Determining the Time Constant of an RC Circuit, Study notes of Engineering Physics

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RC Circuit – Java Applet

RC Circuit

RC Circuit

1 RC Circuit

1.1 Theory

R C

E

S

VR o Vc I

Figure 1:

VR = IR; VC = Q C E − IR −

Q

C

CE − RC

dQ dt

− Q = 0

∫ (^) dQ CE − Q

RC

∫ dt

−ln(CE − Q) = t RC

• k

at t=0 Q = k = −ln(CE)

−ln(CE − Q) =

t RC − ln(CE)

ln(CE − Q) = −

t RC

• ln(CE)

ln(CE − Q) − ln(CE) = − t RC ln

CE − Q

CE

t RC Q = Qo(1 − e−^ RCt ) CQ = CQo(1 − e−^ RCt ) Vc = Eo(1 − e−^ RCt )

1.3 Determining the time constant τ = R 1 C 2 by using an alternative method

during the charging of the capacitor

c E (^) o

Eo 2

t 1 t 2 t (time)

τ

τ

τ > τ 2 1

2

1

V

Figure 3:

Vc = Eo(1. 0 − e−^ RCt ) Vc = Eo − Eoe−^ RCt Eo − Vc Eo

= e−^ RCt

ln

[ E

o −^ Vc Eo

] = −

RC

t

Complete the following table and plot a graph ln

[ (^) E o−Vc Eo

] vs t. Determine the value for R 1 using this graph and compare your value with previous value for R 1 which was determined in section 2.

Eo(volts) Vc(volts) Eo E−oV c ln

[ (^) E o−Vc Eo

] t(sec)

1.4 Determining the time constant τ = R 1 C 2 by using an alternative method

during the discharging of the capacitor

Vc across the capacitor is shown in figure 4.

c E (^) o

V = E ec (^) o^ −t/RC

t (time)

V

Figure 4:

Vc is given by Vc = Eoe−^ RCt

where Eo is the voltage of the source.

lnVc = −

RC

t + lnEo

Complete the following table and plot a graph lnVc vs t. Determine the value for R 1 using this graph and compare your value with previous value for R 1 which was determined in section 2.

Eo(volts) Vc(volts) ln Vc t(sec)