9. Capacitor and Resistor Circuits

ElectronicsLab9.nb

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9. Capacitor and Resistor

Circuits

Introduction

Thus far we have consider resistors in various combinations with a power supply or battery which

provide a constant voltage source or direct current (voltage) DC. Now we start to consider various

combinations of components and much of the interesting behavior depends upon time so we will also

consider AC or alternating current (voltage) sources which are signal generators. The first combination

we consider is a resistor in series with a capacitor and a battery.

The RC Circuit

Consider the resistor-capacitor circuit indicated below:

When the switch is closed, Kirchoff's loop equation for this circuit is

V=

Q

C

+ iR

(1)

for t>0 where both Q[t] and i[t] are functions of time. There are two unknown quantities Q[t] and i[t] in

equation (1) and we need an additional equation namely

ElectronicsLab9.nb

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d

i@tD =

dt

(2)

Q@tD

You can eliminate one of the unknowns between equations (1) and (2) by taking the derivative of equation (1) with respect to time obtaining

0=

1

d

d

Q@tD + R

C dt

dt

(3)

i@tD

and using equation (2) to eliminate the derivative of the charge

0=

1

C

d

i@tD + R

dt

(4)

i@tD

It is easy enough to solve equation (4) since by rearrangement

d

dt

i@tD =

-1

RC

(5)

i@tD

Further

¨¤

1

i

?i =

-1

RC

¨¤ ?t

(6)

Integration yields

LogB

i@tD

i0

F-

t

(7)

RC

where i0 is a constant of integration which we will determine shortly. Using a property of the exponential

function, we obtain from equation (7)

i@tD = i0 ExpB-

t

RC

F

(8)

Initially at t=0, when the switch is closed, the capacitor has zero charge and therefore there is zero

potential across it. The current in the circuit is determined entirely by the battery potential V and the

resistance R through Ohm's law

i@0D R = V

or

i@0D =

V

(9)

R

as initially the capacitor C play no role.

Setting t=0 in equation (8) and using equation (9) yields

i@0D = i0 =

V

R

so we have determined the constant of integration.

finally the solution as

(10)

Utilization of equation (10) in equation (8) yields

ElectronicsLab9.nb

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¨¤ i@tD =

V

R

ExpB-

t

RC

F

(11)

The product RC has units of time and usually is called the time constant t

t=RC

(12)

Graph of the Solution for the current

Suppose the numerical values V=10 volts, R=8,000 ohms, and C=2.5 microfarads as indicated

then

V = 10; R = 8000.; Cap = 2.5 * 10-6 ;

t = R * Cap;

Print@"Time Constant =", t, " sec"D

Time Constant =0.02 sec

IMPORTANT: C is a protect variable assigned to something specific in Mathematica so instead we use

Cap as the symbol for capacitance.

i@t_D :=

V

R

ExpB-

t

t

F

Plot@i@tD, 8t, 0, 4 * t ................
................

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