### Table of Contents

## Step Response

A resistor-capacitor (RC) circuit and a resistor-inductor (RL) circuit are shown in Figure 7-6. These two circuits have some very similar properties that can be used to determine the value of either the inductor or capacitor in the circuit. If the input *(VS)*of the circuit is abruptly stepped from 0 volts to some positive value, the output voltage

(

*VO*) rises in an exponential manner (Figure 7-7).

The mathematical expression for the step response of the circuits shown is The output voltage that results from a voltage step at the input is called the STEP RESPONSE. Notice that the step response does not immediately reach the final

*VS*level, but rises in a more sluggish manner. Theoretically, it will take an infinite amount of time for

*vo*to settle to its final value (which is the same as the final value of

*VS*

*).*The standard method for describing how quickly the circuit responds is the TIME CONSTANT of the circuit.

In one time constant, the step response reaches 63.2 percent of its final value.

**Figure 7-6 RC and RL circuits can be used to determine capacitor and inductor values by measuring the time constant of the circuit, (a) RC circuit, (b) RL circuit.**

The time constant can be computed from the circuit values: An unknown capacitor or inductor whose value is to be determined can be connected to a known resistor in the appropriate circuit. (The resistor's value can be easily measured with an ohmmeter.) The time constant of the circuit is measured and the unknown component value computed.

**Figure**

**7-7**

**The step response input**

*(VS)***and output**

*(VO)***waveforms for the RC and RL circuits shown in Figure 7-6.**

Since the time constant is obviously a time domain parameter, an oscilloscope is required to measure it (Figure 7-8). For the step input, a switch connected to a voltage source might work, but a function generator is usually more practical. The function generator is set to output a fairly low-frequency square wave (about 100 Hz), which acts as a repetitive step voltage. The period of the square wave must be long enough to allow the circuit to settle (within the desired accuracy) to its final value before receiving the next voltage step (rising edge of the next square wave cycle). The amplitude of the square wave is fairly arbitrary, as long as the voltage is large enough to measure easily with the scope, and component ratings are not exceeded.

To a certain extent, the value of the resistor will depend on the value of the capacitor or inductor being measured. At first glance, this may seem like a real problem, since the goal is to measure the capacitor or inductor. However, a small amount of experimentation and experience will simplify the process. An experienced user can make a very rough estimate of the size of capacitor by its physical construc-tion. For example, a polarized electrolytic capacitor is likely to be in 1 µF to 100 µF, while a small ceramic capacitor is more likely to be less than 1 µF. To use the oscilloscope to measure the time constant conveniently, it is recommended that the time constant be kept within the range of 10 msec to 10 µsec. A recommended resistance value to start with is 1 kW. The function generator's output resistance appears in series with the resistance and should be included in the calculations. Figure 7-9 shows the input and output voltages with a suitable time constant and oscilloscope setup.

**Example 7-3**

The step response shown in Figure 7-9 resulted from an RC circuit with a l-kW resistor and an unknown capacitor. The output impedance (resistance) of the source is

50 W. Determine the capacitor value.

**Figure**

**7-8**

**A function generator and an oscilloscope are used to measure the step response of this RC circuit. The function generator's output resistance is in series with**

*R***and will affect the time constant of the circuit.**

**Figure 7-9 An example of an RC circuit's step response. Both input and output waveforms are shown. (See Example 7-3.)**

From Figure 7-9, the step response takes 0.5 msec to reach 63 percent of its final value. Thus, the time constant is 0.5 msec. R is IkW, but the output resistance of the function generator should also be included (in series). These same circuits, except with a sine wave source driving them, can be used in the frequency domain to measure the value of a capacitor or inductor (Figure 7-10). The frequency responses of these two circuits are the same (Figure 7-11). Low-frequency signals are passed from the input to the output with little or no attenuation, while high-frequency signals are attenuated significantly. Hence, it forms a low-pass filter. Normally, the point at which the response has fallen 3 dB (relative to the response at low frequencies) is used to define the filter bandwidth. A loss of 3 dB corresponds to a reduction in output voltage to 70.7 percent of the original value. where

*T*is the time constant of the circuit previously defined. (This is another example of how the time domain and frequency domain concepts are related.)

**Figure**

**7-10**

**The frequency response of the RC and RL circuits can be used to measure a capacitor or an inductor.**

Although the 3-dB point is the classical way of defining where a low-pass filter rolls off, other points such as the 6-dB point may be used. A 6-dB reduction in voltage corresponds to a 50 percent reduction in voltage. This will be a convenient number for measurement use. The equations relating the 3-dB and 6-dB frequencies to the circuit component values are summarized in Table 7-1.

**Figure 7-11 The frequency response of the RC and RLcircuits shown in**

**Figure 7-10.**

To measure an inductor or capacitor, the unknown component is connected in an RC or RL circuit as appropriate. It is assumed that the resistor is known or can be accurately measured independently. The entire frequency response of the circuit does not need to be measured. It is sufficient to tune the sine wave source to a low frequency (usually 100 Hz or so), note the amplitude of the output voltage using an oscilloscope or meter, and then increase the frequency of the sine wave source until the output voltage drops 3 or 6 dB. The 6-dB point is probably more convenient, occurring when the voltage drops to half of its original value. This technique depends on the output of the source being constant with changes in frequency. (If the source amplitude is not constant with frequency, the frequency response must be calculated at each frequency point as the output voltage divided by the input voltage.) Although the entire frequency response does not have to be measured, it is good practice to trace out the response mentally when adjusting the source frequency. There should be no sharp peaks or dips in the response, just a gradual roll-off with increasing frequency. The 3-dB or 6-dB frequency may be determined from the frequency control of the source. Or for more accuracy, a frequency counter can be used to measure the source frequency. After the 3-dB or 6-dB frequency is measured, the value of the inductance or capacitance is calculated using the appropriate equation in Table 7-1.

**TABLE 7-1 EQUATIONS RELATING THE CIRCUIT VALUES TO THE FREQUENCY RESPONSE OF THE RC AND RL LOW-PASS CIRCUITS**

**Example 7-4**

The following gain measurements were made on an RL low-pass circuit having a 500-W resistor. The output impedance of the source is 600 W. Determine the value of the inductor.

The gain is 0.5 at 12 kHz, so the 6-dB equation is the most convenient in this case. Using the appropriate equation from Table 7-1;

*R*in the equation must include both the resistor value and the output impedance of the source.

*R=*500+ 600 = 1100W.

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