Chapter 22 : RC and L/R Time Constants

Example No. 22_1 Page No. 674

In [1]:
# What is the time constant of a 20-H coil having 100 Ohms of series resistance?

# Given data

L = 20.#     # Inductor=20 Henry
R = 100.#    # Resistor=100 Ohms

T = L/R#
print 'The Time Constant = %0.2f Seconds'%T

The Time Constant = 0.20 Seconds

Example No. 22_2 Page No. 674

In [2]:
# An applied dc voltage of 10 V will produce a steady-state current of 100 mA in the 100-Ohms coil. How much is the current after 0.2 s? After 1 s?

# Given data

L = 20.#         # Inductor=20 Henry
R = 100.#        # Resistor=100 Ohms
I = 100.*10**-3#  # Steady-state current=100 mAmps

print 'Since 0.2 sec is one time constant, I is 63% of 100 mA'
I1 = 0.63*I#
print 'The current at 0.2 sec time constant = %0.2f A'%I1

print 'After 1 sec the current reaches its steady state value of 100 mAmps '

Since 0.2 sec is one time constant, I is 63% of 100 mA
The current at 0.2 sec time constant = 0.06 A
After 1 sec the current reaches its steady state value of 100 mAmps

Example No. 22_3 Page No. 675

In [4]:
# If a 1-M Ohms R is added in series with the coil, how much will the time constant be for the higher resistance RL circuit?

# Given data

L = 20.#        # Inductor=20 Henry
R = 1.*10**6#    # Resistor=1 MOhms

T = L/R#
print 'The Time Constant = %0.2e Seconds'%T
print 'i.e 20 us'

The Time Constant = 2.00e-05 Seconds
i.e 20 us

Example No. 22_4 Page No. 676

In [5]:
# What is the time constant of a 0.01-uF capacitor in series with a 1-M Ohmsresistance?

# Given data

C = 0.01*10**-6#     # Capacitor=0.01 uFarad
R = 1*10**6#         # Resistor=1 MOhms

T = C*R#
print 'The Time Constant = %0.2e Seconds'%T

The Time Constant = 1.00e-02 Seconds

Example No. 22_5 Page No. 677

In [6]:
# With a dc voltage of 300 V applied, how much is the voltage across C in Example 22–4 after 0.01 s of charging? After 0.05 s? After 2 hours? After 2 days?

# Given data

C = 0.01*10**-6#     # Capacitor=0.01 uFarad
R = 1.*10**6#         # Resistor=1 MOhms
V = 300.#            # Applied DC=300 Volts

T = C*R#
print 'The Time Constant = %0.2e Seconds'%T

print 'Since 0.01 sec is one time constant, the voltage across C then is 63% of 300 V,'

T1 = 0.63*V#
print 'The Capacitor voltage at 0.01 Sec = %0.2f Volts'%T1

T2 = V
print 'After 5 time constants or 0.05 Sec Capacitor voltage = %0.2f volts '%V

print 'After 2 hours or 2 days the C will be still charged to 300 V if the supply is still connected'

The Time Constant = 1.00e-02 Seconds
Since 0.01 sec is one time constant, the voltage across C then is 63% of 300 V,
The Capacitor voltage at 0.01 Sec = 189.00 Volts
After 5 time constants or 0.05 Sec Capacitor voltage = 300.00 volts 
After 2 hours or 2 days the C will be still charged to 300 V if the supply is still connected

Example No. 22_6 Page No. 678

In [8]:
# If the capacitor is allowed to charge to 300 V and then discharged, how much is the capacitor voltage 0.01 s after the start of discharge? The series resistance is the same on discharge as on charge.

# Given data

C = 0.01*10**-6#     # Capacitor=0.01 uFarad
R = 1.*10**6#         # Resistor=1 MOhms
V = 3000#            # Applied DC=300 Volts

print 'In one time constant, C discharges to 37% of its initial voltage'

V1 = 0.37*V#
print 'The Capacitor voltage after 0.01 sec start of discharge = %0.f volts'%V1

In one time constant, C discharges to 37% of its initial voltage
The Capacitor voltage after 0.01 sec start of discharge = 1110 volts

Example No. 22_7 Page No. 680

In [9]:
# Assume the capacitor is discharging after being charged to 200 V. How much will the voltage across C be 0.01 s after the beginning of discharge? The series resistance is the same on discharge as on charge.

# Given data

C = 0.01*10**-6#     # Capacitor=0.01 uFarad
R = 1*10**6#         # Resistor=1 MOhms
V = 200#            # Capacitor voltage=200 Volts

print 'In one time constant, C discharges to 37% of its initial voltage'

V1 = 0.37*V#
print 'The Capacitor voltage after 0.01 sec start of discharge = %0.f volts'%V1

In one time constant, C discharges to 37% of its initial voltage
The Capacitor voltage after 0.01 sec start of discharge = 74 volts

Example No. 22_8 Page No. 681

In [10]:
# If a 1-M Ohms resistance is added in series with the capacitor 0.01-uF and resistor 1-M Ohms in, how much will the time constant be?

# Given data

C = 0.01*10**-6#     # Capacitor=0.01 uFarad
R = 2*10**6#         # Resistor= 2 MOhms 

T = C*R#
print 'The Time Constant = %0.2e Seconds'%T

The Time Constant = 2.00e-02 Seconds

Example No. 22_9 Page No. 682

In [12]:
from math import log10
# An RC circuit has a time constant of 3 s. The capacitor is charged to 40 V. Then C is discharged. After 6 s of discharge, how much is Vr?

# Given data

RC = 3#     # RC time constant=3 Sec
t = 6#      # Discharge time=6 Sec
Vc = 40#    # Capacitor voltage=40 Volts

A = t/RC#   # constant factor
B = log10(Vc)#

Vr = 10**(B-(A*0.434))#
print 'The Value of Vr = %0.2f Volts'%Vr

The Value of Vr = 5.42 Volts

Example No. 22_10 Page No. 682

In [15]:
from math import log10
# An RC circuit has an R of 10 k Ohms and a C of 0.05 uF. The applied voltage for charging is 36 V. (a) Calculate the time constant. (b) How long will it take C to charge to 24 V?

C = 0.05*10**-6#     # Capacitor=0.05 uFarad
R = 10*10**3#        # Resistor=10 kOhms
V = 36#             # Applied voltage=36 Volts
v = 12#             # Voltage drops from 36 to 12 Volts
A = 2.3#            # Specific factor

T = C*R#
print 'The Time Constant = %0.2e Seconds'%T
print 'i.e 0.5*10**-3 Sec OR 0.5 mSec'

t = A*T*log10(V/v)#
print 'Time required to charge Capacitor upto 24 Volts = %0.2e Seconds'%t
print 'i.e approx 0.549*10**-3 Sec OR 0.549 mSec'

The Time Constant = 5.00e-04 Seconds
i.e 0.5*10**-3 Sec OR 0.5 mSec
Time required to charge Capacitor upto 24 Volts = 5.49e-04 Seconds
i.e approx 0.549*10**-3 Sec OR 0.549 mSec