RC CircuitsRC Circuits

- circuits in which the currents vary in time

- rate of charging a cap depends on C and R of circuit

- differential equations

Quiz:

After the switch is closed, the light from the bulb: A) Is brightest just after the switch is closed, then fades slowly and disappears.

B) Increases gradually to maximum brightness, over a few seconds, then remains steady.

C) Comes to maximum brightness immediately, and remains steady.

D) The bulb remains off. 0.2 F

10

Discharging a CapacitorDischarging a Capacitor

(- sign because q decreases for I > 0 That is, current in circuit equals the decrease of charge on the capacitor)

C Rq

-q

IGiven: R, C, qo (initial charge)

Find: q(t) and I(t) when switch is closed

dtdq

I 2)

1) 0 IRCq

(Kirchhoff’s Loop Rule)

C Rq

-q

I

RCq

dtdq

where: q = q(t)

q(0) = qo

This is a differential equation for the function q(t), subject to the initial condition q(0) = q0 .

We are looking for a function which is proportional to its own first derivative (since dq/dt ~ -q).

Combine 1) and 2) to get:

RC is called the “time constant” or “characteristic time” of the circuit.

Units: 1 Ω x 1 F = 1 second (show this!)

Write (“tau”) = RC, then:

t

oeqtq )( (discharging)

RC

t

οeq q(t)

Solution:

DischargingDischargingq

qo

2 3 t

t

oeqtq

)( RC

t = , q ≈ 0.37 qo = (qo/e)

t = 2 , q ≈ 0.14 qo = (qo/e2)

t = 3 , q ≈ 0.05 qo = (qo/e3)

t ∞ , q 0 = (qo/e∞)

Draw a graph for I(t).

Quiz

A capacitor is charged up to 18 volts, and then connected across a resistor. After 10 seconds, the capacitor voltage has fallen to 12 volts.

Find the time constant RC, and…

What will the voltage be after another 10 seconds (20 seconds total)?

A) 8VB) 6VC) 4VD) 0

Charging a capacitorCharging a capacitor

C is initially uncharged, and the switch is closed at t=0. After a long time, the capacitor has charge Qf .

R

C

t

f e-Qq(t) 1Then, where RC.

Question: What is Qf equal to?

0dq q

Rdt C

Charging a Charging a capacitorcapacitor

t = 0, q=0 t = 3 RC, q 0.95 Qf

t = RC, q 0.63 Qf etc.

t = 2 RC, q 0.86 Qf

q

Qf

2 3 t

t

f eQtq 1)(

RC

Draw a graph of I(t). Why is I=+dq/dt this time?

Example 2Example 2

100 kΩ

12 V2 µF

i) Initial current

ii) Initial voltage across the resistor

iii) Initial voltage across the capacitor

iv) Time for voltage across C to reach 0.63*12V

v) Final voltage across the resistor

vi) Final voltage across the capacitor

The capacitor is initially uncharged. After the switch is closed, find:

Solution

Example:

A 2kΩ and a 3kΩ resistors connected in parallel are connected in series with a 2uF and a 3uF capacitors that are connected in parallel. The power source is 120V.Find the charge on each capacitor as a function of time.

Quiz:

In a simple circuit with a capacitor, resistor and a switch, long time after the switch is closed, the current in the circuit will be:

A) ε/RB) ε/τRC) ε/eRD) zero

““RC” CircuitsRC” Circuits

• a capacitor takes time to charge or discharge through a resistor

• “time constant” or “characteristic time”

= RC

(1 ohm) x (1 farad) = 1 second