Elementary lectures on electric discharges, waves and impulses, and other transients/Lecture I

by Charles Proteus Steinmetz

ELEMENTARY LECTURES ON ELECTRIC
DISCHARGES, WAVES AND IMPULSES,
AND OTHER TRANSIENTS.

LECTURE I.

NATURE AND ORIGIN OF TRANSIENTS.

1. Electrical engineering deals with electric energy and its flow, that is, electric power. Two classes of phenomena are met: permanent and transient, phenomena. To illustrate: Let G in Fig. 1 be a direct-current generator, which over a circuit A connects to a load L, as a number of lamps, etc. In the generator G, the line A, and the load L, a current $i$ flows, and voltages $e$

Fig. 1.

exist, which are constant, or permanent, as long as the conditions of the circuit remain the same. If we connect in some more lights, or disconnect some of the load, we get a different current $i'$, and possibly different voltages $e'$; but again $i'$ and $e'$ are permanent, that is, remain the same as long as the circuit remains unchanged.

Let, however, in Fig. 2, a direct-current generator G be connected to an electrostatic condenser C. Before the switch S is closed, and therefore also in the moment of closing the switch, no current flows in the line A. Immediately after the switch S is closed, current begins to flow over line A into the condenser C, charging this condenser up to the voltage given by the generator. When the ₵condenser C is charged, the current in the line A and the condenser C is zero again. That is, the permanent condition before closing the switch S, and also some time after the closing of the switch, is zero current in the line. Immediately after the closing of the switch, however, current flows for a more or less short time. With the condition of the circuit unchanged: the same generator voltage, the switch S closed on the same circuit, the current nevertheless changes, increasing from zero, at the moment of closing the switch S, to a maximum, and then decreasing again to zero, while the condenser charges from zero voltage to the generator voltage. We then here meet a transient phenomenon, in the charge of the condenser from a source of continuous voltage.

Fig. 2.

Commonly, transient and permanent phenomena are superimposed upon each other. For instance, if in the circuit Fig. 1 we close the switch S connecting a fan motor F, at the moment of closing the switch S the current in the fan-motor circuit is zero. It rapidly rises to a maximum, the motor starts, its speed increases while the current decreases, until finally speed and current become constant; that is, the permanent condition is reached.

The transient, therefore, appears as intermediate between two permanent conditions: in the above instance, the fan motor disconnected, and the fan motor running at full speed. The question then arises, why the effect of a change in the conditions of an electric circuit does not appear instantaneously, but only after a transition period, requiring a finite, though frequently very short, time.

2. Consider the simplest case: an electric power transmission (Fig. 3). In the generator G electric power is produced from mechanical power, and supplied to the line A . In the line A some of this power is dissipated, the rest transmitted into the load L, where the power is used. The consideration of the electric power in generator, line, and load does not represent the entire phenomenon. While electric power flows over the line A , there is a magnetic field surrounding the line conductors, and an electrostatic field issuing from the line conductors. The magnetic field and the electrostatic or “dielectric” field represent stored energy. Thus, during the permanent conditions of the flow of power through the circuit Fig. 3, there is electric energy stored in the space surrounding the line conductors. There is energy stored also in the generator and in the load; for instance, the mechanical momentum of the revolving fan in Fig. 1, and the heat energy of the incandescent lamp filaments. The permanent condition of the circuit Fig. 3 thus represents not only flow of power, but also storage of energy. When the switch S is open, and no power flows, no energy is stored in the system. If we now close the switch, before the permanent condition corresponding to the closed switch can occur,

Fig. 3. the stored energy has to be supplied from the source of power; that is, for a short time power, in supplying the stored energy, flows not only through the circuit, but also from the circuit into the space surrounding the conductors, etc. This flow of power, which supplies the energy stored in the permanent condition of the circuit, must cease as soon as the stored energy has been supplied, and thus is a transient.

Inversely, if we disconnect some of the load L in Fig. 3, and thereby reduce the flow of power, a smaller amount of stored energy would correspond to that lesser flow, and before the conditions of the circuit can become stationary, or permanent (corresponding to the lessened flow of power), some of the stored energy has to be returned to the circuit, or dissipated, by a transient.

Thus the transient is the result of the change of the amount of stored energy, required by the change of circuit conditions, and is the phenomenon by which the circuit readjusts itself to the change of stored energy. It may thus be said that the permanent phenomena are the phenomena of electric power, the transients the phenomena of electric energy.

3. It is obvious, then, that transients are not specifically electrical phenomena, but occur with all forms of energy, under all conditions where energy storage takes place.

Thus, when we start the motors propelling an electric car, a transient period, of acceleration, appears between the previous permanent condition of standstill and the final permanent condition of constant-speed running; when we shut off the motors, the permanent condition of standstill is not reached instantly, but a transient condition of deceleration intervenes. When we open the water gates leading to an empty canal, a transient condition of flow and water level intervenes while the canal is filling, until the permanent condition is reached. Thus in the case of the fan motor in instance Fig. 1, a transient period of speed and mechanical energy appeared while the motor was speeding up and gathering the mechanical energy of its momentum. When turning on an incandescent lamp, the filament passes a transient of gradually rising temperature.

Just as electrical transients may, under certain conditions, rise to destructive values; so transients of other forms of energy may become destructive, or may require serious consideration, as, for instance, is the case in governing high-head water powers. The column of water in the supply pipe represents a considerable amount of stored mechanical energy, when flowing at velocity, under load. If, then, full load is suddenly thrown off, it is not possible to suddenly stop the flow of water, since a rapid stopping would lead to a pressure transient of destructive value, that is, burst the pipe. Hence the use of surge tanks, relief valves, or deflecting nozzle governors. Inversely, if a heavy load comes on suddenly, opening the nozzle wide does not immediately take care of the load, but momentarily drops the water pressure at the nozzle, while gradually the water column acquires velocity, that is, stores energy.

The fundamental condition of the appearance of a transient thus is such a disposition of the stored energy in the system as differs from that required by the existing conditions of the system; and any change of the condition of a system, which requires a change of the stored energy, of whatever form this energy may be, leads to a transient.

Electrical transients have been studied more than transients of other forms of energy because:

(a) Electrical transients generally are simpler in nature, and therefore yield more easily to a theoretical and experimental investigation.

(b) The theoretical side of electrical engineering is further advanced than the theoretical side of most other sciences, and especially:

(c) The destructive or harmful effects of transients in electrical systems are far more common and more serious than with other forms of energy, and the engineers have therefore been driven by necessity to their careful and extensive study.

4. The simplest form of transient occurs where the effect is directly proportional to the cause. This is generally the case in electric circuits, since voltage, current, magnetic flux, etc., are proportional to each other, and the electrical transients therefore are usually of the simplest nature. In those cases, however, where this direct proportionality does not exist, as for instance in inductive circuits containing iron, or in electrostatic fields exceeding the corona voltage, the transients also are far more complex, and very little work has been done, and very little is known, on these more complex electrical transients.

Assume that in an electric circuit we have a transient current, as represented by curve $i$ in Fig. 4 ; that is, some change of circuit condition requires a readjustment of the stored energy, which occurs by the flow of transient current $i$. This current starts at the value $i_{1}$, and gradually dies down to zero. Assume now that the law of proportionality between cause and effect applies; that is, if the transient current started with a different value, $i_{2}$ it would traverse a curve $i'$, which is the same as curve $i$, except that all values are changed proportionally, by the ratio

${\frac {i_{2}}{i_{1}}}$; that is, $i'=i\times {\frac {i_{2}}{i_{1}}}$.

Starting with current $i_{1}$, the transient follows the curve $i$; starting with $i_{2}$, the transient follows the proportional curve $i'$. At some time, $t$, however, the current $i$ has dropped to the value $i_{2}$, with which the curve $i'$ started. At this moment $t$, the conditions in the first case, of current i, are the same as the conditions in the second case, of current $i'$, at the moment $t_{1}$; that is, from $t$ onward, curve $i$ is the same as curve $i'$ from time $t_{1}$ onward. Since

Fig. 4. — Curve of Simple Transient: Decay of Current.

$i'$ is proportional to $i$ from any point $t$ onward, curve $i'$ is proportional to the same curve $i$ from $t_{1}$ onward. Hence, at time $t_{1}$, it is

${\frac {di_{2}}{dt_{1}}}={\frac {di_{1}}{dt_{1}}}\times {\frac {i_{2}}{i_{1}}}$.

But since ${\frac {di_{2}}{dt_{1}}}$ and $i_{2}$ at $t_{1}$ are the same as ${\frac {di}{dt}}$ and $i$ at time $t$, it follows:

${\frac {di}{dt}}={\frac {di_{1}}{dt_{1}}}{\frac {i}{i_{1}}}$,

or,

${\frac {di}{dt}}=-ci$,

where $c=-{\frac {1}{i_{1}}}{\frac {di_{1}}{dt}}={\mbox{constant}}$, and the minus sign is chosen, as ${\frac {di}{dt}}$ is negative.

As in Fig. 4:

${\begin{array}{rcl}{\mbox{tan}}\ \phi &=&-\displaystyle {\frac {di_{1}}{dt}},\\{\overline {a_{1}t_{1}}}&=&i_{1},\\c=-\displaystyle {\frac {1}{i_{1}}}{\frac {di_{1}}{dt}}&=&\displaystyle {\frac {{\mbox{tan}}\ \phi }{\overline {a_{1}t_{1}}}}=\displaystyle {\frac {1}{\overline {t_{1}t_{2}}}};\end{array}}$

that is, $c$ is the reciprocal of the projection $T={\overline {t_{1}t_{2}}}$ on the zero line of the tangent at the starting moment of the transient.

Since	$c={\frac {1}{T'}}$,
	${\frac {di}{i}}=-cdt$;

that is, the percentual change of current is constant, or in other words, in the same time, the current always decreases by the same fraction of its value, no matter what this value is.

Integrated, this equation gives:

	${\begin{array}{rcl}log\ i&=&-ct+C,\\i&=&Ae^{-ct},\end{array}}$
or	$i\ \ \ =\ \ \ Ae^{-{\frac {t}{T}}}$;

that is, the curve is the exponential.

The exponential curve thus is the expression of the simplest form of transient. This explains its common occurrence in electrical and other transients. Consider, for instance, the decay of radioactive substances: the radiation, which represents the decay, is proportional to the amount of radiating material; it is ${\frac {dm}{dt}}=cm$, which leads to the same exponential function.

Not all transients, however, are of this simplest form. For instance, the deceleration of a ship does not follow the exponential, but at high velocities the decrease of speed is a greater fraction of the speed than during the same time interval at lower velocities, and the speed-time curves for different initial speeds are not proportional to each other, but are as shown in Fig. 5. The reason is, that the frictional resistance is not proportional to the speed, but to the square of the speed.

5. Two classes of transients may occur:

1. Energy may be stored in one form only, and the only energy change which can occur thus is an increase or a decrease of the stored energy.

2. Energy may be stored in two or more different forms, and the possible energy changes thus are an increase or decrease of the total stored energy, or a change of the stored energy from one form to another. Usually both occur simultaneously.

An instance of the first case is the acceleration or deceleration of a train, or a ship, etc.: here energy can be stored only as mechanical momentum, and the transient thus consists of an increase of the stored energy, during acceleration, or of a decrease, during

Fig. 5. — Deceleration of Ship. deceleration. Thus also in a low-voltage electric circuit of negligible capacity, energy can be stored only in the magnetic field, and the transient represents an increase of the stored magnetic energy, during increase of current, or a decrease of the magnetic energy, during a decrease of current.

An instance of the second case is the pendulum, Fig. 6: with the weight at rest in maximum elevation, all the stored energy is

Fig. 6. — Double-energy Transient of Pendulum. potential energy of gravitation. This energy changes to kinetic mechanical energy until in the lowest position, a, when all the potential gravitational energy has been either converted to kinetic mechanical energy or dissipated. Then, during the rise of the weight, that part of the energy which is not dissipated again changes to potential gravitational energy, at c, then back again to kinetic energy, at a; and in this manner the total stored energy is gradually dissipated, by a series of successive oscillations or changes between potential gravitational and kinetic mechanical energy. Thus in electric circuits containing energy stored in the magnetic and in the dielectric field, the change of the amount of stored energy — decrease or increase — frequently occurs by a series of successive changes from magnetic to dielectric and back again from dielectric to magnetic stored energy. This for instance is the case in the charge or discharge of a condenser through an inductive circuit.

If energy can be stored in more than two different forms, still more complex phenomena may occur, as for instance in the hunting of synchronous machines at the end of long transmission lines, where energy can be stored as magnetic energy in the line and apparatus, as dielectric energy in the line, and as mechanical energy in the momentum of the motor.

6. The study and calculation of the permanent phenomena in electric circuits are usually far simpler than are the study and calculation of transient phenomena. However, only the phenomena of a continuous-current circuit are really permanent. The alternating-current phenomena are transient, as the e.m.f. continuously and periodically changes, and with it the current, the stored energy, etc. The theory of alternating-current phenomena, as periodic transients, thus has been more difficult than that of continuous-current phenomena, until methods were devised to treat the periodic transients of the alternating-current circuit as permanent phenomena, by the conception of the “effective values,” and more completely by the introduction of the general number or complex quantity, which represents the periodic function of time by a constant numerical value. In this feature lies the advantage and the power of the symbolic method of dealing with alternating-current phenomena, — the reduction of a periodic transient to a permanent or constant quantity. For this reason, wherever periodic transients occur, as in rectification, commutation, etc., a considerable advantage is frequently gained by their reduction to permanent phenomena, by the introduction of the symbolic expression of the equivalent sine wave.

Hereby most of the periodic transients have been eliminated from consideration, and there remain mainly the nonperiodic transients, as occur at any change of circuit conditions. Since they are the phenomena of the readjustment of stored energy, a study of the energy storage of the electric circuit, that is, of its magnetic and dielectric field, is of first importance.