# Understanding and Using Time Constants--RC and RL (ET/D, Apr. 1981)

 Home | Articles | Forum | Glossary | Books Every resistor capacitor, and resistor/inductor, combination has a time constant. They are so common you don't even look for them though they can confuse troubleshooting. Here we'll take some of the mystery out of them. By Bernard B. Daien The words, "time constant summon up vague recollections of resistor-capacitor networks, and time delays, for most technicians. Actually, time constants are used so extensively throughout most electronic circuits that they have become as invisible as "the forest that cannot be seen because of the trees." Time constants occur with resistor-inductor networks, as well as resistor-capacitor networks ... a fact that is infrequently discussed in textbooks. And, since RL and RC networks form either differentiators or integrators, time constants are involved in their design too. You don't remember what an integrator is ...? Well, reading this article is a quick and easy way to become familiar with time constants, integrators, differentiators, and their uses. Fig. 1 Universal Time Constant Graph Fig. 2 Basic RC Network   --------------- Fig. 3 Time Constant Chart Time Constant Chart EXAMPLE FOR USE OF CHART: "How long will it take for a 10 mfd capacitor and 1000 ohm resistor network to charge to 75% of the maximum voltage?" ANSWER, FROM CHART 1.4 = - RC Transposing, then 1.4 RC = T Since R = 1000, and C 10 10-6 Then, 1.4 (1000) (10 x 104) = T = .014 Seconds   ------------ RL Circuits When sudden voltage is applied to an inductor, the resulting current rises slowly, due to the counter electromotive force generated as the expanding magnetic field cuts the turns of the coil. In a perfect inductor, without resistance or other losses, the current would continue to rise forever, at an extremely slow rate, . . first because there would be no resistance to limit it, and also because the counter electromotive force in a perfect coil would be almost equal to the applied voltage. Conversely, in an inductor with appreciable resistance, and other losses, the current would be quickly limited, and stop rising almost immediately. It is therefore self evident that the inductance and resistance have opposite effect . . the inductance causing a slow, gradual rise time, while the resistance shortens the rise time of the current. Expressed as a formula, we can write this as follows: The Time Constant, in seconds Inductance in Henrys Resistance in Ohms Inductance, in Henrys for the current to reach 63% of maximum or, restated in symbols, Tc = This simple formula tells us several interesting things ... The length of time for current to reach 63% of its maximum (steady state) value, is directly proportional to the amount of inductance, and inversely proportional to the amount of resistance. It also says that the time constant is THE RATIO OF THE INDUCTANCE TO THE RESISTANCE, R. This means that if we use a coil of one Henry with a resistance of one Ohm, it will take one second for the current to reach 63% of its steady state value . .. or if we use ten Henrys and ten Ohms, or a hundred Henrys and a hundred Ohms, the time constant will still be one second! As a matter of fact, with an RL time constant there are an infinite number of L's and R's that give the same time constant. Remember, we are dealing only with the time required to reach 63% of the maximum current, (one time constant). At this point you are probably thinking, ... "But what about the time to reach 90%, or even 99% of the maximum current ... or some other times?" Well, the answer to that question is quite simple. During the second time constant, the current will rise 63% of the remaining current .. and so on. To sum up, in the first time constant the current will rise to 63% of its steady state value, with 37% left to increase. During the second time constant the current will increase 63% of the remaining 37%, (which is another 23%) for a total increase, in two time constants of 86%. During the third time constant the current will rise 63% of the remaining 14%, etc., etc. If you do all the calculations you will discover that the current rises to 99% of its final value in five time constants. Simple, isn't it? In order to save you the trouble of doing these repetitive calculations, a universal time constant graph is shown in Figure 1. Since the current falls in the same manner as it rises, the graph shows both rising and falling curves. In using this graph, use a little common sense. Remember that the signal source for the input to the RL network often has resistance, and this must be included. Similarly, don't expect to open an RL circuit with a switch, which is an infinite resistance (open circuit), and get a time constant ... all you will have is a spark coil. You must know what your RL is during charging, and what it is during the discharge ... and the two may be quite different is some circuits. RC Circuits Most technicians appreciate the fact that inductors and capacitors behave "oppositely". This is also true in the matter of time constants. This happens because, in the case of an inductor, we apply a voltage which causes a current to flow, while in the case of the capacitor, we apply a charging current which causes a voltage to appear across the capacitor. With the capacitor, if there is a series resistor in the circuit, the charging current flowing through the resistor will be proportional to the voltage across the resistor, and that voltage will be the difference between the supply voltage and the voltage accumulated on the capacitor. (See Figure 2). Obviously the current flow will decrease as time passes and the capacitor charges up towards the supply voltage .. . but the larger the capacitor, the longer it will take for this to happen. Also the larger the resistor, the longer the time constant. Thus we can state, "The changing time is proportional to the resistance, and is also proportional to the capacitance." Written as a formula, Time Constant i Seconds = Resistance in Ohms X Capacitance in Farads, or, simply, Tc = RC Note that unlike the formula for RL time constants, there is no ratio between C and R. With RC time constants, if R is increased, C must be decreased to maintain the same time constant, and vice versa. Further, unlike the RL circuit, the time constant of an RC circuit increases as R increases. As each time constant elapses, the voltage rises 63% of the remainder, just as with LA circuits, therefore the universal time constant graph in Figure 1 can be used in the same way, remembering of course that in the RL circuit we were talking about the current increase (or decrease), while in RC circuit we are discussing the voltage charge on the capacitor. Figure 3 is a chart derived from the universal time current graph, showing several time constants and their respective percentages of charge (or discharge). Again, this chart is equally useful for RL or RC circuits. Integrators and Differentiators . . . Integrators and differentiators are RL, or RC circuits, which do not change the shape of a pure sine wave. They can only change the amplitude or phase of sine waves. But ... they can change the shape of non sinusoidal waves ... especially pulses, and are therefore used for wave shaping and signal processing . .. such as generating sweep ramp waveforms, or deriving sync pulses from other waveforms (sync separating). Which brings us to the subject of the relationship of time constants and "integrators" and "differentiators". It must be self evident that the values of R and C must affect the operation of the differentiator, which is a basic RC circuit, (shown in Figure 5B). The formula for the output of such a differentiator is change in voltage E_out =Rx Cx change in time or, stated in words, the output of the differentiator is the time constant, RC, multiplied by THE RATE OF CHANGE OF THE INPUT VOLTAGE. (The rate of change is the amount of change, divided by the time in which the change occurred). ------------ Fig. 4 Differentiated Waveforms Differentiated Waveforms INPUT OUTPUT Fig. 5(A) Resistor-inductor differentiator (B) Resistor-capacitor differentiator (A) Resistor-Inductor Differentiator (B) Resistor Capacitor Differentiator ----------- Fig. 6(A) Resistor-inductor integrator (8) Resistor-capacitor integrator INTEGRATORS (A) Resistor-Inductor Integrator (B) Resistor-Capacitor Integrator Fig. 7 Integrator waveforms INTEGRATOR WAVEFORMS Input Output Of course if the rate of change goes to zero (no change), the output goes to zero. You know this circuit as the elementary "sync separator" used to generate the horizontal sync pulses from the composite video signal in TV receivers. There is an optimum value of R and C for each rate of change at the input (an optimum time constant). When R and C are chosen so that the output increases at the rate of 6 db per octave, (doubles each time the frequency is doubled), you have a differentiator. You also have a high pass filter! Of course we are speaking about frequencies that are being employed at the time ... if pulses are being handled, then the frequency is the reciprocal of the period of the pulse, or, 1 F = Since the differentiator responds to the rate of change of the input, we should mention that the rate of change is usually referred to as, "volts per second", or in the case of faster pulses, "volts per microsecond". As you have probably realized by now, the only difference between the RC coupling network in a resistance coupled audio amplifier, and the RC network in a differentiator, is the time constant chosen. An RC network that functions as a coupler at one frequency, is a discriminator at the lower frequencies. To insure a flat frequency response in an RC coupled audio amplifier, the time constant should be greater than the period of a half wave at the lowest frequency to be amplified. To function as a differentiator, the time constant should be one fifth, or less, the period of a half wave. Fig. 8 Integrator output when time constant is much longer than input pulse. The input versus the output waveforms of a differentiator are shown for sine waves, square waves, and triangular waves, in Figure 4. These waveshapes are shown to give the reader "a feeling" for the scope waveforms in differentiator circuits. It should be pointed out that an RL circuit can also be used as a differentiator, as shown in the schematic of Figure 5A. Since an inductor behaves oppositely from a capacitor, the positions of the resistor and inductor are interchanged with respect to the resistor and capacitor in an RC differentiator ... and, as discussed earlier the time constant of the RL differentiator is, L and, as in the RC circuit, should be much less than the period of a half wave at the operating frequency. Now, on to integrators, RL and RC integrators are shown in Figure 6. Figure 7 shows the input versus output waveforms for sine, square, and triangular waveshapes. In order to function as an integrator, the time constants of the circuits in Figure 6 should be longer than the period of a half wave. The integrator will then provide a linear ramp of output from a step function (increase or decrease) at the input, as shown in Figure 8. A positive going step at the input causes a rising ramp, while a negative going step causes a falling ramp. As you may have surmised, integrators are the basis for linear sweep waveform generators. The reason that a long time constant is required can be found by examining the universal time constant graph. Note that the first 30% of the curve is quite linear, while the remainder of the curve becomes progressively more non-linear (curved). If the output of the integrator is to be a linear ramp, the time constant must be long compared to the period of the input pulse, so that we will always be operating in the portion of the time constant curve that is confined to the linear 30% (or less). The integrator is really a form of low pass filter, with a time constant such that the output decreases at the rate of 6 db per octave as the frequency is increased. This is just opposite to the action of the differentiator discussed earlier. More Time Constants A.M. broadcast receivers use time constants . . . in the A.G.C. line there is a low pass filter, with a time constant much longer than the lowest demodulated audio frequency. The output of this low pass RC filter ramps up and down, as the rectified carrier voltage step functions up and down when the station is changed. It's an integrator. The RC coupling in the audio amplifier of the same receiver has a time constant longer than the lowest desired audio frequency. More complex equipment uses "signal conditioning" (signal processing . .. which means that the signal is amplified, or attenuated .. differentiated or integrated . . AGC'd or clipped or limited ... expanded or compressed, etc., etc., in order to put the signal into the proper form and amplitude for the following circuits. As you can readily understand, the use of RC and RL circuits plays a very important role in signal processing ... and, since RC and RL circuits are made of passive (non-amplifying) components, they attenuate the signal. These losses can accumulate to an undesirable degree. It is therefore common to use amplifiers to offset the losses. Op-amps are generally used in conjunction with RC and RL time constant circuitry in order to make practical designs. Following articles covering op amps will illustrate representative op amp RL and RC circuits for a variety of practical applications. (source: Electronic Technician/Dealer)
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