1. Time Response of Reactive Circuits
DC/AC Fundamentals: A Systems Approach
Thomas L. Floyd
David M. Buchla
Time Response of Reactive Circuits
Chapter 15

2. The RC Integrator Ch.15 Summary
An RC integrator approximates the mathematical process of integration. Integration is a summing process, and a basic integrator can produce an output that is a running sum of the input under certain conditions. VS R C
Vout
A basic RC integrator circuit is simply a capacitor in series with a resistor and the source. The output is taken across the capacitor.

3. The RC Integrator Ch.15 Summary
When a pulse is applied to the input of an RC integrator, the capacitor charges and discharge in response to the pulses.
Switch closes
When the input goes HIGH, the pulse generator acts like a battery in series with a switch and the capacitor charges. R
The output is an exponentially rising curve. C

4. The RC Integrator Ch.15 Summary
When the pulse goes low, the small internal impedance of the generator makes it look like a closed switch has replaced the battery. R C
Switch closes
The pulse generator now acts like a closed switch and the capacitor discharges.
The output is an exponentially falling curve.

5. The RC Integrator Ch.15 Summary 220 ms
Waveforms for the RC integrator depend on the time constant (t) of the circuit. If the time constant is short compared to the period of the input pulses, the capacitor will fully charge and discharge. For an RC circuit, t = RC. The output will reach 63% of the final value in 1t. R C
The output reaches its steady state in about 5t.
What is t if R = 10 kW and C = mF?
220 ms

6. The RC Integrator Ch.15 Summary
If t is increased, the waveforms approach the average dc level as in the last waveform. The output will appear triangular but with a smaller amplitude.
Vin t
Vout t
Vout
Alternatively, the input frequency can be increased (T shorter). The waveforms will again approach the average dc level of the input. t
Vout t

7. The RC Differentiator Ch.15 Summary
An RC differentiator approximates the mathematical process of differentiation; a process that finds a rate of change. A basic differentiator can produce an output that equals the rate of change of its input voltage under certain conditions. VS R C
Vout
A basic RC differentiator circuit is simply a resistor in series with a capacitor and the source. The output is taken across the resistor.

8. The RC Differentiator Ch.15 Summary
When a pulse generator is connected to the input of an RC differentiator, the capacitor appears as an instantaneous short to the rising edge and passes it to the resistor.
The capacitor looks like a short to the rising edge because voltage across C cannot change instantaneously.
During this first instant, the output follows the input.

9. The RC Differentiator Ch.15 Summary
After the initial edge has passed, the capacitor charges and the output voltage decreases exponentially.
The voltage across C is the traditional charging waveform.
The output decreases as the pulse levels off.

10. The RC Differentiator Ch.15 Summary
The falling edge is a rapid change, so it is passed to the output because the capacitor voltage cannot change instantaneously. The type of response shown happens when t is much less than the pulse width (t<< tw).
After dropping to a negative value, the output voltage increases exponentially as the capacitor discharges.
The voltage across C when the input goes low decreases exponentially.

11. The RC Differentiator Ch.15 Summary tw 5t = tw 5t >> tw
The output shape is dependent on the ratio of t to tw.
Vin tw
5t = tw
When 5t = tw, the pulse has just returned to the baseline when it repeats.
5t >> tw
If t is long compared to the pulse width, the output does have time to return to the original baseline before the pulse ends. The resulting output looks like a pulse with “droop”.

12. The RL Integrator Ch.15 Summary
Like the RC integrator, an RL integrator is a circuit that approximates the mathematical process of integration. Under equivalent conditions, the waveforms look like the RC integrator. For an RL circuit, t = L/R. VS R L
Vout
A basic RL integrator circuit is a resistor in series with an inductor and the source. The output is taken across the resistor.
1.0 ms
What is the time constant if R = 22 kW and L = 22 mH?

13. The RL Integrator Ch.15 Summary
When the pulse generator output goes high, a voltage immediately appears across the inductor in accordance with Lenz’s law. The instantaneous current is zero, so the resistor voltage is initially zero.
The induced voltage across L opposes the initial rise of the pulse. L + -
The output is initially zero because there is no current. VS R
0 V

14. The RL Integrator Ch.15 Summary
During the peak of the input pulse, the inductor voltage decreases exponentially and current increases. As a result, the voltage across the resistor increases exponentially. As in the case of the RC integrator, the output is 63% of its final value in one time constant.
The induced voltage across L decreases. L + - VS
The output voltage increases as current builds in the circuit. R

15. The RL Integrator Ch.15 Summary
When the pulse goes low, a reverse voltage is induced across L opposing the change. The inductor voltage initially is a negative voltage that is equal and opposite to the generator; then it exponentially returns to 0 V.
The induced voltage across L initially opposes the change in the source voltage. L - +
The output voltage decreases as the magnetic field around L collapses. VS R

16. The RL Differentiator Ch.15 Summary
An RL differentiator also approximates the mathematical process of differentiation. It can produce an output that is the rate of change of the input under certain conditions. VS L R
Vout
A basic RL differentiator circuit is an inductor in series with a resistor and the source. The output is taken across the inductor.

17. The RL Differentiator Ch.15 Summary
When a pulse generator is connected to the input of an RL differentiator, the inductor has a voltage induced across it that opposes the source; initially, no current is in the circuit.
VR = 0 V
Current is initially zero, so VR = 0 V. R
During this first instant, the inductor develops a voltage that is equal to the source voltage. VS + L -

18. The RL Differentiator Ch.15 Summary
After the rising edge of the input voltage completes its transition, current increases in the circuit. Eventually, the current reaches a steady state value given by Ohm’s law.
The voltage across R increases as current increases. R VS
The output decreases as the pulse levels off. + L -

19. The RL Differentiator Ch.15 Summary
Next, the falling edge of the pulse causes a (negative) voltage to be induced across the inductor that opposes the change. The current decreases as the magnetic field collapses.
The voltage across R decreases as current decreases. R VS -
The output decreases initially and then increases exponentially. L +

20. The RL Differentiator Ch.15 Summary
As in the case of the RC differentiator, the output shape is dependent on the ratio of t to tw. tw
Vin
When 5t = tw, the pulse has just returned to the baseline when it repeats.
5t = tw
If t is long compared to the pulse width, the output looks like a pulse with “droop”.
5t >> tw

21. Application Ch.15 Summary
An application of an integrator is to generate a time delay. The voltage at B rises as the capacitor charges until the threshold circuit detects that the capacitor has reached a predetermined level.
SW closes
Vout
Vin A R B VA
Threshold circuit SW VB
Threshold C
Vout
Time delay

22. Key Terms Ch.15 Summary Integrator
A circuit producing an output that approaches the mathematical integral of the input.
Time constant
A fixed time interval, set by R and C, or R and L values, that determines the time response of a circuit.
Transient time
An interval equal to approximately five time constants.

23. Key Terms Ch.15 Summary Steady state
The equilibrium condition of a circuit that occurs after an initial transient time.
Differentiator
A circuit producing an output that approaches the mathematical derivative of the input.

24. Quiz Ch.15 Summary The circuit shown is a. An integrator
b. A high-pass filter
c. Both of the above
d. None of the above

25. Quiz Ch.15 Summary The circuit shown is a. An integrator
b. A low-pass filter
c. Both of the above
d. None of the above

26. Ch.15 Summary
Quiz
3. Initially, when the pulse from the generator rises,
the voltage across R will be
a. Equal to the inductor voltage
b. One-half of the inductor voltage
c. Equal to VS
d. Zero VS

27. Quiz Ch.15 Summary 4. After an RL integrator has reached steady state
from an input pulse, the output voltage will be equal to
a. 1/2 VS
b VS
c. VS
d. Zero

28. Ch.15 Summary
Quiz
5. The time constant for an RL integrator is given by
the formula
a. t = L / R
b. t = 0.35RL
c. t = R / L
d. t = LR

29. Ch.15 Summary
Quiz
6. The input and output waveforms for an integrator are shown. From the waveforms, you can conclude that
a. t = tw
b. t >> tw
c. t << tw
d. None of the above
are correct
Vin
Vout tw

30. Quiz Ch.15 Summary 7. If a 20 kW resistor is in series with a 0.1 mF
capacitor, the time constant is
a. 200 ms
b. 0.5 ms
c. 1.0 ms
d. None of the above

31. Ch.15 Summary
Quiz
8. After a single input transition from 0 to 10 V, the
output of a differentiator will return to 0 V in
a. Less than one time constant
b. One time constant
c. Approximately five time constants
d. Approximately ten time constants

32. Quiz Ch.15 Summary 9. An interval equal to approximately five time
constants is called
a. Transient time
b. Rise time
c. Time delay
d. Charge time

33. Quiz Ch.15 Summary Assume a time delay is set by an RC integrator. If
the threshold is set at 63% of the final pulse height, the time delay will be equal to
a. 1t
b. 2t
c. 3t
d. 5t
SW closes
Threshold
Time delay VA VB
Vout

34. Answers Ch.15 Summary 1. b 2. c 6. b 3. d 7. d 4. c 8. c 5. a 9. a