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Transient response of RC and RL circuits
ENGR 40M lecture notes — July 26, 2017 Chuan-Zheng Lee, Stanford University
Resistor–capacitor (RC) and resistor–inductor (RL) circuits are the two types of first-order circuits: circuits either one capacitor or one inductor. In many applications, these circuits respond to a sudden change in an input: for example, a switch opening or closing, or a digital input switching from low to high. Just after the change, the capacitor or inductor takes some time to charge or discharge, and eventually settles on its new steady state. We call the response of a circuit immediately after a sudden change the transient response, in contrast to the steady state.
A first example
Consider the following circuit, whose voltage source provides vin(t) = 0 for t < 0, and vin(t) = 10 V for t ≥ 0.
R
C
vout
A few observations, using steady state analysis. Just before the step in vin from 0 V to 10 V at t = 0, vout(0−) = 0 V. Since vout is across a capacitor, vout just after the step must be the same: vout(0+) = 0 V. Long after the step, if we wait long enough the circuit will reach steady state, then vout(∞) = 10 V.
What happens in between? Using Kirchoff’s current law applied at the top-right node, we could write
10 V − vout R
= C
dvout dt
Solving this differential equation, then applying the initial conditions we found above, would yield (in volts)
vout(t) = 10 − 10 e−^ RCt .
τ = RC 2 τ 3 τ 4 τ
vout(t)
vin(t)
t
v
What’s happening? Immediately after the step, the current flowing through the resistor—and hence the capacitor (by KCL)—is i(0+) = 10 V R. Since i = C dv dtout , this current causes vout to start rising. This, in turn, reduces the current through the resistor (and capacitor), 10 V− Rv out. Thus, the rate of change of vout decreases as vout increases. The voltage vout(t) technically never reaches steady state, but after about 3RC, it’s very close.
Transient response equation
It turns out that all first-order circuits respond to a sudden change in input with some sort of exponential decay, similar to the above. Therefore, we don’t solve differential equations every time we see a capacitor or an inductor, and we won’t ask you to solve any.
Instead, we use the following shortcut: In any first-order circuit, if there is a sudden change at t = 0, the transient response for a voltage is given by
v(t) = v(∞) + [v(0+) − v(∞)]e−t/τ^ ,
where v(∞) is the (new) steady-state voltage; v(0+) is the voltage just after time t = 0; τ is the time constant, given by τ = RC for a capacitor or τ = L/R for an inductor, and in both cases R is the resistance seen by the capacitor or inductor.
The transient response for a current is the same, with i(·) instead of v(·):
i(t) = i(∞) + [i(0+) − i(∞)]e−t/τ^.
What do we mean by the “resistance seen by the capacitor/inductor”? Informally, it means the resistance you would think the rest of the circuit had, if you were the capacitor/inductor. More precisely, you find it using these steps:
- Zero out all sources (i.e. short all voltage sources, open all current sources)
- Remove the capacitor or inductor
- Find the resistance of the resistor network whose terminals are where the capacitor/inductor was
About the time constant
The time constant τ (the Greek letter tau) has units of seconds (verify, for both RC and R/L), and it governs the “speed” of the transient response. Circuits with higher τ take longer to get close to the new steady state. Circuits with short τ settle on their new steady state very quickly.
More precisely, every time constant τ , the circuit gets 1 − e−^1 ≈ 63% of its way closer to its new steady state. Memorizing this fact can help you draw graphs involving exponential decays quickly.
After 3τ , the circuit will have gotten 1 − e−^3 ≈ 95% of the way, and after 5τ , more than 99%. So, after a few time constants, for practical purposes, the circuit has reached steady state. Thus, the time constant is itself a good rough guide to “how long” the transient response will take.
Of course, mathematically, the steady state is actually an asymptote: it never truly reaches steady state. But, unlike mathematicians, engineers don’t sweat over such inconsequential details.
τ (^2) τ 3 τ 4 τ 5 τ
v(∞)
v(0+)
˜63% of [v(0+)−v(∞)]
˜95% of [v(0+)−v(∞)]
t
v(t)
Solution to Example 3
When t < 10 ms. When vin(t) = 0, the NMOS transistor is off, so the circuit reduces to this:
5 V
20 kΩ
330 nF
vout
20 kΩ
Since vin(t) = 0 for all t < 10 ms, the circuit is in steady state at t = 10 ms−. Furthermore, that 20 kΩ resistor on the right doesn’t do anything while the transistor is off. So the circuit reduces to this:
5 V
20 kΩ
vout
from which we see that vout(t) = 5 V at steady state while the transistor is off, including when t < 10 ms.
When 10 ms < t < 20 ms. During this period, vin(t) = 5 V, so the transistor is on (vGS = 5 V − 0 V = 5 V). Then we have this circuit:
+ 5 V
20 kΩ
330 nF
vout
20 kΩ
The voltage vout is the voltage across a capacitor, which can’t change instantaneously, so vout(10 ms+) = vout(10 ms−) = 5 V. (Caution! The logic isn’t that the output voltage can’t change instantaneously, it’s the the voltage across the capacitor can’t.)
To find the v(∞) term of the equation, for this part, we find the steady state value of the circuit while the transistor is on—in other words, the value that vout(t) would converge to if the transistor stayed on forever. (The circuit can’t predict the future, so it doesn’t know that the transistor will turn off soon.) To find the steady state value, we replace the capacitor with an open circuit again:
5 V
20 kΩ
20 kΩ^ vout
Now we can see that there is just a voltage divider, and vout(∞) in this state would be 2 .5 V.
The time constant is τ = RC, where R is the resistance seen by the capacitor. To find this, we short (zero) the voltage source and imagine measuring the resistance from the capacitor:
20 kΩ
20 kΩ capacitor was here
And now we can see that it’s just two 20 kΩ resistors in parallel, yielding R = 10 kΩ. Then
τ = RC = 10 kΩ · 330 nF = 3.3 ms.
Putting it all together, for 10 ms < t < 20 ms:
vout(t) = vout(∞) + [vout(10 ms+) − vout(∞)]e−^
t−10 ms τ
= 2.5 + [5 − 2 .5]e−^
t− 3 .10 ms3 ms (V) = 2.5 + 2. 5 e−^
t− 3 .10 ms3 ms (V)
When t > 20 ms. Here, the transistor has turned off again. Because the voltage across the capacitor can’t change instantaneously, vout(20 ms+) = vout(20 ms−), and vout(20 ms−) falls into the equation we just found:
vout(20 ms−) = 2.5 + 2. 5 e−^
20 ms 3 .3 ms−10 ms
= 2.62 V.
The vout(∞) term is just the steady state when the transistor is off, which we’ve already found above; it was 5 V.
As for the time constant, if we short the voltage source and remove the capacitor while the transistor is off, we get this:
20 kΩ
capacitor was here
20 kΩ
