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Experiment :C-3Study of exponential relaxation time constant of a RC circuit.

Submitted to Mr.Golam Dastegir Al-Quaderi and

Dr. Ratan Chandra Gosh

Submitted by Md. Mehedi Hassan Batch-9;Group A Second Year, Roll–SH 236 Student of Physics Department, Uinversity of Dhaka. [Dated April 16,2011]1

Experiment : Study of exponential relaxation time constant of a RC circuit. Theory : The time required to charge a capacitor to 63% (actually 63.2%) of full charge or to discharge it to 37% (actually 37.8%) of its initial voltage is known as the time constant of discharges exponentially the circuit. as q = if q

q

o

o

et/τ is the initial charge of a capacitor then it , here τ = RC is the characteristic time or relaxation time. It is convenient to measure time for the quantity to drop one half of of its initial value and from this half time

we can rewrite:

t1/2

= e−t

1/2/τ

= 1/2 or, t1/2

= τln2 = τ0.693

or, τ = t1/2

/0.693 (1) Apparatus : An oscilloscope. signal generator, a resistor, a capacitor and a circuit board.

Figure 1.1 Charging & discharging of a Capacitor.

Figure 1.2 Circuit diagram of RC circuit.

2

naac Chara

Purc

20 10

Table -1:For charging of the capacitorFrequency X X

c

f

y

T

B

= t1/2

= Time Mean 1/(f

y

.X

c

) T

B

X Constant τ = 1.44 × t

1/2 Hz div div Hz ms ms ms ms

50 0.40 10.80 49.35 1.89 0.75 1.09 100 0.65 10.00 97.70 1.00 0.65 0.95 200 1.30 9.90 195.40 0.50 0.67 0.96 0.91 300 1.20 6.80 293.10 0.50 0.60 0.86 400 1.00 5.20 390.80 0.49 0.49 0.71 500 0.82 4.19 480.50 0.48 0.40 0.57 600 0.69 3.33 586.20 0.51 0.35 0.50 700 0.61 3.00 683.90 0.48 0.29 0.43 800 0.58 2.60 781.60 0.49 0.29 0.41 900 0.49 2.30 879.30 0.49 0.24 0.35 1000 0.42 2.05 977.70 0.49 0.21 0.30 2000 1.00 4.40 1954.0 0.11 0.11 0.17 5000 0.90 3.50 4885.0 0.05 0.045 0.064 10000 0.42 1.78 9770.0 0.057 0.024 0.034

Table -2: For discharging of the capacitorFrequency X X

c

f

y

T

B

= t1/2

= Time Mean 1/(f

y

.X

c

) T

B

X Constant τ = 1.44 × t

1/2 Hz div div Hz ms ms ms ms

50 0.40 10.80 49.35 1.89 0.75 1.09 100 0.65 10.00 97.70 1.00 0.65 0.95 200 1.20 9.90 195.40 0.50 0.60 0.86 0.90 300 1.10 6.80 293.10 0.50 0.55 0.79 400 0.90 5.20 390.80 0.49 0.44 0.63 500 0.78 4.19 480.50 0.48 0.38 0.55 600 0.70 3.33 586.20 0.51 0.35 0.51 700 0.60 3.00 683.90 0.48 0.28 0.41 800 0.58 2.60 781.60 0.49 0.28 0.41 900 0.50 2.30 879.30 0.49 0.24 0.35 1000 0.45 2.05 977.70 0.49 0.22 0.32 2000 1.00 4.40 1954.0 0.11 0.11 0.17 5000 0.85 3.50 4885.0 0.05 0.045 0.064 10000 0.42 1.78 9770.0 0.057 0.024 0.034

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Calculation :When dial frequency=100 Hz Then true frequency, f

y

= (100 × 0.977Hz) = 97.7Hz

and X = 0.65, X

c

= 0.65

We know, T

B

= 1/(f

y

.X

c

) = 1/(97.7 × 10.0)ms

=0.001s =1ms

Half time, t1/2

= T

B

.X =1 × 0.065ms

=0.65ms

Time constant, τ = 1.44 × t1/2

=(1.44 × 0.65)ms =0.95ms

Mean τ=(1.09 + 0.95 + 0.96 + 0.86 + 0.71)/5 ms = 0.91ms

Result : Time constant of given RC circuit is 0.91 ms. This result(avearge) is not ac- curate because when e took higher frequency the capacitor is not fully charged and thus our result has some errors. For higher frequency current the capacitor failed to charge fully because the Time period (T) is becoming lower .The orig- inal Time constant for this RC circuit is 1.0ms which was in our experiment , when the frequency is under 300 Hz. Discussion : We are taking the half time of the circuit,but it is not the actual t1/2

be- cause for high frequency the capacitor failed to reach its maximum value(fully charged).On the contrary we are an arbitrary charged capacitor as fully charged and thus t1/2

became lower than the true t1/2

.As a result the time constant (τ) also becoming lower and lower when frequency is increasing.

At higher frequency the time period (T) becomes smaller than actual and the signal doesn’t have enough time to reach final destination. However there is another cause of error, that is the tolerence of the given resistor. The resistence is 10kΩ and the golden color denote 5% tolerence so

∆τ =∆ RC=± 5% (RC)=±RC/20

So the correctedτshould be:

τ=τ±∆τ=[0.90 ± (0.90/20)]ms = 0.945ms or 0.855ms.

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