Basic Electric Circuits Session 18

Basic Electric Circuits – © 2020 Mouli Sankaran Email: mouli.sankaran@yahoo.com 2
Session 18: Focus
 Properties of Exponential Response
◦ Time constant (τ) Definition
 Source Free RC Circuit
 v(t) of RC Circuit
 Time constant of RC Circuit
 Problem
 Home work Problem 1

Properties of Exponential Response

RL Circuit Response
 We have found that the inductor
current in RL circuit is
represented by:
 At t = 0, the current has value
i(0) = I0, but as time increases,
the current decreases and
approaches zero
 Since the function we are plotting
is e−Rt/L, the curve will not change
if R/L remains unchanged.
 Thus, the same curve must be
obtained for every series RL
circuit having the same R/L ratio

How much time for i(t) to become zero?
 Since the curve is extended from
t = 0 to , let us use the
following assumption to find the
time taken for i(t) to become
zero
 First, let us find the initial rate
of decay at t = 0
 Then find the time that would be required for the current to
drop to zero if it continued to drop at its initial rate
 The initial rate of decay is found by evaluating the derivative
at zero time

Time Constant of RL Circuit
 We designate the value of time it
takes for i/I0 to drop from unity
to zero, assuming a constant
rate of decay, by the Greek letter
τ (tau).
 This value of time τ is called the
time constant
 An equally important interpretation of the time constant τ is
obtained by determining the value of i (t)/I0 at t = τ
,,thus in one time constant the response
has dropped to 36.8 percent of its initial value
Slope = -R/L

Time taken for i(t) to become zero
 How long does it take for the current to decay to zero?
 About five time constants later:
◦ At τ = 5, current will be close to 0.67% of original value (I0)

Source Free
RC Circuits

Source Free RC Circuit
 Circuits based on resistor-capacitor (RC)
combinations are more common than their resistor-
inductor (RL) circuits.
 The principal reasons for this are
◦ The smaller losses present in a physical capacitor, lower
cost
◦ Better agreement between the simple mathematical
model and the actual device behaviour
◦ And also smaller size and lighter weight, both of which are
particularly important for integrated-circuit applications.

Source Free RC Circuit: Derivation
 Let the initial energy stored in the
capacitor be v(0) = V0
 The nodal equation for the circuit is:
 Division by C yields:
 This is similar to the RL equation form:
 Similar to the i(t) for RL circuit:
 v(t) for the RC circuit can be derived as:

Time constant of RC Circuit
 v(t)
 Time constant (τ) can be derived as:

Problem 1: RC Time constant
 Find the v(t) labeled at t = 200 μsecs
 At t <= 0, the circuit will be as shown in (b)
 The capacitor would have charged to a maximum of
9V and zero current would be flowing through 4Ω
 When the switch is opened at t = 0,

Problem 1: RC Time constant … contd.
 Find the v(t) labeled at t = 200 μsecs
 The equation for v(t) is given as:

Home work: Problem 1: RC Time constant
 Find the v(t) labeled at t = 0 and t = 160 μsecs
50 V and 18.39 V

Session 17: Summary
 Properties of Exponential Response
◦ Time constant (τ) Definition
 Source Free RC Circuit
 v(t) of RC Circuit
 Time constant of RC Circuit
 Problem
 Home work Problem 1

References
Ref 1 Ref 2

Basic Electric Circuits Session 18

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Basic Electric Circuits Session 18