductor of one centimeter length and one square centimeter cross-sectional area.

So-called insulators have resistivity values as follows:

Fiber: approximately [math]10^{12}[/math]
Paraffin oil: approximately [math]10^{13}[/math]
Paraffin: ranges from approximately [math]10^{14}[/math] to [math]10^{18}[/math]
Mica: approximately [math]10^{14}[/math]
Glass: ranges from approximately [math]10^{14}[/math] to [math]10^{18}[/math]
Rubber: approximately [math]10^{10}[/math]
Air: practically infinite in resistivity.

In the wide gap between the highest resistivity of metal alloys, about [math]150 \times 10^{-6}[/math], and the lowest resistivity of electrolytes, about [math]1[/math], we find the following:

Carbon:

Metallic carbon: as low as [math]100 \times 10^{-6}[/math]
Amorphous dense carbon: starting at [math]0.04[/math] and higher
Anthracite carbon: exhibits very high resistivity

Silicon and silicon alloys:

Cast silicon: varying resistivity
Ferro silicon: ranges from [math]0.04[/math] down to [math]50 \times 10^{-6}[/math]

The resistivity of arcs and of Geissler tube discharges is of about the same magnitude as electrolytic resistivity.

The resistivity, represented as [math]\rho[/math], is usually a function of temperature. It rises slightly with an increase in temperature in metallic conductors but decreases in electrolytic conductors. Only with a few materials, such as silicon, does the temperature variation of [math]\rho[/math] become so significant that [math]\rho[/math] can no longer be considered approximately constant for all currents which produce a considerable temperature rise in the conductor. Such materials are commonly referred to as pyroelectrolytes.

Six. The inductance, represented as [math]L[/math], is proportional to the section and inversely proportional to the length of the magnetic circuit surrounding the conductor. It can be expressed as

[math]L = \mu \frac{A}{l}[/math],

where [math]\mu[/math] is a constant of the material filling the space surrounding the conductor and is called the magnetic permeability.

As in general, neither section nor length is constant in different parts of the magnetic circuit surrounding an electric conductor, the magnetic circuit has, as a rule, to be calculated piecemeal or by integration over the space occupied by it.

The permeability, represented as [math]\mu[/math], is constant and equals unity, or very closely [math]\mu = 1[/math], for all substances, with the exception of a few materials referred to as magnetic materials, such as iron, cobalt, nickel, and similar substances. In these materials, permeability is significantly higher, reaching values as high as six thousand and, under certain conditions in iron, as high as thirty thousand.

In magnetic materials, the permeability [math]\mu[/math] is not constant but varies with the magnetic flux density, which is the number of lines of magnetic force per unit section, represented as [math]\Theta[/math], and decreases rapidly for high values of [math]\Theta[/math].

In such materials, the term [math]\mu[/math] becomes inconvenient, and the inductance, represented as [math]L[/math], is calculated using the relation between the magnetizing force, given in ampere-turns per unit length of the magnetic circuit (referred to as field intensity), and magnetic induction.

The magnetic induction, represented as [math]Q[/math], in magnetic materials is the sum of the space induction, represented as [math]H[/math], corresponding to unit permeability, plus the metallic induction, represented as [math]B'[/math], which reaches a finite limiting value. That is,

[math]Q = H + B'[/math].

The limiting values, or so-called saturation values, of [math]B'[/math] are approximately as follows, in lines of magnetic force per square centimeter:

Iron: 21,000
Cobalt: 12,000
Nickel: 6,000
Magnetite: 5,000
Manganese alloys: up to 5,000

The inductance, represented as [math]L[/math], is a constant of the circuit if the space surrounding the conductor contains no magnetic material. However, if magnetic material exists in the space surrounding the conductor, the inductance becomes more or less variable with the current, represented as [math]i[/math].

In such cases, as the current [math]i[/math] increases, the inductance [math]L[/math] first slightly increases, then reaches a maximum, and subsequently decreases, approaching as a limiting value the inductance it would have if no magnetic material were present in the space surrounding the conductor.

Electrostatic Capacity and Dielectric Constant

Seven. The capacity, represented as [math]C[/math], is proportional to the section and inversely proportional to the length of the electrostatic field of the conductor. This relationship is expressed as

[math]C = \kappa \frac{A}{l}[/math],

where [math]\kappa[/math] is a constant of the material filling the space surrounding the conductor. This constant is referred to as the dielectric constant, specific capacity, or permittivity.

Usually, the section and length of different parts of the electrostatic circuit vary, and the capacity therefore must be calculated piecemeal or by integration.

The dielectric constant, represented as [math]\kappa[/math], varies only over a relatively narrow range across different materials. It is approximately as follows:

[math]\kappa = 1[/math] in a vacuum, in air, and in other gases.
[math]\kappa[/math] ranges from 2 to 3 in oils, paraffins, fibers, and similar materials.
[math]\kappa[/math] ranges from 3 to 4 in rubber and gutta-percha.
[math]\kappa[/math] ranges from 3 to 5 in glass and mica.
[math]\kappa[/math] can reach values as high as 7 to 8 in organic compounds of heavy metals, such as lead stearate, and approximately 12 in sulfur.

The dielectric constant, [math]\kappa[/math], is practically constant for all voltages, represented as [math]e[/math], up to that voltage at which the electrostatic field intensity, or the electrostatic gradient (measured as volts per centimeter), exceeds a certain value represented as [math]\delta[/math]. This value, [math]\delta[/math], depends on the material and is called the dielectric strength or disruptive strength of the material.

At this potential gradient, the medium breaks down mechanically by puncture and ceases to insulate, allowing electricity to pass and equalizing the potential gradient.

The approximate disruptive strengths, represented as [math]\delta[/math] and measured in volts per centimeter, are as follows:

Air: 30,000
Oils: 250,000 to 1,000,000
Mica: up to 4,000,000

The capacity, represented as [math]C[/math], of a circuit is constant up to the voltage, [math]e[/math], at which, in some part of the electrostatic field, the dielectric strength is exceeded. At this point, disruption occurs, rendering part of the surrounding space conductive. This increases the effective size of the conductor and thereby increases the capacity, [math]C[/math]. Eight. Of the amount of energy consumed in creating the electric field of the circuit, not all is returned upon the disappearance of the electric field. A part of this energy is consumed by conversion into heat or by changing the electric field in other ways. That is, the conversion of electric energy into and from electromagnetic and electrostatic stress is not entirely complete, and energy loss occurs. This is especially notable with the magnetic field in so-called magnetic materials and with the electrostatic field in non-homogeneous dielectrics.

The energy loss associated with the production and reconversion of the magnetic component of the field can be represented by an effective resistance, denoted as [math]r'[/math], which adds to the resistance of the conductor, [math]r_0[/math], and thereby increases it.

The energy loss in the electrostatic field can be represented by an effective resistance, denoted as [math]r”[/math], which acts as a shunt across the circuit and consumes an energy current, denoted as [math]i”[/math], in addition to the current, [math]i[/math], in the conductor. Typically, instead of using the effective resistance [math]r”[/math], its reciprocal is used, and the energy loss in the electrostatic field is represented by a shunted conductance, denoted as [math]g[/math].

In its most general form, the electric circuit therefore contains the following constants:

Inductance, represented as [math]L[/math], stores energy equal to [math]\frac{1}{2} L i^2[/math].
Capacity, represented as [math]C[/math], stores energy equal to [math]\frac{1}{2} C e^2[/math].
Resistance, represented as [math]r = r_0 + r'[/math], consumes power equal to [math]i^2 r[/math]. This can be further broken down as [math]i^2 r_0 + i^2 r'[/math].
Conductance, represented as [math]g[/math], consumes power equal to [math]e^2 g[/math].

Here, [math]r_0[/math] represents the resistance of the conductor, [math]r'[/math] represents the effective resistance accounting for power loss in the magnetic field [math]L[/math], and [math]g[/math] represents the power loss in the electrostatic field [math]C[/math].

Electromagnetic and Electrostatic Fields in Electric Circuits

Nine. If, among the three components of the electric field—namely, the electromagnetic stress, the electrostatic stress, and the power gradient—any one of them equals zero, a second component must also equal zero. In other words, either all three components are present, or only one exists.

Electric systems in which the magnetic component of the field is absent, while the electrostatic component may be considerable, are exemplified by an electric generator or a battery operating on an open circuit, or by an electrostatic machine. In such systems, the disruptive effects caused by high voltage are therefore most pronounced, while the power is negligible, and phenomena of this character are usually referred to as “static.”

Of approximately equal magnitude are the electromagnetic energy, represented as [math]\frac{1}{2} L i^2[/math], and the electrostatic energy, represented as [math]\frac{1}{2} C e^2[/math], in high-potential long-distance transmission circuits, telephone circuits, and condenser discharges. This also applies to many phenomena resulting from lightning or other disturbances. In such cases, all three circuit constants—[math]r[/math], [math]L[/math], and [math]C[/math]—are of essential importance.

Transient Phenomena and Energy Readjustment

Ten. In an electric circuit with negligible inductance, [math]L[/math], and negligible capacity, [math]C[/math], no energy is stored, and changes in the circuit can occur instantly without any disturbance or intermediary transient condition.

In a circuit containing only resistance and capacity, as in a static machine, or only resistance and inductance, as in a low- or medium-voltage power circuit, electric energy is stored primarily in one form. In these cases, a change in the circuit, such as opening the circuit, cannot occur instantly but happens more gradually, as the energy must first be stored or discharged.

In a circuit containing resistance, inductance, and capacity—capable of storing energy in two different forms—the mechanical change of circuit conditions, such as opening the circuit, can occur instantly. In this scenario, the internal energy of the circuit adjusts to the changed conditions through a transfer of energy between the static and magnetic forms, and vice versa. After the change in circuit conditions, a transient phenomenon, often oscillatory in nature, occurs as the stored energy readjusts.

These transient phenomena, involving the readjustment of stored electric energy when circuit conditions change, require careful study in cases where the stored energy is sufficiently large to cause serious damage. This is analogous to the readjustment of stored energy in mechanical motion. For example, while instantly stopping a slowly moving light carriage may be harmless, the instant stoppage of a fast-moving railway train, such as in a collision, typically results in disastrous consequences.

Similarly, in electric systems with small amounts of stored energy, sudden changes in circuit conditions may be safe, but in high-potential power systems with significant stored electric energy, any sudden change in circuit conditions requiring a rapid energy adjustment is likely to be destructive.

Where electric energy is stored in only one form, there is typically little danger, as the circuit naturally protects itself against sudden changes by retarding the change through energy adjustment. However, in circuits where energy is stored both electrostatically and magnetically, the mechanical change of circuit conditions, such as the opening of the circuit, can occur instantly. In such cases, the stored energy surges back and forth between its electrostatic and magnetic forms.

In the following sections, the phenomena resulting from the stored energy and its readjustment in circuits storing energy in only one form—most commonly as electromagnetic energy—will be considered first. Subsequently, the general problem of circuits storing energy both electromagnetically and electrostatically will be addressed.

Electric systems in which the electrostatic component of the field is absent, while the electromagnetic component is considerable, are exemplified by the short-circuited secondary coil of a transformer. In such systems, no potential difference exists and, consequently, no electrostatic field, as the generated electromotive force is entirely consumed at the place of generation. The electrostatic component is also practically negligible in all low-voltage circuits.

The effect of resistance on the flow of electric energy in industrial applications is limited to fairly narrow ranges. Since the resistance of the circuit consumes power, thereby lowering the efficiency of electric transmission, it is uneconomical to allow excessive resistance. Conversely, lowering the resistance too much requires greater expenditure on conductor material, which may also be uneconomical.

As a result, the relative resistance, defined as the ratio of the power lost in the resistance to the total power, typically lies between 2% and 20%.

The situation is different for the inductance, denoted as [math]L[/math], and the capacity, denoted as [math]C[/math]. Of the two forms of stored energy—magnetic energy, equal to [math]\frac{1}{2} L i^2[/math], and electrostatic energy, equal to [math]\frac{1}{2} C e^2[/math]—one is often so small that it can be neglected compared with the other. Thus, an electric circuit can often be approximated as containing resistance and inductance only, or resistance and capacity only.

In the so-called electrostatic machine and its applications, capacity and resistance frequently dominate the considerations.

In all lighting and power distribution circuits—whether direct current or alternating current—such as the 110V and 220V lighting circuits, the 500V railway circuits, or the 2000V primary distribution circuits, the relatively low voltage renders the electrostatic energy, represented as [math]\frac{1}{2} C e^2[/math], very small compared with the electromagnetic energy. Therefore, in such circuits, the capacity, [math]C[/math], can often be neglected for most purposes, and the circuit can be treated as containing resistance and inductance only.

Chapter Two

Introduction

Eleven. In the investigation of electrical phenomena, currents and potential differences—whether continuous or alternating—are usually treated as stationary phenomena. This means it is assumed that after establishing the circuit, sufficient time has passed for the currents and potential differences to reach their final or permanent values. These values are constant for continuous current or constant periodic functions of time for alternating current. However, at the very moment of establishing the circuit, the currents and potential differences have not yet reached their permanent values. That is, the electrical conditions of the circuit are not yet normal or permanent, and a certain amount of time is required for the electrical conditions to adjust.

Current Rise in a Resistive and Inductive Circuit

Twelve. For example, a continuous electromotive force, represented as [math]e_0[/math], impressed upon a circuit with resistance [math]r[/math], produces and maintains in the circuit a current, represented as [math]i_0[/math], calculated as

[math]i_0 = \frac{e_0}{r}[/math].

At the moment of closing the circuit of electromotive force [math]e_0[/math] on resistance [math]r[/math], the current in the circuit is initially zero. Therefore, after closing the circuit, the current, represented as [math]i[/math], must rise from zero to its final value, [math]i_0[/math]. If the circuit contained only resistance and no inductance, this rise would occur instantly, with no transition period. However, every circuit contains some inductance.

The inductance, represented as [math]L[/math], of the circuit signifies [math]L[/math] interlinkages of the circuit with lines of magnetic force produced by a unit current in the circuit, or [math]i L[/math] interlinkages for a current [math]i[/math]. That is, in establishing a current [math]i_0[/math] in the circuit, a magnetic flux equal to [math]i_0 L[/math] must be produced.

A change in the magnetic flux, represented as [math]i L[/math], surrounding a circuit generates an electromotive force in the circuit. This electromotive force, represented as [math]e[/math], is equal to the time derivative of [math]i L[/math], or

[math]e = L \frac{di}{dt}[/math].

This opposes the impressed electromotive force, represented as [math]e_0[/math], and therefore reduces the electromotive force available to produce the current, thereby lowering the current. As a result, the current cannot instantly reach its final value but rises to it gradually. Thus, between the moment the circuit is started and the establishment of a permanent condition, a transition period occurs.

Charging a Condenser in a Circuit with Resistance and Inductance

At the moment of closing the circuit, an infinite current would occur, charging the condenser instantly to the potential difference [math]e_0[/math]. If [math]r[/math] represents the resistance of the direct-current circuit containing the condenser, and the circuit contains no inductance, the current starts at the value

[math]i = \frac{e_0}{r}[/math].

That is, in the first moment after closing the circuit, all the impressed electromotive force is consumed by the current in the resistance, as no charge and therefore no potential difference exists at the condenser.

With increasing charge of the condenser and therefore increasing potential difference at its terminals, less electromotive force is available for the resistance, and the current decreases. Ultimately, the current becomes zero when the condenser is fully charged.

If the circuit also contains inductance, represented as [math]L[/math], then the current cannot rise instantly but does so gradually. In the moment after closing the circuit, the potential difference at the condenser is still zero and rises at a rate such that the increase of magnetic flux, represented as [math]i L[/math], in the inductance produces an electromotive force,

[math]L \frac{di}{dt}[/math],

which consumes the impressed electromotive force.

Gradually, the potential difference at the condenser increases with its increasing charge. The current, and thus the electromotive force consumed by the resistance, also increases. As less electromotive force becomes available for the inductance, the current increases more slowly until it ultimately ceases to rise and reaches a maximum. At this point, the inductance consumes no electromotive force, and all the impressed electromotive force is consumed by the current in the resistance and by the potential difference at the condenser.

The potential difference at the condenser continues to rise with its increasing charge, leaving less electromotive force available for the resistance. Consequently, the current decreases and eventually becomes zero when the condenser is fully charged.

During the decrease of current, the decreasing magnetic flux, represented as [math]i L[/math], in the inductance produces an electromotive force that assists the impressed electromotive force and somewhat retards the decrease of current.

Similarly, for the same reasons, if the impressed electromotive force, [math]e_0[/math], is withdrawn but the circuit remains closed, the current, represented as [math]i[/math], does not instantly disappear but gradually decreases.

Figure One: Rise and Decay of Continuous Current in an Inductive Circuit

An example is given by the exciting current of an alternator field or a circuit with the following constants:

Resistance, [math]r[/math] = 12 ohms
Inductance, [math]L[/math] = 6 henries
Electromotive force, [math]e_0[/math] = 240 volts

The horizontal axis represents time in seconds.

Electrostatic Charge in a Condenser

Thirteen. If an electrostatic condenser with a capacity represented as [math]C[/math] is connected to a continuous electromotive force, [math]e_0[/math], no current exists in the stationary condition of this direct current circuit. A very small current may leak through the insulation or the dielectric of the condenser, but the condenser is charged to the potential difference [math]e_0[/math] and contains the electrostatic charge, represented as [math]Q[/math], which is

[math]Q = C e_0[/math].

At the moment of closing the circuit of electromotive force, [math]e_0[/math], on the capacity, [math]C[/math], the condenser initially contains no charge. That is, the potential difference at the terminals of the condenser is zero. If there were no resistance and no inductance in the circuit, in the first moment after closing the circuit, the current would be infinitely large, charging the condenser instantly to the potential difference [math]e_0[/math].

Chapter Two

Introduction

If the resistance is very small, the current immediately after closing the circuit rises very rapidly, quickly charging the condenser. However, at the moment when the condenser is fully charged to the impressed electromotive force, [math]e_0[/math], a current still exists. This current cannot stop instantly because the decrease in current, and thus the decrease in its associated magnetic flux, represented as [math]i L[/math], generates an electromotive force.

Figure Two: Charging a Condenser Through a Circuit with Resistance and Inductance

Constant Potential. Logarithmic Charge: High Resistance

This electromotive force maintains the current or retards its decrease. As a result, electricity continues to flow into the condenser for a period of time even after it is fully charged. When the current ultimately ceases, the condenser becomes overcharged, meaning that the potential difference at the condenser terminals exceeds the impressed electromotive force, [math]e_0[/math]. Consequently, the condenser partially discharges, causing electricity to flow in the opposite direction, or out of the condenser.

Similarly, the reverse current, due to the inductance of the circuit, overshoots and discharges the condenser further than the impressed electromotive force, [math]e_0[/math]. After the discharge current ceases, a charging current—now smaller than the initial charging current—resumes. Through a series of oscillations, involving overcharging and undercharging, the condenser gradually stabilizes, and the current eventually diminishes to zero.

Figure Three: Charging a Condenser Through a Circuit with Resistance and Inductance

Constant Potential. Oscillating Charge: Low Resistance

Figure Three illustrates the oscillating charge of a condenser through an inductive circuit with a continuous impressed electromotive force, [math]e_0[/math]. The current is represented as [math]i[/math], and the potential difference at the condenser terminals as [math]e[/math], with time plotted along the horizontal axis. The circuit constants are:

Resistance, [math]r = 40[/math] ohms
Inductance, [math]L = 100[/math] millihenries
Capacity, [math]C = 10[/math] millifarads
Electromotive force, [math]e_0 = 1000[/math] volts

In such a continuous-current circuit containing resistance, inductance, and capacity connected in series, the current at the moment of closing the circuit, as well as the final current, equals zero, but a current exists immediately after closing the circuit as a transient phenomenon. This temporary current may either steadily increase and then decrease again to zero, or it may consist of a number of alternations with successively decreasing amplitudes, resulting in an oscillating current.

If the circuit contains neither resistance nor inductance, the current flowing into the condenser would theoretically be infinite.

That is, with low resistance and low inductance, the charging current of a condenser can become enormous. Although only transient, this current requires careful consideration and investigation. If the resistance is very low and the inductance is appreciable, the overcharging of the condenser may raise its voltage above the impressed electromotive force, [math]e_0[/math], sufficiently to cause disruptive effects.

Alternating Current and Transient Response

Fourteen. If an alternating electromotive force, represented as

[math]e = E \cos \theta[/math],

is impressed upon a circuit with such constants that the current lags by 45 degrees, the current is represented as

[math]i = I \cos (\theta – 45^\circ)[/math].

If the circuit is closed at the moment when [math]\theta = 45^\circ[/math], the current should theoretically be at its maximum value. However, it is initially zero.

Since in a circuit containing inductance (that is, in practically any circuit) the current cannot change instantly, the current instead gradually rises from zero as its initial value to the permanent value of the sine wave represented as [math]i[/math].

This gradual approach of the current from the initial value, in the present case zero, to the final value of the curve represented as [math]i[/math], can either be gradual, as shown by the curve [math]i_1[/math] in Figure Four, or occur through a series of oscillations with gradually decreasing amplitudes, as shown by curve [math]i_2[/math] in Figure Four.

Figure Four: Starting of an Alternating-Current Circuit Having Inductance

Fifteen. The general solution of an electric current problem, therefore, includes, in addition to the permanent term—whether constant or periodic—a transient term. This transient term disappears after a time that depends on the circuit conditions, ranging from an extremely small fraction of a second to several seconds.

These transient terms appear whenever the circuit conditions are altered, such as when closing the circuit, opening the circuit, or changing load, impedance, or other parameters of the circuit.

In general, for a circuit containing resistance and inductance but no capacity, the transient terms of current and voltage are neither sufficiently large nor of long duration to cause harmful or even appreciable effects. It is mainly in circuits containing capacity that transient terms can result in excessive values of current and potential difference, potentially causing serious consequences. Thus, the investigation of transient terms is primarily concerned with the effects of electrostatic capacity.

Transient Terms and Energy Storage in Circuits

Sixteen. No transient terms originate from resistance alone. Only those circuit constants that represent the storage of energy—magnetically by the inductance, [math]L[/math], or electrostatically by the capacity, [math]C[/math]—give rise to transient phenomena. The more the resistance predominates, the less severe and shorter in duration the transient term becomes.

When closing a circuit that contains inductance, capacity, or both, the energy stored in the inductance and capacity must first be supplied by the impressed electromotive force before the circuit conditions can stabilize. That is, immediately after closing an electric circuit, or more generally, after any change in the circuit conditions, the impressed electromotive force—or rather the source producing it—must, in addition to maintaining the circuit, supply the energy required to store energy in the inductance and capacity. As a result, a transient term appears immediately following any change in the circuit conditions.

If the circuit contains only one energy-storing constant, such as inductance or capacity, the transient term, which connects the initial and stationary conditions of the circuit, can only be a steady logarithmic term or a gradual approach. An oscillation can occur only if two energy-storing constants, such as capacity and inductance, are present. These allow energy to surge back and forth between the two, resulting in overreaching and oscillations.

Successive Transients in Repetitive Systems

Seventeen. Transient terms may occur periodically and in rapid succession, such as when rectifying an alternating current by synchronously reversing the connections of the alternating impressed electromotive force with the receiver circuit. This reversal can be achieved mechanically or without moving apparatus, such as with unidirectional conductors like arcs.

With each half wave, the circuit reversal initiates a transient term. Frequently, this transient term has not yet disappeared, and in some cases, it has not even greatly diminished when the next reversal starts another transient term. These transient terms can dominate to such an extent that the current primarily consists of a series of successive transient terms.

Oscillatory Discharge of a Condenser through an Inductance

Eighteen. If a condenser is charged through an inductance, and the condenser is shunted by a spark gap set for a lower voltage than the impressed electromotive force, the spark gap discharges as soon as the condenser charge reaches a certain value. This discharge initiates a transient term. The condenser then recharges, discharges again, and so on. Through successive charges and discharges of the condenser, a series of transient terms is produced. This series recurs at a frequency determined by the circuit constants and the ratio of the disruptive voltage of the spark gap to the impressed electromotive force.

Such a phenomenon, for instance, occurs when a weak spot appears in the cable insulation of a high-potential alternating-current system, allowing a spark discharge to pass to the ground. This discharge occurs in shunt to the condenser formed by the cable conductor and the cable armor or ground.

Significance of Transient Phenomena in Electric Circuits

Nineteen. In most cases, the transient phenomena occurring in electric circuits immediately after a change in circuit conditions are insignificant due to their short duration. However, they require serious consideration in the following cases:

(a) When they reach excessive values.

For example:

Connecting a large transformer to an alternator may result in a large initial current value that could cause damage.
Short-circuiting a large alternator may produce a momentary short-circuit current that is far greater than the stationary short-circuit current. While the stationary current may represent little power, the momentary current could exceed the capacity of automatic circuit-opening devices, leading to damage due to its high power.
In high-potential transmission systems, the potential differences produced by transient terms may far exceed normal voltage levels, potentially causing disruptive effects.
The frequency or steepness of the wavefront of these transients may create destructive voltages across inductive components, such as reactors, end turns of transformers and generators, and similar parts.

(b) Lightning and high-potential surges

These are, by nature, essentially transient phenomena, often of oscillating character.

(c) Generation of high-frequency currents

The periodic production of transient terms with oscillating character is a primary method of generating electric currents of very high frequency, as used in wireless telegraphy and similar technologies.

(d) Alternating-current rectifying apparatus

In these systems, where the direction of current in part of the circuit is reversed with each half wave, the current is made unidirectional. In such cases, the stationary condition of the current in the alternating portion of the circuit is usually never reached, and the transient term often becomes of primary importance.

(e) Telegraphy

The current in the receiving apparatus depends essentially on transient terms. In long-distance cable telegraphy, the stationary condition of the current is never approached, and the speed of telegraphy is determined by the duration of the transient terms.

(f) Analogous space-dependent phenomena

Phenomena of the same character occur, but with space instead of time as the independent variable. These include:

The distribution of voltage and current in a long-distance transmission line.
The phenomena occurring in multigap lightning arresters.
The transmission of current impulses in telephony.
The distribution of alternating current in a conductor such as the rail return of a single-phase railway.
The distribution of alternating magnetic flux in solid magnetic material.

Some of the simpler forms of transient terms are investigated and discussed in the following pages.

Transients in Time

Chapter Three

Inductance and Resistance in Continuous-Current Circuits

Inductance in Continuous-Current Circuits

Twenty. In continuous-current circuits, the inductance does not appear in the equations describing the stationary condition. If [math]e_0[/math] represents the impressed electromotive force, [math]r[/math] represents the resistance, and [math]L[/math] represents the inductance, the permanent value of the current is given by

[math]i_0 = \frac{e_0}{r}[/math].

For this reason, less care is taken to minimize inductance in direct-current circuits compared to alternating-current circuits, where inductance typically causes a voltage drop. Direct-current circuits generally have higher inductance, especially when used to produce magnetic flux, as in solenoids, electromagnets, or machine fields.

Any change in the conditions of a continuous-current circuit, such as a change in electromotive force or resistance, which leads to a change in current from one value, [math]i_0[/math], to another value, [math]i_1[/math], results in the appearance of a transient term connecting the initial and final current values. The equation describing this transient term includes the inductance.

Counting time, [math]t[/math], from the moment the change begins, let [math]e_0[/math] denote the impressed electromotive force, [math]r[/math] the resistance, and [math]L[/math] the inductance.

The current in the permanent or stationary condition after the change in circuit conditions is represented as

[math]i_1 = \frac{e_0}{r}[/math].

Let [math]i_0[/math] represent the current in the circuit before the change, which occurs at time [math]t = 0[/math]. Denoting the current during the change as [math]i[/math], the electromotive force consumed by the resistance, [math]r[/math], is

[math]i r[/math].

The electromotive force consumed by the inductance, [math]L[/math], is

[math]L \frac{di}{dt}[/math],

where [math]i[/math] is the current in the circuit.

Starting of a Continuous-Current Lighting Circuit or Non-Inductive Load

Twenty-One. Let the impressed electromotive force of the circuit, [math]e_0[/math], be 125 volts, and the current in the circuit under stationary conditions, [math]i_1[/math], be 1000 amperes.

The effective resistance of the circuit is calculated as

[math]r = \frac{e_0}{i_1} = 0.125 \text{ ohms}[/math].

Assuming a 10% voltage drop in feeders and mains, or 12.5 volts, gives a resistance, [math]r_0[/math], of 0.0125 ohms for the supply conductors. In such large conductors, the inductance can be estimated as 10 millihenries per ohm. Hence,

[math]L = 0.125 \text{ millihenries} = 0.000125 \text{ henries}[/math].

The current at the moment of starting is zero, [math]i_0 = 0[/math]. The general equation for the current in the circuit is obtained by substitution and is expressed as

[math]i = 1000 \left( 1 – e^{-1000 t} \right)[/math].

To determine the time required for the current to reach half its full value, or [math]i = 500[/math] amperes, substitute into the equation:

[math]500 = 1000 \left( 1 – e^{-1000 t} \right)[/math].

Solving,

[math]e^{-1000 t} = 0.5[/math].

Taking the natural logarithm,

[math]t = 0.00069 \text{ seconds}[/math].

Similarly, for the current to reach 90% of its full value, or [math]i = 900[/math] amperes, the time is calculated as

[math]t = 0.0023 \text{ seconds}[/math].

This shows that the current is established in the circuit in an almost negligible time, a fraction of a hundredth of a second.

Excitation of a Motor Field

Twenty-Two. In a continuous-current shunt motor, let the impressed electromotive force, [math]e_0[/math], be 250 volts, and the number of poles be 8.

Assume the magnetic flux per pole, [math]\Phi_0[/math], is 12.5 megalines, and the ampere-turns per pole required to produce this magnetic flux is represented as [math]\mathcal{F} = 9000[/math].

If 1000 watts are used for the excitation of the motor field, the exciting current, [math]i_1[/math], is calculated as

[math]i_1 = \frac{1000}{250} = 4 \text{ amperes}[/math].

From this, the resistance of the total motor field circuit is

[math]r = \frac{e_0}{i_1} = 62.5 \text{ ohms}[/math].

To produce [math]\mathcal{F} = 9000[/math] ampere-turns with [math]i_1 = 4[/math] amperes requires

[math]\frac{\mathcal{F}}{i_1} = 2250[/math] turns per field spool, or a total of 18,000 turns.

Eighteen thousand turns interlinked with [math]\Phi_0 = 12.5[/math] megalines results in a total number of interlinkages for [math]i_1 = 4[/math] amperes of

[math]n \Phi_0 = 225 \times 10^9[/math].

This is equivalent to

[math]562.5 \times 10^9[/math] interlinkages per unit current or 10 amperes.

Thus, the inductance of the motor field circuit is

[math]L = 562.5 \text{ henries}[/math].

The constants of the circuit are:

[math]e_0 = 250[/math] volts
[math]r = 62.5[/math] ohms
[math]L = 562.5[/math] henries
[math]i_0 = 0[/math], representing the current at [math]t = 0[/math].

Substituting these values into the equation of the exciting current yields:

[math]i = 4 \left( 1 – e^{-0.1111 t} \right)[/math].

Half excitation of the field is reached after

[math]t = 6.23 \text{ seconds}[/math].

Ninety percent of full excitation, or [math]i = 3.6[/math] amperes, is reached after

[math]t = 20.8 \text{ seconds}[/math].

This indicates that such a motor field requires significant time after closing the circuit before it approaches full excitation, allowing the armature circuit to be safely closed.

Effect of Circuit Configuration on Excitation Time

Now assume the motor field is redesigned or reconnected to consume only part of the impressed electromotive force, such as half, with the remainder consumed in non-inductive resistance. This can be achieved by connecting the field spools in pairs in parallel.

Under these conditions, the resistance and inductance of the motor field are reduced to one-quarter, but the same amount of external resistance must be added to consume the impressed electromotive force.

The new circuit constants are:

[math]e_0 = 250[/math] volts
[math]r = 31.25[/math] ohms
[math]L = 140.6[/math] henries
[math]i_0 = 0[/math].

The equation for the exciting current becomes:

[math]i = 8 \left( 1 – e^{-0.2222 t} \right)[/math].

This configuration allows the current to rise much more rapidly, reaching half its value after

[math]t = 3.11 \text{ seconds}[/math]

and 90% of its value after

[math]t = 10.4 \text{ seconds}[/math].

Inductance and Resistance in Continuous-Current Circuits

This is the equation of the field current during the time in which the motor armature gradually comes to rest.

At the moment when the motor armature stops, or for [math]t = t_1[/math], the current is given by

[math] i_2 = i_1 e^{-\frac{t_1}{2L}}. [/math]

This is the same value that the current would have if the armature were permanently at rest, without the assistance of the electromotive force generated by rotation, at the time [math]t = \frac{t_1}{2}[/math].

The rotation of the motor armature therefore reduces the rate of decrease of the field current, effectively doubling the time required to reach the value [math]i_2[/math] compared to the case without rotation.

These equations no longer apply for [math]t > t_1[/math], that is, after the armature has come to rest. This is because they are based on the speed equation

[math] S \left( 1 – \frac{t}{t_1} \right), [/math]

which applies only up to [math]t = t_1[/math]. For [math]t > t_1[/math], the speed is zero, not negative as given by the speed equation.

At the moment [math]t = t_1[/math], a break occurs in the field discharge curve. After this time, the current [math]i[/math] decreases according to Equation Three, which is given as

[math] i = i_2 e^{-\frac{r}{L} (t – t_1)}. [/math]

Substituting the value of [math]i_2[/math], the equation becomes

[math] i = i_1 e^{-\frac{r}{L} \frac{(t – t_1)}{2}}. [/math]

Numerical Values for Field Discharge

Using numerical values, the equations are:

For [math] t < t_1 [/math]:
[math] i = 4 e^{-0.001388 t^2}. [/math]
For [math] t = t_1 [/math], where [math] t_1 = 40 [/math] seconds:
[math] i = 0.436 [/math] amperes.
For [math] t > t_1 [/math]:
[math] i = 4 e^{-0.1111 (t – 20)}. [/math]

Hence, the field current decreases to half its initial value after a time of 22.15 seconds and to one-tenth of its initial value after 40.73 seconds.

Figure Five: Field Discharge Current

Figure Five illustrates the field discharge current as Curve One, determined using Equations Nineteen, Twenty, and Twenty-One. Curve Two represents the current calculated by the equation

[math] i = 4 e^{-0.1111 t}, [/math]

which describes the discharge of the field with the armature at rest or when short-circuited upon itself, without assistance from the electromotive force of rotation of the armature.

Additionally, Figure Five includes Curve Three, which represents the beginning of the field discharge current for an inductance [math] L = 4200 [/math] henries. In this case, the discharge current is described by the equation

[math] i = 4 e^{-0.0001855 t^2}. [/math]

In this scenario, the decrease of the field current is very slow, with the field decreasing to half its value in 47.5 seconds.

Self-Excitation of Direct-Current Generator

Twenty-Four. In the preceding discussion, the inductance [math]L[/math] of the machine has been assumed to remain constant, meaning that the magnetic flux [math]\Phi[/math] is proportional to the exciting current [math]i[/math]. For higher values of [math]\Phi[/math], this assumption is not even approximately valid.

Self-Excitation of a Direct-Current Generator

The self-excitation of a direct-current generator, whether shunt wound or series wound, is the feature that the voltage of the machine gradually builds up from the value given by residual magnetism to its full value. This process depends on the disproportionality of the magnetic flux relative to the magnetizing current. When analyzing this phenomenon, the inductance cannot be assumed as constant.

When examining circuits where the inductance, [math]L[/math], varies with the current, it is preferable to avoid the term “inductance” altogether and instead introduce the magnetic flux, [math]\Phi[/math].

The magnetic flux, [math]\Phi[/math], varies with the magnetizing current, [math]i[/math], according to an empirical curve known as the magnetic characteristic or saturation curve of the machine. This relationship can approximately be represented, within the range considered here, by a hyperbolic curve, as first demonstrated by Fröhlich in 1882:

[math] \Phi = \frac{\phi i}{1 + b i}. [/math]

Here,

[math]\phi[/math] represents the magnetic flux per ampere, in megalines, at low density.
[math]\frac{\phi}{b}[/math] gives the magnetic saturation value or maximum magnetic flux, in megalines.

The reciprocal of this expression is

[math] \frac{i}{\Phi} = 1 + \frac{b i}{\phi}. [/math]

This expression can be regarded as the magnetic exciting reluctance of the machine field circuit, which appears here as a linear function of the exciting current, [math]i[/math].

Machine Constants for a Shunt-Wound Commutating Machine

Consider the same shunt-wound commutating machine discussed earlier, having the following constants:

Field resistance: [math] r = 62.5 [/math] ohms
Magnetic flux per pole at normal magnetomotive force: [math] \Phi_0 = 12.5 [/math] megalines
Normal magnetomotive force per pole: [math] \mathcal{F} = 9000 [/math] ampere-turns
Total field turns: [math] n = 18000 [/math] turns
Field turns per pole: [math] \frac{18000}{8} = 2250 [/math] turns
Current for full excitation: [math] i_1 = 4 [/math] amperes

Assuming that at full excitation ([math] \Phi_0 [/math]), the magnetic reluctance has already increased by 50% above its initial value, that is,

[math] 1 + b i_1 = 1.5, [/math]

giving

[math] b = 0.125. [/math]

Since [math] i_1 = 4 [/math] amperes produces [math] \Phi_0 = 12.5 [/math] megalines, it follows from the equations that

[math] \phi = 4.69. [/math]

Thus, the magnetic characteristic of the machine is approximated by the equation:

[math] \Phi = \frac{4.69 i}{1 + 0.125 i}. [/math]

Electromotive Force in the Field Circuit

Let [math] e_c [/math] represent the electromotive force generated by the rotation of the armature per megaline of field flux.

This electromotive force, [math] e_c [/math], is proportional to the speed and depends on the machine constants. At the speed assumed earlier, where

[math] \Phi_0 = 12.5 [/math] megalines
[math] e_0 = 250 [/math] volts

it follows that

[math] e_c = \frac{e_0}{\Phi_0} = 20 [/math] volts.

In the field circuit of the machine, the impressed electromotive force, or the electromotive force generated in the armature by its rotation through the magnetic field, is

[math] e = e_c \Phi = 20 \Phi. [/math]

The electromotive force consumed by the field resistance, [math]r[/math], is

[math] i r = 62.5 i. [/math]

The electromotive force consumed by the field inductance, or generated in the field coils by the rise of magnetic flux [math] \Phi [/math], is

[math] n \frac{d\Phi}{dt} \times 10^{-2} = 180 \frac{d\Phi}{dt}. [/math]

Here, [math] \Phi [/math] is given in megalines, and [math] e_0 [/math] is in volts.

Self-Excitation of a Direct-Current Generator

The differential equation of the field circuit is given as

[math] e_c \Phi = i r + \frac{n}{100} \frac{d\Phi}{dt}. [/math]

Since this equation contains the derivative of [math] \Phi [/math], it is more convenient to make [math] \Phi [/math], and not [math] i [/math], the dependent variable. Substituting for [math] i [/math] from the equation

[math] i = \frac{\Phi}{\phi – b \Phi}, [/math]

we obtain

[math] e_c \Phi = \frac{\Phi r}{\phi – b \Phi} + \frac{n}{100} \frac{d\Phi}{dt}. [/math]

Rearranging,

[math] \frac{100 dt}{n} = \frac{(\phi – b \Phi) d\Phi}{\Phi (\phi e_c – r – b e_c \Phi)}. [/math]

This equation is integrated by resolving into partial fractions using the identity

[math] \frac{\phi – b \Phi}{\Phi (\phi e_c – r – b e_c \Phi)} = \frac{A}{\Phi} + \frac{B}{\phi e_c – r – b e_c \Phi}. [/math]

Expanding,

[math] \phi – b \Phi = A (\phi e_c – r) – A b e_c \Phi – B \Phi. [/math]

Equating coefficients,

[math] A = \frac{\phi}{\phi e_c – r}, \quad B = \frac{b r}{\phi e_c – r}. [/math]

Substituting these coefficients,

[math] \frac{100 dt}{n} = \frac{\phi d\Phi}{(\phi e_c – r) \Phi} + \frac{b r d\Phi}{(\phi e_c – r)(\phi e_c – r – b e_c \Phi)}. [/math]

Integrating,

[math] \frac{100 t}{n} = \frac{\phi}{\phi e_c – r} \ln \Phi – \frac{r}{e_c (\phi e_c – r)} \ln (\phi e_c – r – b e_c \Phi) + C. [/math]

Determining the Integration Constant

The integration constant, [math] C [/math], is calculated using the residual magnetic flux of the machine, which represents the remanent magnetism of the field poles at the moment of start.

Assume that at time [math] t = 0 [/math], magnetic flux [math] \Phi = \Phi_r = 0.5 [/math] megalines, the residual magnetism.

Substituting into the equation:

[math] 0 = \frac{\phi}{\phi e_c – r} \ln \Phi_r – \frac{r}{e_c (\phi e_c – r)} \ln (\phi e_c – r – b e_c \Phi_r) + C. [/math]

Solving for [math] C [/math] and substituting back,

[math] \frac{100 t}{n} = \frac{\phi}{\phi e_c – r} \ln \frac{\Phi}{\Phi_r} – \frac{r}{e_c (\phi e_c – r)} \ln \frac{\phi e_c – r – b e_c \Phi}{\phi e_c – r – b e_c \Phi_r}. [/math]

Rearranging,

[math] t = \frac{n}{100 e_c (\phi e_c – r)} \left[ \phi e_c \ln \frac{\Phi}{\Phi_r} – r \ln \frac{\phi e_c – r – b e_c \Phi}{\phi e_c – r – b e_c \Phi_r} \right]. [/math]

Rewriting in Terms of the Electromotive Force

Substituting [math] e = e_c \Phi [/math] and [math] e_m = e_c \Phi_r [/math], where [math] e_m [/math] is the electromotive force generated in the armature by rotation in the residual magnetic field,

[math] t = \frac{n}{100 e_c (\phi e_c – r)} \left[ \phi e_c \ln \frac{e}{e_m} – r \ln \frac{\phi e_c – r – b e}{\phi e_c – r – b e_m} \right]. [/math]

Using numerical values:

[math] n = 18000 [/math] turns
[math] \phi = 4.69 [/math] megalines
[math] b = 0.125 [/math]
[math] e_c = 20 [/math] volts
[math] r = 62.5 [/math] ohms
[math] \Phi_r = 0.5 [/math] megalines
[math] e_m = 10 [/math] volts

the time equation becomes

[math] t = 26.8 \ln \Phi – 17.9 \ln (31.25 – 2.5 \Phi) + 79.6. [/math]

In terms of electromotive force [math] e [/math],

[math] t = 26.8 \ln e – 17.9 \ln (31.25 – 0.125 e) – 0.98. [/math]

Figure Six shows the electromotive force, [math] e [/math], as a function of time, [math] t [/math]. Under the conditions assumed, it takes several minutes for the electromotive force of the machine to build up to approximately its full value.

Condition for Self-Excitation in a Shunt Generator

The phenomenon of self-excitation in shunt generators is therefore a transient phenomenon, which may have a very long duration.

From the equations, it follows that

[math] e = \frac{\phi e_c – r}{b} = 250 [/math] volts.

This represents the electromotive force to which the machine builds up when time approaches infinity, corresponding to the stationary condition.

For the machine to be self-exciting, the condition

[math] \phi e_c – r > 0 [/math]

must be satisfied.

This implies that the field winding resistance must satisfy the inequality

[math] r < \phi e_c. [/math]

Substituting numerical values gives

[math] r < 93.8 [/math] ohms.

The time required for the machine to build up decreases with increasing [math] e_c [/math] (which corresponds to increasing speed) and increases with increasing [math] r [/math] (which represents increasing field resistance).

Self-Excitation of a Direct-Current Series Machine

Twenty-Five. The phenomenon of self-excitation in a series machine, such as a railway motor, is of particular interest. When using a railway motor as a brake by closing its circuit upon a resistance, its effectiveness depends on how quickly it can build up as a generator.

Assume a four-pole railway motor designed for

[math] e_0 = 600 [/math] volts
[math] i_1 = 200 [/math] amperes

At a current [math] i = i_1 [/math], the magnetic flux per pole of the motor is

[math] \Phi_0 = 10 [/math] megalines,

and 8000 ampere-turns per field pole are required to produce this flux. This results in

[math] 40 [/math] exciting turns per pole, or a total of

[math] n = 160 [/math] turns.

Estimating an 8% loss in the conductors of the field and armature at 200 amperes gives a motor circuit resistance of

[math] r_0 = 0.24 [/math] ohms.

To limit the current to full load value, [math] i_1 = 200 [/math] amperes, the total circuit resistance must be

[math] r = 3 [/math] ohms,

which implies an external resistance of 2.76 ohms.

Self-Excitation of a Direct-Current Series Machine

Assuming a residual magnetism of ten percent, the residual magnetic flux is

[math] \Phi_r = 1 [/math] megaline.

The electromotive force generated by this residual magnetism is

[math] e_m = e_c \Phi_r = 60 [/math] volts.

Substituting into the earlier equation with the following values:

Turns in the field circuit: [math] n = 160 [/math]
Magnetic flux per ampere: [math] \phi = 0.15 [/math] megalines
Saturation coefficient: [math] b = 0.01 [/math]
Electromotive force per megaline: [math] e_c = 60 [/math] volts
Circuit resistance: [math] r = 3 [/math] ohms
Residual magnetic flux: [math] \Phi_r = 1 [/math] megaline
Electromotive force due to residual magnetism: [math] e_m = 60 [/math] volts

The time equation becomes

[math] t = 0.04 \ln e – 0.01333 \ln (600 – e) – 0.08. [/math]

For e = 300 volts, which corresponds to 50% excitation, the time required is

[math] t = 0.072 [/math] seconds.

For e = 540 volts, which corresponds to 90% excitation, the time required is

[math] t = 0.117 [/math] seconds.

Thus, the motor excites itself as a series generator almost instantly, within a small fraction of a second.

Minimum Electromotive Force for Self-Excitation

The minimum value of [math] e_c [/math] required for self-excitation is given by

[math] e_c = \frac{r}{\phi} = 20 [/math] volts.

This corresponds to one-third of full speed.

If this series motor, with its field and armature windings connected in reverse generator position, is short-circuited upon itself, where

[math] r = 0.24 [/math] ohms,

the equation becomes

[math] t = 0.0274 \ln e – 0.00073 \ln (876 – e) – 0.1075. [/math]

In this case, self-excitation is practically instantaneous, with e = 300 volts reached after

[math] t = 0.044 [/math] seconds.

For e = 300 volts, the current is

[math] i = \frac{e}{r} = 1250 [/math] amperes.

The power is

[math] P = e i = 375 [/math] kilowatts.

This indicates that a series motor short-circuited in generator position instantly stops.

If short-circuited upon itself with

[math] r = 0.24 [/math] ohms,

the motor still builds up at

[math] e_c = \frac{r}{\phi} = 1.6 [/math] volts.

At full-load speed,

[math] e_c = 60 [/math] volts.

Thus,

[math] e_c = 1.6 [/math] volts corresponds to 2.67% of full-load speed, meaning the motor acts as a brake down to 2.67% of its full speed.

Limitations of the Approximation

It should be noted, however, that the parabolic equation used for the magnetic characteristic is only an approximation.

The results derived from the parabolic approximation of the magnetic characteristic are therefore approximate and should be interpreted accordingly.

One of the most significant transient phenomena in direct-current circuits is the reversal of current in the armature coil short-circuited by the commutator brush in a commutating machine.

For a detailed analysis of this phenomenon, refer to:

“Theoretical Elements of Electrical Engineering, Part Two, Section B.”

Chapter Four

Inductance and Resistance in Alternating-Current Circuits

Twenty-Six

In alternating-current circuits, the inductance, denoted as [math] L [/math], or as it is commonly expressed, the reactance, denoted as [math] x [/math], is given by

[math] x = 2 \pi f L, [/math]

where [math] f [/math] is the frequency. The inductance appears in both the transient and permanent terms.

At the moment when [math] \theta = 0 [/math], let the electromotive force be expressed as

[math] e = E \cos (\theta – \theta_0), [/math]

which is applied to a circuit with resistance [math] r [/math] and inductance [math] L [/math], where the inductive reactance is

[math] x = 2 \pi f L. [/math]

Let time [math] \theta = 2 \pi f t [/math] be measured from the moment of closing the circuit, and let [math] \theta_0 [/math] be the phase of the applied electromotive force at that instant.

In this scenario:

The electromotive force consumed by the resistance is[math] i r, [/math]where [math] i [/math] is the instantaneous value of the current.
The electromotive force consumed by the inductance [math] L [/math] is proportional to [math] L [/math] and the rate of change of the current, expressed as[math] L \frac{di}{dt}. [/math]

By substituting [math] \theta = 2 \pi f t [/math] and [math] x = 2 \pi f L [/math], the electromotive force consumed by the inductance becomes

[math] x \frac{di}{d\theta}. [/math]

Thus, since

[math] e = E \cos (\theta – \theta_0), [/math]

the differential equation of the problem is given by

[math] E \cos (\theta – \theta_0) = i r + x \frac{di}{d\theta}. [/math]

The solution to this equation is

[math] i = I \cos (\theta – \delta) + A e^{-a \theta}, [/math]

where [math] e [/math] is the base of natural logarithms, approximately equal to [math] 2.7183 [/math].

Solution and System of Equations

Substituting the proposed solution into the differential equation gives

[math] E \cos (\theta – \theta_0) = I r \cos (\theta – \delta) + A r e^{-a \theta} – I x \sin (\theta – \delta) – A a x e^{-a \theta}. [/math]

Rearranging terms results in:

[math] \Big( E \cos \theta_0 – I r \cos \delta – I x \sin \delta \Big) \cos \theta [/math]
[math] + \Big( E \sin \theta_0 – I r \sin \delta + I x \cos \delta \Big) \sin \theta [/math]
[math] – A e^{-a \theta} (a x – r) = 0. [/math]

Since this equation must hold for any value of [math] \theta [/math], the coefficients of [math] \cos \theta [/math], [math] \sin \theta [/math], and [math] e^{-a \theta} [/math] must independently equal zero.

This gives the following system of equations:

[math] E \cos \theta_0 – I r \cos \delta – I x \sin \delta = 0. [/math]
[math] E \sin \theta_0 – I r \sin \delta + I x \cos \delta = 0. [/math]
[math] a x – r = 0. [/math]

From the last equation, it follows that

[math] a = \frac{r}{x}. [/math]

Substituting this into the earlier equations, we obtain:

The lag angle [math] \theta_1 [/math] is given by[math] \tan \theta_1 = \frac{x}{r}. [/math]
The impedance [math] z [/math] of the circuit is[math] z = \sqrt{r^2 + x^2}. [/math]

The equations for the coefficients become:

[math] E \cos \theta_0 – I z \cos (\delta – \theta_1) = 0, [/math]

[math] E \sin \theta_0 – I z \sin (\delta – \theta_1) = 0. [/math]

From these, we find:

[math] I = \frac{E}{z}, [/math]

[math] \delta = \theta_0 + \theta_1. [/math]

Final Solution

Substituting these results into the general solution of the differential equation, we obtain:

[math] i = \frac{E}{z} \cos (\theta – \theta_0 – \theta_1) + A e^{-\frac{r}{x} \theta}. [/math]

Here, A remains undetermined and must be determined using the initial conditions of the circuit.

For theta equal to zero, the initial condition [math] i = 0 [/math] is applied.

Substituting into the solution gives:

[math] 0 = \frac{E}{z} \cos (\theta_0 + \theta_1) + A. [/math]

Thus,

[math] A = -\frac{E}{z} \cos (\theta_0 + \theta_1). [/math]

Substituting this value of [math] A [/math] into the expression for the current gives the general equation:

[math] i = \frac{E}{z} \left( \cos (\theta – \theta_0 – \theta_1) – e^{-\frac{r}{x} \theta} \cos (\theta_0 + \theta_1) \right). [/math]

If at the initial moment, when [math] \theta = 0 [/math], the current is not zero but equals [math] i_0 [/math], substituting this condition into the general equation yields:

[math] i_0 = \frac{E}{z} \cos (\theta_0 + \theta_1) + A. [/math]

From this, [math] A [/math] is determined as:

[math] A = i_0 – \frac{E}{z} \cos (\theta_0 + \theta_1). [/math]

Substituting this into the general expression for the current, we obtain:

[math] i = \frac{E}{z} \left( \cos (\theta – \theta_0 – \theta_1) – \left( \cos (\theta_0 + \theta_1) – i_0 \frac{z}{E} \right) e^{-\frac{r}{x} \theta} \right). [/math]

Twenty-Seven

The equation for the current contains two components:

A permanent term:[math] \frac{E}{z} \cos (\theta – \theta_0 – \theta_1), [/math]which represents the steady-state current.
A transient term:[math] \frac{E}{z} e^{-\frac{r}{x} \theta} \cos (\theta_0 + \theta_1). [/math]

The transient term diminishes more rapidly as the resistance [math] r [/math] increases and the reactance [math] x [/math] decreases.

This transient term reaches its maximum value if the circuit is closed at the moment when

[math] \theta_0 = -\theta_1, [/math]

which corresponds to the maximum of the steady-state current. Under this condition, the transient term is:

[math] \frac{E}{z} e^{-\frac{r}{x} \theta}. [/math]

The transient term vanishes if the circuit is closed at the moment when

[math] \theta_0 = 90^\circ – \theta_1, [/math]

which corresponds to the instant when the steady-state current passes through zero.

As an example, Figure Seven illustrates the starting of the current under conditions of maximum transient term, where

[math] \theta_0 = -\theta_1, [/math]

in a circuit with the following constants:

The ratio of reactance to resistance equals 0.1, approximately corresponding to a lighting circuit, where the permanent value of the current is reached in a small fraction of a half wave.
The ratio of reactance to resistance equals 0.5, corresponding to the starting of an induction motor with a rheostat in the secondary circuit.
The ratio of reactance to resistance equals 1.5, corresponding to an unloaded transformer or to the starting of an induction motor with a short-circuited secondary.
The ratio of reactance to resistance equals 10, corresponding to a reactive coil.

In the last case, where

[math] \frac{x}{r} = 10, [/math]

a series of successive waves is shown in Figure Eight, demonstrating the very gradual approach to the permanent condition.

Figure Nine illustrates the current in a circuit where

[math] \frac{x}{r} = 1.5, [/math]

for various instances of circuit closure at angles of 0^\circ, 30^\circ, 60^\circ, 90^\circ, 120^\circ, and 150^\circ, respectively, behind the zero value of the permanent current.

The instantaneous values of the current are represented as the vertical distances from:

The sine wave:
[math] I \cos (\theta – \theta_0), [/math]
To the exponential curve:
[math] I e^{-\frac{r}{x} \theta} \cos \theta_0, [/math]

which starts from the initial value of the permanent current.

In polar coordinates, the sine wave

[math] I \cos (\theta – \theta_0) [/math]

corresponds to a circle, while the exponential curve

[math] I e^{-\frac{r}{x} \theta} \cos \theta_0 [/math]

represents an exponential or loxodromic spiral.

As a general observation, the transient term in alternating-current circuits that contain both resistance and inductance is typically significant only in circuits with iron, where hysteresis and magnetic saturation introduce additional complexities to the phenomenon.

This term is also of importance in circuits involving unidirectional or periodically recurring changes, such as rectifiers. Some of these cases are examined in the following chapters.

The permanent value of the current is represented in Figure Seven by the dotted line.

Instead of viewing the current wave in Figure Nine as the superposition of:

The permanent term:
[math] I \cos (\theta – \theta_0), [/math]
And the transient term:
[math] -I e^{-\frac{r}{x} \theta} \cos \theta_0, [/math]

the current wave can alternatively be represented directly by the permanent term:

[math] I \cos (\theta – \theta_0). [/math]

This approach involves considering the zero line of the diagram as being deflected exponentially to the curve

[math] I e^{-\frac{r}{x} \theta} \cos \theta_0, [/math]

as shown in Figure Ten.

In this representation, the instantaneous values of the current correspond to the vertical distances from the deflected zero line to the curve of the permanent term.

———————–correction

Note: On the above page, B should be understood as magnetic induction, not Q or any other letter—which was read in error. B, magnetic induction in the material, is simply the product of mu, the permeability of the material, and H, the density of magnetic lines of force in air. Since the density increases due to magnetic induction, this increase in density is expressed as the constant mu.

Please check other related videos on my channel. Links are provided in the description.

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Hence, the electromotive force, denoted as [math] e_0 [/math], is equal to the product of current and resistance plus the product of inductance and the rate of change of current with respect to time:

[math] e_0 = i r + L \frac{di}{dt}. [/math]

Rewriting this equation, the ratio of resistance to inductance multiplied by the change in time equals the change in current divided by the difference between the instantaneous current and the final current:

[math] \frac{r}{L} dt = \frac{di}{i_1 – i}. [/math]

This relationship integrates into a logarithmic expression:

[math] -\frac{r}{L} t = \ln (i_1 – i) + C, [/math]

where C is the integration constant.

By substitution, this simplifies to show that the instantaneous current is equal to the final current plus the product of the difference between the initial and final currents and the exponential of the negative ratio of resistance to inductance multiplied by time:

[math] i = i_1 + (i_0 – i_1) e^{-\frac{r}{L} t}. [/math]

The counter-electromotive force, denoted as [math] e_1 [/math], is proportional to the product of resistance and the difference between the initial and final currents multiplied by the exponential term, reaching its maximum at the initial time:

[math] e_1 = (i_0 – i_1) r e^{-\frac{r}{L} t}. [/math]

In conclusion, the electromotive force generated during a change in an inductive circuit increases with the speed of the circuit break, potentially causing insulation damage in highly inductive circuits when the break occurs quickly.

As an example, these principles can be applied to analyze typical circuits.

Section Forty-Four:

The charge and discharge of a condenser through an inductive circuit produces periodic currents of a frequency depending upon the circuit constants.

The range of frequencies which can be produced by electrodynamic machinery is rather limited: synchronous machines or ordinary alternators can give economically and in units of larger size frequencies from ten to one hundred twenty-five cycles. Frequencies below ten cycles are available by commutating machines with low-frequency excitation. Above one hundred twenty-five cycles, the difficulties rapidly increase, due to the great number of poles, high peripheral speed, high power required for field excitation, poor regulation due to the massing of the conductors, which is required because of the small pitch per pole of the machine, etc., so that one thousand cycles probably is the limit of generation of constant potential alternating currents of appreciable power and at fair efficiency.

For smaller powers, by using capacity for excitation, inductor alternators have been built and are in commercial service for wireless telegraphy and telephony, for frequencies up to one hundred thousand and even two hundred thousand cycles per second.

Still, even going to the limits of peripheral speed, and sacrificing everything for high frequency, a limit is reached in the frequency available by electrodynamic generation. It becomes of importance, therefore, to investigate whether by the use of the condenser discharge, the range of frequencies can be extended.

Since the oscillating current approaches the effect of an alternating current only if the damping is small, that is, the resistance low, the condenser discharge can be used as high-frequency generator only by making the circuit of as low resistance as possible. This, however, means limited power. When generating oscillating currents by condenser discharge, the load put on the circuit, that is, the power consumed in the oscillating-current circuit, represents an effective resistance, which increases the rapidity of the decay of the oscillation, and thus limits the power.

When approaching the critical value, it also lowers the frequency. This is obvious, since the oscillating current is the dissipation of the energy stored electrostatically in the condenser, and the higher the resistance of the circuit, the more rapidly this energy is dissipated, meaning the faster the oscillation dies out.

Section Forty-Six:

With a resistance of the circuit sufficiently low to give a fairly well-sustained oscillation, the frequency is, with sufficient approximation:

[math] f = \frac{1}{2 \pi \sqrt{L C}}, [/math]

where:

[math] f [/math] is the frequency,
[math] L [/math] is the inductance, and
[math] C [/math] is the capacity.

The constants, capacity [math] C [/math], inductance [math] L [/math], and resistance [math] r [/math], have no relation to the size or bulk of the apparatus.

For instance, a condenser of one millifarad, built to withstand continuously a potential of ten thousand volts, is far larger than a two-hundred-volt condenser of one hundred millifarads capacity.

The energy which the former is able to store is:

[math] W = \frac{C e^2}{2}, [/math]

or fifty joules, while the latter stores only two joules, and therefore the former is twenty-five times as large.

A reactive coil of zero point one henry inductance, designed to carry continuously one hundred amperes, stores:

[math] W = \frac{L i^2}{2}, [/math]

or five hundred joules. A reactive coil of one thousand times the inductance, or one hundred henries, but with a current-carrying capacity of one ampere, stores only five joules and is therefore only about one one-hundredth the size of the former.

A resistor of one ohm, carrying continuously one thousand amperes, is a ponderous mass, dissipating one thousand kilowatts. A resistor having a resistance a million times as large, or one megohm, may be nothing more than a lead pencil scratch on a piece of porcelain.

Therefore, the size or bulk of condensers and reactors depends not only on the values of [math] C [/math] and [math] L [/math], but also on the voltage and current which can be applied continuously. That is, it is approximately proportional to the energy stored:

[math] W = \frac{C e^2}{2} \quad \text{for capacitors}, [/math]

and

[math] W = \frac{L i^2}{2} \quad \text{for inductors}. [/math]

The condenser consumes ten amperes. Its capacitive reactance is calculated as the terminal voltage divided by the current, which is one thousand ohms.

The capacity, denoted as [math] C [/math], is calculated using the formula:

[math] C = \frac{1}{2 \pi f X_c}, [/math]

where [math] X_c [/math] is the capacitive reactance.

Substituting the values gives a capacity of two point six five millifarads.

Designing the reactor for different currents, and consequently different voltages, results in different values of inductance, denoted as [math] L [/math], and therefore different frequencies of oscillation, denoted as [math] f [/math].

The table presents the characteristics of the reactive coil and its associated parameters:

Reactive Coil:

Current, measured in amperes.

Voltage, denoted as eee sub zero.

Reactance, given by the voltage divided by the current.

Inductance, expressed as the reactance divided by two times pi times the frequency.

Frequency of oscillation, calculated as one divided by two times pi times the square root of the product of inductance and capacitance.

Oscillating current, measured in amperes.

Oscillating power, measured in kilovolt-amperes.

The tabular data is as follows: For one ampere and a voltage of one hundred thousand volts, the reactance is one hundred thousand ohms, the inductance is two hundred sixty-five henries, the frequency of oscillation is six hertz, the oscillating current is one ampere, and the oscillating power is ten kilovolt-amperes.

For ten amperes and a voltage of ten thousand volts, the reactance is one thousand ohms, the inductance is two point six five henries, the frequency of oscillation is sixty hertz, the oscillating current is ten amperes, and the oscillating power is one hundred kilovolt-amperes.

For one hundred amperes and a voltage of one thousand volts, the reactance is ten ohms, the inductance is two point six five times ten raised to the power of negative two henries, the frequency of oscillation is six hundred hertz, the oscillating current is one hundred amperes, and the oscillating power is one thousand kilovolt-amperes.

For one thousand amperes and a voltage of one hundred volts, the reactance is zero point one ohms, the inductance is two point six five times ten raised to the power of negative four henries, the frequency of oscillation is six thousand hertz, the oscillating current is one thousand amperes, and the oscillating power is ten thousand kilovolt-amperes.

For ten thousand amperes and a voltage of ten volts, the reactance is zero point zero zero one ohms, the inductance is two point six five times ten raised to the power of negative six henries, the frequency of oscillation is sixty thousand hertz, the oscillating current is ten thousand amperes, and the oscillating power is one hundred thousand kilovolt-amperes.

For one hundred thousand amperes and a voltage of one volt, the reactance is zero point zero zero zero zero one ohms, the inductance is two point six five times ten raised to the power of negative eight henries, the frequency of oscillation is six hundred thousand hertz, the oscillating current is one hundred thousand amperes, and the oscillating power is one million kilovolt-amperes.

As indicated, with equal kilovolt-ampere capacities for the condenser and reactive coil, a wide range of frequencies can be generated, from low commercial frequencies to several hundred thousand cycles. At frequencies between five hundred and two thousand cycles, the use of iron in the reactive coil is restricted to the inner core. Above these frequencies, iron is not used because hysteresis and eddy currents would excessively damp the oscillation, requiring larger reactive coils.

Assuming a ninety-six percent efficiency of the reactive coil and ninety-nine percent efficiency of the condenser, the resistance rrr is approximately zero point zero five times the reactance. This means rrr equals zero point zero five times the square root of inductance divided by capacitance.

The energy of discharge, expressed in volt-ampere-seconds, is calculated as the square of the voltage divided by two times the resistance, multiplied by the square root of the product of inductance and capacitance, and equals ten times the square of the voltage multiplied by the capacitance. The power factor is given as zero point zero five.

The energy stored in the capacity is expressed as:

The energy [math] W_0 [/math] equals the square of the voltage [math] e_0 [/math], multiplied by the capacitance [math] C [/math], divided by two, measured in joules.

The critical resistance is given by:

The resistance [math] r_1 [/math] equals two multiplied by the square root of the inductance [math] L [/math] divided by the capacitance [math] C [/math].

The ratio of the resistance [math] r [/math] to the critical resistance [math] r_1 [/math] is:

The ratio of [math] \frac{r}{r_1} = 0.025 [/math].

The decrement of the oscillation is:

The decrement [math] \Delta [/math] equals [math] 0.92 [/math], which indicates that the decay of the wave is very slow under no-load conditions.

Assuming an external effective resistance, denoted as [math] r’ [/math], equal to three times the internal resistance [math] r [/math], gives an electrical efficiency of seventy-five percent. The total resistance becomes:

The total resistance, given as [math] r + r’ [/math], equals [math] 0.2 \cdot x [/math], where [math] x [/math] is the reactance.

The ratio of total resistance to critical resistance is:

The ratio of [math] \frac{r + r’}{r_1} = 0.1 [/math].

The decrement of the oscillation in this case is:

The decrement [math] \Delta [/math] equals [math] 0.73 [/math], indicating a fairly rapid decay of the wave.

At high frequencies, electrostatic, inductive, and radiation losses significantly increase the resistance, resulting in lower efficiency and a more rapid decay of the wave.

The frequency of oscillation is not directly influenced by the size of the apparatus, which is the kilovolt-ampere capacity of the condenser and reactor. For instance, if the size is reduced by a factor of [math] n [/math], and the apparatus is designed for the same voltage, the current for both condenser and reactor is reduced to [math] \frac{1}{n} [/math]. This means:

The condensive reactance becomes [math] n [/math] times greater, and the capacitance [math] C [/math] is reduced to [math] \frac{1}{n} [/math]. The inductance [math] L [/math] increases in proportion to [math] n [/math], ensuring that the product of the capacitance [math] C [/math] and inductance [math] L [/math], and thereby the frequency, remains constant. The power output of the oscillating currents, however, is reduced to [math] \frac{1}{n} [/math].

The maximum frequency achievable is limited by the mechanical dimensions of the circuit. For example, a condenser with a bulk of ten to twenty cubic feet results in a minimum length of the discharge circuit of approximately ten feet. Ten feet of conductor of a large size has an inductance of at least [math] 2 \times 10^{-6} [/math] henry. Consequently, the frequency of oscillation is limited to about sixty thousand cycles per second, even without a reactive coil, assuming a straight discharge path.

To estimate the highest frequency possible:

The minimum length of the discharge circuit is determined by the gap between the condenser plates. The smallest condenser capacity is formed by two spheres, as small plates tend to increase capacity due to edge effects. The minimum diameter of the spheres is [math] 1.5 [/math] times their distance, since smaller diameters result in brush discharges preceding spark discharges.

Assume the voltage [math] e_0 [/math] equals [math] 10,000 \sqrt{2} [/math] volts. The spark gap length between the spheres is approximately [math] 0.3 [/math] inches, requiring the sphere diameter to be [math] 0.45 [/math] inches. The oscillating circuit length then becomes approximately [math] 0.5 [/math] inches, with an inductance of [math] 0.125 \times 10^{-7} [/math] henry.

The capacity of the spheres relative to one another is approximately [math] 50 \times 10^{-8} [/math] millifarads. This results in a frequency of oscillation calculated as follows:

The frequency [math] f [/math] equals

[math] f = \frac{1}{2 \pi \sqrt{L C}} [/math]

giving a value of approximately two billion cycles per second.

For a voltage [math] e_0 [/math] of [math] 10,000 \sqrt{2} [/math] volts, the following values apply:

The voltage

[math] e_1 = 10,000 e^{-\frac{r}{2L} t} [/math] volts.

The current

[math] i = 2.83 e^{-\frac{r}{2L} t} [/math] amperes.

The power

[math] p_1 = 28.3 e^{-\frac{r}{L} t} [/math] kilovolt-amperes.

Reducing the size and spacing of the spheres proportionally, along with proportionally lowering the voltage or increasing the dielectric strength of the gap by raising the air pressure, enables the generation of even higher frequencies.

However, the power of oscillation decreases as the frequency increases due to the reduction in size, which reduces the storage capacity of both the capacitance and inductance.

At frequencies in the range of billions of cycles per second, the effective resistance becomes very large, leading to rapid damping. An oscillating system consisting of two spheres separated by a gap would need to be charged by induction, or the spheres could be charged separately and then brought closer together. Alternatively, the spheres could be made part of a series of spheres separated by gaps and connected across a high-potential circuit, such as in some types of lightning arresters.

It is evident that the highest frequency of oscillation with appreciable power that can be generated by a condenser discharge reaches billions of cycles per second, which is significantly higher than the frequencies attainable through electrodynamic machinery.

For instance, at a frequency of five billion cycles per second, the wavelength is approximately six centimeters. This is only a few octaves lower than the lowest frequencies observed as heat radiation or infrared light.

The average wavelength of visible light is approximately [math] 55 \times 10^{-6} [/math] centimeters, corresponding to a frequency of [math] 5.5 \times 10^{14} [/math] cycles per second. To achieve this frequency, the spheres would need to be approximately [math] 10^{-5} [/math] centimeters in diameter, which approaches molecular dimensions.

Oscillating Current Generator

A system consisting of a constant impressed electromotive force [math] e [/math], charging a condenser [math] C [/math] through a circuit of inductance [math] L [/math] and resistance [math] r [/math], with a discharge circuit of the condenser [math] C [/math] comprising an air gap in series with a reactor of inductance [math] L_0 [/math] and a resistor of resistance [math] r_0 [/math], acts as a generator of oscillating current under specific conditions.

The air gap must be set for a voltage [math] e_0 [/math] such that it discharges before the voltage of the condenser [math] C [/math] reaches its maximum value. Additionally, the resistance [math] r_0 [/math] must satisfy the condition that:

[math] r_0^2 < \frac{4L_0}{C} [/math],

ensuring the condenser discharge is oscillatory.

In such a system, as illustrated in the diagram of Figure sixteen, during the charging of the condenser, as soon as the terminal voltage across the condenser, and thus across the spark gap, reaches the value [math] e_0 [/math], the condenser discharges through the spark gap. This causes the potential difference across the condenser to fall to zero. Afterward, the condenser recharges to the potential difference [math] e_0 [/math], discharges again, and so on. This process produces a series of oscillating discharges in the circuit comprising [math] L_0 [/math] and [math] r_0 [/math] at intervals determined by the time required to charge the condenser [math] C [/math] via the reactor [math] L [/math] and resistor [math] r [/math] up to the potential difference [math] e_0 [/math] under the influence of the impressed electromotive force [math] e [/math].

The resistance [math] r [/math] should be minimized to ensure efficient energy transformation. The inductance [math] L [/math] must be sufficiently large so that the time required to charge the condenser [math] C [/math] to the potential [math] e_0 [/math] is adequate for the oscillating discharge through [math] L_0 [/math] and [math] r_0 [/math] to decay and for the spark gap [math] e_0 [/math] to reopen. This reopening involves dissipating the conducting products of the discharge within the gap [math] e_0 [/math].

The dissipation of these products takes significant time, but directing an air blast at the spark gap [math] e_0 [/math] can facilitate a more rapid recurrence of the discharge. The velocity of the air blast must be high enough to remove the ionized air or metallic vapors formed by the discharge faster than the time it takes for the condenser to recharge.

For instance, if the spark gap [math] e_0 [/math] is set for a voltage of twenty thousand volts, corresponding to approximately [math] 0.75 [/math] inches, the motion of the air blast during successive discharges must be significantly greater than [math] 0.75 [/math] inches—preferably three to six inches. With one thousand discharges per second, this requires an air velocity [math] v [/math] between [math] 250 [/math] and [math] 500 [/math] feet per second.

For five thousand discharges per second, the air velocity [math] v [/math] must range from [math] 1250 [/math] to [math] 2500 [/math] feet per second, which corresponds to an air pressure approximately calculated as:

[math] p = 14.7 \left( 1 + 2 v^2 \times 10^{-7} \right)^{3.5 – 1} [/math].

This yields air pressures of [math] 0.66 [/math] to [math] 2.75 [/math] pounds per square inch for the first case and [math] 23 [/math] to [math] 230 [/math] pounds per square inch for the second case.

While the condenser charge may be oscillatory or logarithmic, efficiency necessitates a low value of resistance, denoted as [math] r [/math], which results in an oscillatory charge.

With a discharge frequency in [math] L_0 [/math] and [math] r_0 [/math] that is significantly higher than the charge frequency, the duration of the discharge is brief compared to the charge period. In this case, the oscillating currents consist of a sequence of oscillations separated by relatively extended periods of rest. Therefore, the current in [math] L [/math] does not change appreciably during the discharge duration. At the conclusion of the condenser charge, the current in the reactor [math] L [/math] equals the current in [math] L [/math] at the start of the subsequent condenser charge.

The charging current of the condenser, represented by [math] C [/math], in the circuit [math] L [/math], transitions from [math] i_0 [/math] at the beginning of the charge—when the condenser electromotive force, denoted as [math] e_0 [/math], equals zero—to the same value [math] i_0 [/math] at the end of the charge, where the condenser electromotive force [math] e_1 = e_0 [/math].

Section Fifty

Counting the time [math] t [/math] from the moment the condenser charge begins, the terminal conditions are as follows:

At time [math] t = 0 [/math], the current [math] i = i_0 [/math], and the voltage [math] e_1 = 0 [/math] at the beginning of the condenser charge.

At time [math] t = t_0 [/math], the current [math] i = i_0 [/math], and the voltage [math] e_1 = e_0 [/math] at the end of the condenser charge.

For the condenser discharge through the circuit containing [math] L_0 [/math] and [math] r_0 [/math], the time [math] t’ [/math] is counted from the moment the condenser discharge begins. Thus, [math] t’ = t – t_0 [/math].

The terminal condition for the discharge is as follows:

At [math] t’ = 0 [/math], the current [math] i = 0 [/math], and the voltage [math] e_1 = e_0 [/math].

The voltage [math] e_0 [/math] is the value at which the discharge occurs across the spark gap. The time [math] t_0 [/math] represents the interval required for the voltage [math] e_1 [/math] to build up from zero to [math] e_0 [/math], sufficient to break down the spark gap.

Assuming the period of oscillation for the condenser charge through [math] L [/math] and [math] r [/math] is much larger than the period of oscillation for the condenser discharge through [math] L_0 [/math] and [math] r_0 [/math], the equations governing these processes are:

(A) Condenser Discharge:

[math] i = \frac{2 e_0}{q_0} e^{-\frac{r_0}{2L_0} t’} \sin \left( \frac{q_0}{2L_0} t’ \right) [/math].

[math] e_1 = e_0 e^{-\frac{r_0}{2L_0} t’} \left( \cos \left( \frac{q_0}{2L_0} t’ \right) + \frac{r_0}{q_0} \sin \left( \frac{q_0}{2L_0} t’ \right) \right) [/math].

Where [math] q_0 = \sqrt{\frac{4L_0}{C} – r_0^2} [/math].

(B) Condenser Charge:

The current [math] i [/math] is given by:

[math] i = e^{-\frac{r}{2L}t} \left[ i_0 \cos \left( \frac{q}{2L} t \right) + \frac{2(e – r i_0)}{q} \sin \left( \frac{q}{2L} t \right) \right] [/math].

The voltage [math] e_1 [/math] is given by:

[math] e_1 = e – e^{-\frac{r}{2L}t} \left[ e \cos \left( \frac{q}{2L} t \right) + \frac{r – \frac{r^2 + q^2}{2} i_0}{q} \sin \left( \frac{q}{2L} t \right) \right] [/math],

where

[math] q = \sqrt{\frac{4L}{C} – r^2} [/math].

Substituting the Terminal Conditions:

For [math] t = t_0 [/math], [math] i = i_0 [/math], and [math] e_1 = e_0 [/math]:

[math] i_0 = e^{-\frac{r}{2L} t_0} \left[ i_0 \cos \left( \frac{q}{2L} t_0 \right) + \frac{2(e – r i_0)}{q} \sin \left( \frac{q}{2L} t_0 \right) \right] [/math],

[math] e_0 = e – e^{-\frac{r}{2L} t_0} \left[ e \cos \left( \frac{q}{2L} t_0 \right) + \frac{r e – \frac{r^2 + q^2}{2} i_0}{q} \sin \left( \frac{q}{2L} t_0 \right) \right] [/math].

Convenient Substitutions:

Let

[math] s = \frac{r}{2L} t_0 [/math],

[math] \phi = \frac{q}{2L} t_0 [/math],

[math] a = \frac{r}{q} [/math].

Then, the initial current [math] i_0 [/math] is:

[math] i_0 = \frac{2e}{q} e^{-s} \frac{\sin \phi}{1 – e^{-s} \cos \phi + a e^{-s} \sin \phi} [/math].

Substituting this expression for [math] i_0 [/math] into the voltage equation for [math] e_0 [/math] gives:

[math] e_0 = e \frac{1 – 2 e^{-s} \cos \phi + e^{-2s}}{1 – e^{-s} \cos \phi + a e^{-s} \sin \phi} [/math].

The Two Equations:

Equations (75) and (76) allow the calculation of two of the three quantities: the initial current [math] i_0 [/math], the initial voltage [math] e_0 [/math], and the time [math] t_0 [/math]. The time [math] t_0 [/math], representing the condenser charge, appears in the exponential function as [math] s [/math] and in the trigonometric function as [math] \phi [/math].

Since, in an oscillating-current generator with fair efficiency—when the resistance [math] r [/math] is as small as possible—the parameter [math] s [/math] is a small quantity, the exponential term can be expanded into a series:

[math] e^{-s} = 1 – s + \frac{s^2}{2} – \text{higher-order terms}. [/math]

Substituting this series expansion into equation (75) and omitting all terms higher than the square of [math] s [/math], we find:

[math] i_0 = \frac{2e}{q} \frac{(1 – s + \frac{s^2}{2}) \sin \phi}{1 – \cos \phi + s \cos \phi – \frac{s^2}{2} \cos \phi + a \sin \phi – a s \sin \phi} [/math].

To simplify, the numerator and denominator are multiplied by [math] (1 + \frac{s}{2}) [/math], and after rearranging:

[math] i_0 = \frac{2e}{q} \frac{\sin \phi}{\frac{2+s}{2} – \cos \phi + a \sin \phi} [/math].

Rearranging further:

[math] i_0 = \frac{2e}{q} \frac{\sin \phi}{\frac{2s}{2 – s} + \frac{2 \sin^2 \phi}{2} + a \sin \phi} [/math].

Substituting the series expansion into equation (76), omitting all terms higher than the square of [math] s [/math], multiplying numerator and denominator by [math] (1 + \frac{s}{2}) [/math], and rearranging:

[math] e_0 = 2e \frac{\frac{2 \sin^2 \phi}{2} + \frac{s^2}{2}}{\frac{2s}{2 – s} + \frac{2 \sin^2 \phi}{2} + a \sin \phi} [/math].

Substituting the Expression for Time [math] t_0 [/math]:

The initial current, denoted as [math] i_0 [/math], is given by:

[math] i_0 = \frac{2e}{q} \frac{\sin \left(\frac{q}{2L} t_0 \right)}{\frac{2r t_0}{4L} – r t_0 + 2 \sin^2 \left(\frac{q}{4L} t_0 \right) + \frac{r}{q} \sin \left(\frac{q}{2L} t_0 \right)} [/math].

Similarly, the initial voltage, denoted as [math] e_0 [/math], is given by:

[math] e_0 = 2e \frac{2 \sin^2 \left(\frac{q}{4L} t_0 \right) + \frac{r^2 t_0^2}{8L^2}}{\frac{2r t_0}{4L} – r t_0 + 2 \sin^2 \left(\frac{q}{4L} t_0 \right) + \frac{r}{q} \sin \left(\frac{q}{2L} t_0 \right)} [/math].

These provide approximate equations for [math] i_0 [/math] and [math] e_0 [/math] as functions of [math] t_0 [/math], representing the time of condenser charge.

Fifty-One:

The time [math] t_0 [/math] during which the condenser charges increases with increasing [math] e_0 [/math]. This means increasing the length of the spark gap in the discharge circuit. Initially, this increase is almost proportional, but as [math] e_0 [/math] approaches [math] 2e [/math], the increase becomes slower.

As long as [math] e_0 [/math] remains appreciably below [math] 2e [/math], approximately less than [math] 1.75e [/math], the time [math] t_0 [/math] remains relatively short. During this time, the charging current, denoted as [math] i [/math], which increases from [math] i_0 [/math] to a maximum and then decreases back to [math] i_0 [/math], does not vary significantly. It remains approximately constant, with an average value only slightly above [math] i_0 [/math]. Consequently, the power supplied by the impressed electromotive force, [math] e [/math], to the charging circuit can be approximately represented as:

[math] p_0 = e_0 [/math].

The condenser discharge is intermittent, consisting of a series of oscillations, separated by periods of rest. These periods of rest are significantly longer than the duration of the oscillations, during which the condenser recharges.

Discharge Current of the Condenser:

From equation sixty-six, the current [math] i [/math] is given by:

[math] i = \frac{2e_0}{q_0} e^{-\frac{r_0}{2L_0} t} \sin \left(\frac{q_0}{2L_0} t \right) [/math].

Since this oscillation recurs at intervals of [math] t_0 [/math] seconds, the effective value, or the square root of the mean square of the discharge current, is expressed as:

[math] i_1 = \sqrt{\frac{1}{t_0} \int_0^{t_0} i^2 dt } [/math].

Long before [math] t = t_0 [/math], the current [math] i [/math] is practically zero. Therefore, as the upper limit of the integral, infinity can be chosen instead of [math] t_0 [/math].

Substituting equation sixty-six into the expression for [math] i_1 [/math] and taking constant terms out of the square root gives:

[math] i_1 = \frac{2e_0}{q_0} \sqrt{\frac{1}{2t_0} \left( \int_0^\infty e^{-\frac{r_0}{L_0} t} dt – \int_0^\infty e^{-\frac{r_0}{L_0} t} \cos \left(\frac{q_0}{L_0} t \right) dt \right)} [/math].

Now, solving the first integral:

[math] \int_0^\infty e^{-\frac{r_0}{L_0} t} dt = \frac{L_0}{r_0} [/math].

Next, using fractional integration for the second integral:

[math] \int_0^\infty e^{-\frac{r_0}{L_0} t} \cos \left(\frac{q_0}{L_0} t \right) dt = \frac{L_0 r_0}{r_0^2 + q_0^2} [/math].

Substituting these results into the earlier expression for [math] i_1 [/math]:

[math] i_1 = e_0 \sqrt{\frac{2L_0}{t_0 r_0 (r_0^2 + q_0^2)}} [/math].

Since [math] q^2 = \frac{4L}{C} – r^2 [/math], substituting this into the expression for [math] i_1 [/math] gives:

[math] i_1 = e_0 \sqrt{\frac{C}{2t_0 r_0}} [/math].

Finally, denoting the frequency by [math] f_1 [/math] as the reciprocal of [math] t_0 [/math]:

[math] f_1 = \frac{1}{t_0} [/math].

The frequency of the condenser charge, also known as the number of complete trains of discharge oscillations per second, is given by:

[math] i_1 = e_0 \sqrt{\frac{C f_1}{2r_0}} [/math].

This indicates that the effective value of the discharge current is proportional to the condenser potential, [math] e_0 [/math], proportional to the square root of the capacity, [math] C [/math], and proportional to the square root of the frequency of charge, [math] f_1 [/math]. It is inversely proportional to the square root of the resistance, [math] r_0 [/math], of the discharge circuit. However, it does not depend on the inductance, [math] L_0 [/math], of the discharge circuit and, therefore, does not depend on the frequency of the discharge oscillation.

Power of the Discharge:

The power of the discharge, [math] p_1 [/math], is given by:

[math] p_1 = i_1^2 r_0 = f_1 \frac{e_0^2 C}{2} [/math].

Since [math] \frac{e_0^2 C}{2} [/math] represents the energy stored in the condenser with capacity [math] C [/math] at potential [math] e_0 [/math], and [math] f_1 [/math] represents the frequency, or number of discharges of this energy per second, this equation is intuitive and directly follows.

Effective Value of the Discharge Current:

From this equation, the effective value of the discharge current can be calculated directly as:

[math] i_1 = \sqrt{\frac{p_1}{r_0}} = e_0 \sqrt{\frac{C f_1}{2r_0}} [/math].

Ratio of Effective Discharge Current to Mean Charging Current:

The ratio of the effective discharge current, [math] i_1 [/math], to the mean charging current, [math] i_0 [/math], is given as follows:

[math] \frac{i_1}{i_0} = \frac{e_0}{i_0} \sqrt{\frac{C f_1}{2r_0}} [/math].

Substituting equations eighty and eighty-one into this expression gives:

[math] \frac{i_1}{i_0} = q \sqrt{\frac{C f_1}{2r_0}} \frac{2 \sin^2 \left(\frac{q}{4L} t_0 \right) + \frac{r^2 t_0^2}{8L}}{\sin \left(\frac{q}{2L} t_0 \right)} [/math].

The magnitude of the current ratio can be approximated by neglecting the resistance, [math] r [/math], compared with the fraction of four times [math] L [/math], divided by [math] C [/math]. Substituting [math] q [/math] as the square root of the fraction of four times [math] L [/math], divided by [math] C [/math], and replacing the sine function with its arcs gives the following:

[math] \frac{i_1}{i_0} = \frac{1}{\sqrt{2 r_0 C f_1}} [/math].

This result indicates that the ratio of currents is inversely proportional to the square root of the resistance in the discharge circuit, the capacity, and the frequency of charge.

Now consider the example:

Assume an oscillating-current generator is being used to feed a Tesla transformer for operating X-ray tubes or directly supplying an iron arc, which is a condenser discharge between iron electrodes, for the production of ultraviolet light.

The constants of the charging circuit are given as follows:
The impressed electromotive force, [math] e [/math], is fifteen thousand volts.
The resistance, [math] r [/math], is ten thousand ohms.
The inductance, [math] L [/math], is two hundred fifty henrys.
The capacity, [math] C [/math], is equal to [math] 2 \times 10^{-8} [/math] farads, which equals [math] 0.02 [/math] millifarads.

The constants of the discharge circuit are divided into two cases:

(a) When operating a Tesla transformer:
The estimated resistance, [math] r_0 [/math], is twenty ohms (effective).
The estimated inductance, [math] L_0 [/math], is [math] 60 \times 10^{-6} [/math] henry, which equals [math] 0.06 [/math] millihenrys.

(b) When operating an ultraviolet arc:
The estimated resistance, [math] r_0 [/math], is five ohms (effective).
The estimated inductance, [math] L_0 [/math], is [math] 4 \times 10^{-6} [/math] henry, which equals [math] 0.004 [/math] millihenrys.

For the charging circuit, we calculate:
[math] q [/math], the characteristic impedance, is 223,400 ohms.
The ratio of [math] r [/math] to [math] q [/math] equals [math] 0.0448 [/math].
The ratio of [math] q [/math] to twice [math] L [/math] equals [math] 446.8 [/math].
The ratio of [math] r [/math] to twice [math] L [/math] equals [math] 20 [/math].
The ratio of [math] L [/math] to [math] r [/math] equals [math] 0.025 [/math].

Thus, the equations for the charging current and voltage are as follows:

The current, [math] i_0 [/math], equals:

[math] i_0 = 0.1344 \frac{\sin (446.8 , t_0)}{ \frac{2 t_0}{0.1} – t_0 + 2 \sin^2 (223.4 , t_0) + 0.0448 \sin (446.8 , t_0) } [/math].

The voltage, [math] e_0 [/math], equals:

[math] e_0 = 30,000 \frac{ 2 \sin^2 (223.4 , t_0) + 200 , t_0^2 }{ \frac{2 t_0}{0.1} – t_0 + 2 \sin^2 (223.4 , t_0) + 0.0448 \sin (446.8 , t_0) } [/math].

Figure seventeen illustrates the initial current, denoted as [math] i_0 [/math], and the initial voltage, denoted as [math] e_0 [/math], plotted as ordinates with the time of charge, denoted as [math] t_0 [/math], represented as the abscissas.

The frequency of the charging oscillation is given by the equation:

[math] f = \frac{q}{4 \pi L} = 71.2 [/math] cycles per second.

For [math] i_0 = 0.365 [/math] amperes, substituting into equations sixty-nine and seventy provides the following expressions:

The current, [math] i [/math], equals:

[math] i = e^{-20 t} \left[ 0.365 \cos (446.8 , t) + 0.118 \sin (446.8 , t) \right] \text{ amperes} [/math].

The voltage, [math] e_1 [/math], equals:

[math] e_1 = 15,000 \left[ 1 – e^{-20 t} \left( \cos (446.8 , t) – 2.67 \sin (446.8 , t) \right) \right] \text{ volts} [/math].

These are the equations describing the condenser charge.

From these equations, the values of [math] i [/math] and [math] e_1 [/math] are plotted in figure eighteen, with time [math] t [/math] as the abscissas.

As shown, the current [math] i [/math] reaches the initial current [math] i_0 = 0.365 [/math] amperes again at the time [math] t_0 = 0.0012 [/math] seconds, which corresponds to 30.6 time degrees or approximately one-twelfth of a period. At this instant, the condenser electromotive force, [math] e_1 [/math], equals [math] e_0 [/math], which is 22,300 volts. By setting the spark gap to 22,300 volts, the duration of the condenser charge is 0.0012 seconds. In other words, discharge oscillations occur every 0.0012 seconds, or 833 times per second.

With this spark gap setting, the charging current at the beginning and the end of the condenser charge is 0.365 amperes. The average charging current is calculated as 0.3735 amperes at an impressed electromotive force of 15,000 volts, which results in a power consumption of 5.6 kilovolt-amperes.

Assume that the electromotive force at the condenser terminals at the end of the charge is equal to 22,300 volts. Consider two cases for the discharge of the condenser:

Case (a): The condenser discharges into a Tesla transformer.
Case (b): The condenser discharges into an iron arc.

Figure eighteen illustrates the condenser charge in an oscillating-current generator.

In Case (a), the Tesla transformer is an oscillating-current transformer without iron. It consists of a primary coil with 20 turns and a secondary coil with 360 turns, both immersed in oil. Although the actual ohmic resistance of the discharge circuit is only 0.1 ohms, the effective resistance is significantly increased due to several factors: the load on the secondary of the Tesla transformer, the dissipation of energy into space by brush discharge, and the increase in resistance caused by unequal current distribution within the conductor. Therefore, the effective resistance in the discharge circuit is much greater than the actual ohmic resistance.

Following the given estimated values for the Tesla transformer case, the resistance of the discharge circuit, denoted as [math] r_0 [/math], is twenty ohms, the inductance, denoted as [math] L_0 [/math], is [math] 60 \times 10^{-6} [/math] henry, and the capacity, denoted as [math] C [/math], is [math] 2 \times 10^{-8} [/math] farads.

Calculations yield:

[math] q_0 [/math], the characteristic impedance of the circuit, equals 108 ohms.
The ratio of [math] r_0 [/math] to [math] q_0 [/math] equals [math] 0.186 [/math].
The value of [math] q_0 [/math] divided by twice [math] L_0 [/math] equals [math] 0.898 \times 10^6 [/math].
The value of [math] r_0 [/math] divided by twice [math] L_0 [/math] equals [math] 0.1667 \times 10^6 [/math].

Using these values, the discharge current [math] i [/math] is given as:

[math] i = 415 , e^{-0.1667 \times 10^6 , t} \sin(0.898 \times 10^6 , t) [/math] amperes.

The condenser potential [math] e_1 [/math] is given as:

[math] e_1 = 22,300 , e^{-0.1667 \times 10^6 , t} \left[ \cos(0.898 \times 10^6 , t) + 0.186 \sin(0.898 \times 10^6 , t) \right] [/math] volts.

The frequency of oscillation, denoted as [math] f_0 [/math], is calculated as:

[math] f_0 = \frac{0.898 \times 10^6}{2 \pi} = 143,000 [/math] cycles per second.

Figure nineteen illustrates the current [math] i [/math] and the condenser potential [math] e_1 [/math] during the discharge, with time [math] t [/math] represented as the horizontal axis. As observed, the discharge frequency is significantly higher compared to the frequency of charge. The duration of discharge is very short, and the damping is considerable, with a decrement value of 0.55, causing the oscillation to decay rapidly.

The oscillating current during the discharge is extraordinarily large in comparison to the charging current. With a mean charging current of 0.3735 amperes and a maximum charging current of 0.378 amperes, the maximum discharge current reaches 315 amperes, which is 813 times greater than the charging current. The effective value of the discharge current, as calculated from equation eighty-seven, is 14.4 amperes, or nearly 40 times the charging current.

Section Fifty-Three

Case (b): When the condenser discharges directly through an ultraviolet or iron arc, in a straight path, and assuming the resistance of the discharge circuit, denoted as [math] r_0 [/math], to be five ohms and the inductance, denoted as [math] L_0 [/math], to be [math] 4 \times 10^{-6} [/math] henry, the following calculations are made:

[math] q_0 [/math], the characteristic impedance, equals 27.84 ohms.
The ratio of [math] r_0 [/math] to [math] q_0 [/math] equals [math] 0.1795 [/math].
The value of [math] q_0 [/math] divided by twice [math] L_0 [/math] equals [math] 3.48 \times 10^6 [/math].
The value of [math] r_0 [/math] divided by twice [math] L_0 [/math] equals [math] 0.625 \times 10^6 [/math].

The discharge current, [math] i [/math], is expressed as:

[math] i = 1600 , e^{-0.625 \times 10^6 , t} \sin(3.48 \times 10^6 , t) [/math] amperes.

The condenser potential, [math] e_1 [/math], is given as:

[math] e_1 = 22,300 , e^{-0.625 \times 10^6 , t} \left[ \cos(3.48 \times 10^6 , t) + 0.1795 \sin(3.48 \times 10^6 , t) \right] [/math] volts.

The frequency of oscillation, [math] f_0 [/math], is calculated as:

[math] f_0 = \frac{3.48 \times 10^6}{2 \pi} = 562,000 [/math] cycles per second.

This shows that the frequency of oscillation is even higher, exceeding half a million cycles per second. The maximum discharge current exceeds 1,000 amperes, but the duration of the discharge is even shorter, with the oscillations decaying more rapidly.

The effective value of the discharge current, from equation eighty-seven, is 28.88 amperes, or 77 times the charging current. A hot-wire ammeter placed in the discharge circuit under these conditions measured 29 amperes.

As observed, with a very small supply current of 0.3735 amperes at an impressed voltage of 15,000 volts, the discharge circuit produces a maximum voltage of 22,300 volts—nearly 50% higher than the impressed voltage—and generates a very large current, with an effective value many times larger than the supply current.

Typically, instead of a constant impressed electromotive force, [math] e [/math], a low-frequency alternating electromotive force is used, as it is more conveniently generated by a step-up transformer. In this scenario, the discharges of the condenser occur not at constant intervals of [math] t_0 [/math] seconds, but only during those parts of each half-wave when the electromotive force is sufficient to overcome the gap [math] e_0 [/math]. Moreover, the intervals between discharges are shorter at the maximum of the electromotive force wave and longer at its beginning and end.