Note 15 Resistor-Capacitor Circuits
Resistor-Capacitor Circuits
A capacitor in a circuit does the following.
a. With no charge on it, the capacitor acts like a wire because there is no counter potential acting
against the battery.
b. When fully charged, the capacitor acts like a break because the potential established by the
charges in the capacitor already acts against the battery.
current
+ + + +
+
+ +++ +
V
no current
+
uncharged
capacitor
+ +++ +
V
+ +++ +
charged
capacitor
c. During the time between the above two cases, while charging using a constant voltage V, the
amount of charge on a capacitor is described by the following.
q(t) = CV(1 − e −t/RC )
The function looks like this.
charge
time
It has the following limits.
⎛
1 ⎞
as t → 0, q(t) = CV ⎜⎜ 1 − t/RC ⎟⎟⎟ → CV(1 − 1) = 0
⎜⎝
⎟⎠
e
⎛
1 ⎞
as t → ∞, q(t) = CV ⎜⎜ 1 − t/RC ⎟⎟⎟ → CV(1 − 0) = CV
⎜⎝
⎟⎠
e
The constant RC is called the time constant. It measures how fast the capacitor charges and
discharges. The larger the time constant the slower the circuit charges and discharges.
d. While discharging, the charge on a capacitor is described by this function.
q(t) = Qoe −t/RC
The constant Qo is the initial charge on the capacitor which is related to the initial voltage as well.
Qo = CVo
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The function looks like this.
charge
time
This function has the following limits.
as t → 0, q(t) = Qo
1
e
as t → ∞, q(t) = Qo
t/RC
→ Qo
1
e
t/RC
1
= Qo
1
→0
Example
The circuit below is turned on at time = 0 s.
500 Ω
5V
200 µF
What charge will be on the capacitor 0.1 second after the switch is closed?
At t = 0.1 s,
−6
q(0.1) = C ΔV(1 − e −0.1/(500)(200×10 ) ) = (200 ×10−6 )(5)(0.632) = 6.32 ×10−4 C = 0.632 mC
What charge will be on the capacitor “A long time” after the switch is closed?
A long time means infinity.
q(t → ∞) = C ΔV = (200 µF)(5 V ) = 10−3 C = 1 mC
How much time does it take the capacitor to reach a charge that is 80% of the maximum capacity?
When the charge is at 80% of the maximum,
q(t) = (0.80)(1 mC ) = (1 mC )(1 − e −t/0.1 )
0.80 = 1 − e −t/RC
⇒ e −t/0.1 = 0.20
⇒
−
t
= ln(0.20) = −1.6094
0.1
t = 0.161 s
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