Transients and Time Constants
September 1944 Radio News

September 1944 Radio News

[Table of Contents] Wax nostalgic about and learn from the history of early electronics. See articles from Radio & Television News, published 1919-1959. All copyrights hereby acknowledged.

Here is a very nice introduction / refresher on RC (resistor | capacitor) and RL (resistor | inductor) time constants and the waveforms they produce based on the applied signal and component values. Author Tatz presented the information in a 1944 issue of Radio News magazine. Only basic mathematical formulas are used, and the illustrations make the subject of RC time constants understandable even without the equations. Don't be put off by the vacuum tubes in Figure 1 because that oscillator schematic is only given as a means for generating square wave and rectangular wave inputs to the RC and RL test circuits, in case you don't have a signal generator handy. I remember in college how much more meaningful the integrated and differentiated waveforms were once Laplace transforms were introduced for circuit analysis.

Transients and Time Constants

By Abraham Tatz

An analysis of the effect of inductive and capacitive circuits having various time constants upon sinusoidal and non-sinusoidal waveforms.

In testing the frequency response of amplifiers and transformers, square waves have found great application because of their high harmonic content. The distortion of the square wave in the output of the circuit tested yields a picture of the lack of uniform response for the various frequencies involved among the harmonics. This process of distortion to a new wave shape has now become a tool in the hands of the research engineer in the new fields of electronics and television. The new wave shapes needed for this work include narrow trigger pulses for synchronization, and linear saw-tooth waves for sweep circuits. This desirable distortion of square waves can be accomplished by a simple coupling unit of resistance and capacitance, or resistance and inductance, in either case utilizing the frequency discriminating quality of the reactance to change the shape of an applied voltage wave.

The original generation of a square, or rectangular waveform usually requires a vacuum-tube circuit for the production of the original distortion and hence the creation of the harmonics. A circuit that can be used to generate a symmetrical square wave or an asymmetrical rectangular wave is shown in Fig. 1. The input is a sine wave of sufficient amplitude (25 volts peak) to overdrive the first triode, yielding a square wave on the plate when the cathode resistor R₁ is set at minimum. The second triode is an amplifier, also overdriven, which re-squares the output of the first tube. R₁ can be adjusted for symmetry of the output square wave. When R₁ is set at maximum the surge of tube current through it on the positive signal swing accumulates a heavy charge on C₁ sufficient to bias the tube for Class C operation. The output pulse is then amplified by the second triode to yield a rectangular wave at this bias setting.

Transients in RC Circuits

To examine the effect of a resistance-capacitance circuit upon a square wave it is helpful to regard the square wave as a sudden application of d.c. voltage lasting for the duration limited by the positive alternation, and then a sudden removal of the applied voltage. In Fig. 2, upon throwing the switch to position 1, the battery will force current through the circuit until the condenser is fully charged to 100 volts. The charging of the condenser is not instantaneous, but takes place in a measurable time, which is lengthened by an increase in the size of either R or C. The formula for the charging curve of condenser voltage is

E_c = E_a (1 - (1/e^t/RC)

where E_a is the applied d.c., t is in seconds, R in ohms, C in farads, and e is a mathematical constant, 2.718. The formula yields the following information:

1) At initial time t = 0, E_c equals zero.

2) At time t = RC seconds, E_c = 63% of E_a. This time (RC seconds) is called the time constant of the circuit, the product of Rand C, and is defined as the time required for the condenser to acquire 63% of the applied d.c. voltage.

3) At time t = 5 RC the condenser voltage practically reaches the applied voltage.

4) The charging curve is linear up to the time t = 0.3 RC, or 30% of the first time constant.

5) The initial slope is such that the condenser would be fully charged in one RC if the initial charging rate had been maintained.

During this charging time the resistor voltage, initially 100 volts because of the heavy charging current, drops to zero in 5 RC time constants, when current flow ceases because the condenser is fully charged. E_r drops 63% in the first time constant.

Upon throwing the switch to position 2 the condenser will discharge, taking five time constants to do so. The discharge curve follows the formula

E_c = E_a/e^t/RC

During the discharge the resistor voltage is in the opposite direction because of reversed current flow. Here E_r is initially negative 100 volts and falls back to zero in 5 RC's, the end of the discharge.

Distortion by Time Constants

Now consider the square wave of Fig. 3a. It shoots from zero to 100 volts and back, much as the plate voltage wave of a triode driven to cut-off and saturation. Its frequency is 50 cycles per second, and the time between points A and B, the duration of a half-cycle of positive d.c. voltage, is 0.01 second or 10,000 microseconds. The circuit to which the square wave is applied has a time constant of 1,000 microseconds (100,000 ohms times 0.01 μfd.). The condenser becomes fully charged in 5 RC's, halfway between A and B, and fully discharged halfway between Band C. Note that the condenser wave follows the applied square wave but lags behind it. The resistor voltage, always the difference between the applied voltage and the condenser voltage, is a positive peaked wave between A and B (condenser charging) and a negative peaked wave between Band C (condenser discharging). At all times E_a = E_r + E_c whether E. is 100 volts or zero. The resistor voltage is pure a.c., equal areas each side of the zero axis, whereas the condenser wave is all above the zero axis.

The time constant of Fig. 3a is termed "short," allowing the condenser a full charge or discharge within the time of an alternation. The resistor wave is the useful output, finding its application as a trigger pulse. Note that the peaks can be made more narrow by using a shorter time constant. A time constant is short only in relation to the frequency of the applied wave. When the resistor voltage is obtained as output from an RC circuit of short time constant the process is called "differentiation," and in this case distorts a square wave into a peaked wave.

The distortion in the output is due to poor low frequency response. The square wave has an infinite number of odd harmonics, sine waves whose frequencies are odd multiples of the fundamental fifty cycles per second. The highest frequency harmonics cause the wave shape to be further removed from the appearance of the fundamental sine wave. The reactance of the condenser in the differentiating circuit of Fig.  3a is greatest for the low frequency harmonics, causing the output across the resistor to have an exaggerated component of high frequency harmonics. Thus, the peaked wave is further removed from the sinusoidal appearance than the square wave.

The rectangular wave of Fig. 3b can be differentiated by the same time constant, resulting in the peaked wave shown. Here the time constant must be short enough to allow the condenser to become fully charged within the narrow portion of the applied wave. For example, if the time between A and B is less than 5,000 microseconds the time constant must be shorter than the one used in Fig. 3b to obtain a peaked wave output.

When the square wave is applied to a circuit whose time constant is relatively long, never allowing the condenser to become fully charged or discharged, distortion can result in a useful voltage across the condenser. This process is called "integration" and is illustrated in Fig. 3c. Here the fifty cycles per second square wave is applied to a time constant of 50,000 microseconds. The time between A and B is merely two tenths of one time constant, so the condenser voltage rises and falls only slightly, and always within the linear portion of its charging or discharging curve. The condenser output is called a triangular wave, whose operating axis is the same as the axis of the input square wave, namely fifty volts. Several cycles after it starts, the triangle settles to a fluctuation between 45 volts and 55 volts. This amplitude could be increased by making the time constant smaller, but the wave might then become nonlinear, especially since the discharge path is usually through a heavily conducting tube of low resistance.

Examining the output triangle in the light of harmonic content of the input square wave, it is seen that the condenser voltage is greatest for the lowest harmonic frequencies. Hence the output contains a smaller proportion of high frequency harmonics, resulting in a wave more nearly approximating the sinusoidal appearance.

The rectangular wave may be similarly integrated by the same time constant circuit (Fig. 3d), resulting in a sawtooth wave of condenser voltage. This is the most useful process of integration, yielding a voltage wave form used as a linear sweep for deflection of the beam in a cathode-ray tube. Note that neither portion of the saw-tooth is an instantaneous "fly-back," necessitating a blanking circuit to light up the tube during only the sweep time. The great advantage of this sweep circuit is that its frequency is rigidly controlled by the input wave being integrated.

Further Applications of BC Circuits

When the triangular wave is obtained in Fig. 3c, the resistor voltage wave is a somewhat distorted square wave. The sides are steep, but the positive portion falls off as the condenser charges up, and the negative portion falls off to the extent that the condenser discharges within the time of one alternation. If the time constant is made very long the triangle will become of negligible amplitude, and the resistor voltage will be substantially square. This long time constant circuit has application as the coupling circuit of Fig. 4a. Here the time constant of one million microseconds yields an output resistor wave which is a replica of the input. This coupling circuit is used to pass any harmonic wave with a minimum of distortion. The reactance of the condenser is small for practically all harmonics, avoiding distortion as the wave shape is coupled to a succeeding stage of amplification.

The triangular wave can be differentiated by using the short time constant circuit of 100 microseconds in Fig. 4b. Here the voltage across the 0.001 μfd. condenser is another triangle, following the applied voltage very closely and always lagging it. Between times A and B the resistor voltage (E_r = E_a - E_c) is a constant positive value between times B and C the resistor voltage is a constant negative value. Hence the output E_r is an a.c. square wave. The constant rate of change of applied voltage causes a steady current to flow through the resistor during charging, and a steady current flow in the opposite direction during the discharging time.

The same time constant may be used to differentiate the sawtooth wave (Fig. 4c). The resistor output is a rectangular wave whose positive portion is of greater amplitude because the slope of the sawtooth portion which produced it, A to B, is steeper than the slope from B to C.

The processes of differentiating and integrating will change the. shape of any input wave containing harmonics. For example, the triangular wave will be integrated by the circuit of Fig. 4d. The condenser output is smoothed out as compared to the input and because of the further removal of high frequency harmonics, has very nearly the appearance of a sine wave. However, it still contains harmonics, and is called a parabolic wave. Upon being differentiated this wave would yield a triangle, whereas a pure sine wave, containing no harmonics against which the reactance may discriminate, would yield another sine wave of diminished amplitude. The integration of a saw-tooth wave, just as in Fig. 4d, would result in a parabolic wave which is as symmetrical. Mathematically, it is said that an infinite number of integrations of a harmonic wave will yield a sine wave. Practically, further integrations approach a sine wave.

As a summary of these processes of distortion the chart of Fig. 5 shows the interrelation of differentiation and integration. Note that two successive opposite operations result in cancellation of the distortion. For example, the square wave differentiated yields a peaked wave; the peaked wave integrated yields a square wave of the same frequency. Actually, the new square waves usually will not be as perfectly formed as the original unless the peaks are very narrow, but it does work. It must be pointed out that each operation of wave shaping by distortion involves a loss of amplitude in the output. Therefore, a string of successive differentiations or integrations will require intermediate amplification, using a Class A amplifier.

Inductive Reactance for Wave Shaping

Just as the reactance of the condenser in a resistance-capacitance circuit can produce desirable distortion effects upon a wave of high harmonic content, so can the reactance of a coil in a circuit of inductance and resistance yield analogous results for wave shaping. The previous discussion of differentiating and integrating can be applied to this inductive circuit by deriving the definition of a time constant with the same introductory example of a sudden application of a battery and a sudden removal of the d.c. voltage. In Fig. 6 the closing of switch number 1 will allow current to flow in the circuit. The back voltage induced in the coil by the sudden current change prevents the current from rising instantaneously. Instead it rises to its full value, 2 ma. (assuming no coil resistance) along the curve shown. The curve is therefore a picture of the resistor voltage in the circuit. The formula for this rise of resistor voltage is:

E_r = E_a [1-(1/e^R_t/L)]

where E_a is .the battery voltage applied, R is in ohms, L in Henries, t in seconds, and e is a mathematical constant, 2.718.

Analysis of this formula yields the following information:

1) At time t = 0, E_r = 0

2) At time t = L/R, = 63% of E_a

3) At time t = 5(L/R), E_r practically equals E_a

The term L/R is the time constant of the circuit, defined as the time in seconds for the current to reach 63% of its total possible rise toward a steady state. Here again five time constants are required for a steady state to be reached.

The coil voltage E_L is always the difference between the applied voltage and the resistor voltage. It is initially 100 volts, and becomes zero after five time constants. Remember that a coil voltage depends only upon changes in current. Initially, the current change is greatest; therefore E_L is greatest. After five time constants the circuit has become steady and E_L becomes zero.

Closing switch 2 will cause the current in the circuit to die out. Again the decay is not instantaneous, but requires five time constants for E_r to become zero. The sudden change to a decreasing current puts a voltage across the coil of opposite polarity because the change is in the opposite direction. E_L becomes zero when the current finally becomes a steady zero value. The formula for this decay is

EQUATION HERE

The time constant for this circuit is L in henries divided by R in ohms, with the answer in seconds. Increasing the size of the inductance in the circuit of Fig.  6 will increase the time for the current to become a steady 2 ma. Keeping L constant, decreasing R to 25,000 ohms will cause more time to be taken for the current to reach its higher final value, 4 ma. Hence a time constant is made long by a large L and a small R.

Wave Shaping Application of L/R Circuits

The fifty cycles per second square wave may now be applied to the circuit of Fig. 7a. Here the time constant of 1,000 microseconds is short, for it allows the resistor voltage to reach a final steady value within the time of one-half cycle. The coil voltage is a peaked wave, a.c. This circuit is differentiating, yielding the same output wave shape as the differentiating RC circuit. Distortion of the square wave into the peaked wave is due to the loss of low frequency harmonics, for these low frequencies will meet less coil reactance and will yield a smaller coil voltage as compared to the high frequency output response. The rectangular wave used in Fig. 3b can be applied to the same L/R circuit used above, yielding peaked waves across the coil. 

Since the circuit differentiates, it might be expected that a triangular wave applied to it would yield a square wave across the coil. This is true, and is shown in Fig. 7b. Note that the current wave, E_r follows the applied triangle closely but always lags it. A current lag is natural to an inductive circuit. Similarly, the sawtooth wave of Fig. 4c, differentiated by the L/R circuit above, will result in a rectangular voltage across the coil.

It will be remembered that integration, passing from a square wave to a triangle, involves the use of a long time constant. A long L/R time constant needs a large inductance and small resistance. The time constant must be sufficiently long to allow only the linear portion of the rise and fall of resistor voltage to take effect. When the square wave is applied to the circuit of Fig. 7c, whose time constant is 50,000 microseconds, the output across the resistor is a triangular wave. Here the high frequency harmonics are blocked from the output, leaving the low harmonics to make up the triangle, the next step toward a sine wave. The rectangular wave of Fig. 7d results in a sawtooth resistor wave, after integration by the same long time constant.

Conclusions

Although wave shaping can be produced by either an RC or L/R circuit, several important differences exist. The principal. advantages of the RC circuit are:

1) It is simpler, lighter and cheaper, therefore more prevalent.

2) The differentiating circuit can be used as coupling between two vacuum tube stages, because the input condenser effectively blocks the d.c. voltage of a previous tube from the succeeding grid circuit.

3) The differentiating RC circuit automatically insures a low impedance output across the resistor, which can be made small for a shorter time constant.

4) The integrating circuit insures a high impedance output because the output contains mostly the low harmonic frequencies across the condenser.

The L/R circuit enjoys the following advantages:

1) The differentiating circuit yields a high impedance output, only the higher frequencies predominating across the coil output.

2) The integrating circuit results in a low impedance output, for this output is taken across a resistor whose value is relatively small.

3) The principal advantage of the L/R circuit is that the triangular or sawtooth wave obtained in the output of the integrating circuit is a resistor voltage, hence a linear change in current. This linear rise or fall of current finds its greatest application in magnetic sweep circuits using coils as the mechanism to deflect the electron beam of a cathode-ray tube. This linear change in current guarantees a linear change in the magnetic field deflecting the electron beam.

Posted June 28, 2021