Introduction to
Electronic Circuits
DC Circuit Analysis:
Transient Response of RL Circuits
DC Circuits: Transient Analysis
Up until this point, we have been looking at the “Steady State” response of
DC circuits. “Steady-State” implies that nothing has changed in the circuit
in a long time – no power has been turned on or off, no switches have been
flipped, no lightening bolts have hit the circuit, and so on.
In DC “Steady State” conditions, we can treat inductors as short circuits and
capacitors as open circuits. However, when a switch has just been “flipped”
from on to off or off to on, we can no longer make these assumptions about
inductors and capacitors. The time-dependent or DC “Transient” response
of circuits must then be calculated using known behaviors of inductors (V =
L di/dt) and capacitors (i = C dv/dt) and the appropriate mathematics/
calculus.
75Ω
+
-
20V
t=0
20mH
50Ω
+
1mF Vc
-
DC Circuits: Transient Analysis
Our discussion of transient response is divided into several sections:
1.  Those circuits that contain only resistors and capacitors (RC circuits)
2.  Those circuits that contain only resistors and inductors (RL circuits)
3.  Those circuits that contain resistors, capacitors, and inductors (RLC
circuits)
75Ω
+
-
20V
20mH
+
1mF Vc
50Ω
-
t=0
and
1.  Those circuits responding to power removed from the circuit (Natural
Response)
2.  Those circuits responding to power delivered/applied to the circuit (Step
Response)
The circuit above is an RLC circuit and a Natural Response
DC Circuits: Transient Analysis
In this lecture, we will look at the following circuits:
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power removed from the circuit (Natural Response)
3.  Those circuits responding to power delivered/applied to the circuit (Step Response)
a
R1
V
t=0
+
-
b
R
L
This circuit is an RL circuit
which goes into its Natural Response at t > 0
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power removed from the circuit (Natural Response)
a
R'
V
b
t=0
+
R
-
L
Right before t = 0 (at t = 0-):
The circuit is presumed to have had the switch in
position A for "a long time" and is in DC Steady State;
the inductor is acting like a short circuit
R'
L
R
Voltage
V
0
0
Current
V/R'
V/R'
0
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and capacitors (RL circuits)
2.  Those circuits responding to power removed from the circuit (Natural Response)
a
R'
V
b
t=0
+
R
-
L
Right after t = 0 (at t = 0+):
The switch moves to position B. Only the inductor current cannot
change instantaneously from t = 0- to t = 0+. All other
parameters are calculated based on the inductor current.
R'
L
R
Voltage
0
-RV/R'
RV/R'
Current
0
V/R'
V/R'
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power removed from the circuit (Natural Response)
a
R'
V
b
t=0
+
R
-
L
At t = infinity:
The switch has been in position B for a long time. The
inductor is again acting as a short circuit.
R'
L
R
Voltage
0
0
0
Current
0
0
0
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power removed from the circuit (Natural Response)
a
R'
V
+
-
b
t=0
L
For t > 0 in general,
KVL gives:
+
R
i
+
-
VL + VR = 0
Ldi/dt + iR = 0
Ldi/dt = -iR
Solving this KVL equation as a first order differential equation:
Ldi/dt = -iR
(1/i) di = -R/L dt
Taking the integral of both sides gives:
ln i = -t/(L/R)
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power removed from the circuit (Natural Response)
a
R'
V
b
t=0
+
-
-
L
i
+
The limits of integration
are:
+
-
R 1.  t goes from 0 to t
2.  i goes from i(t = 0+)
to i(t), where i(t = 0+)
is V/R'
ln i = -t/(L/R)
Evaluating this equation across the limits of integration gives:
ln i – ln (V/R') = -t / (L/R)
ln [ i / (V/R') ] = -t / (L/R)
i = (V/R') e-t/(L/R)
and
i/(V/R') = e-t/(L/R)
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power removed from the circuit (Natural Response)
a
R'
V
b
t=0
+
-
-
L
+
i
+
-
Knowing the inductor
current as a function
of time, we can
R calculate any
remaining
parameters in the
circuit as a function
of time.
The inductor current is i(t) = (V/R') e-t/(L/R)
The resistor current is also i(t) = (V/R') e-t/(L/R)
The resistor voltage is v(t) = (RV/R') e-t/(L/R)
The inductor voltage is v(t) = -(RV/R') e-t/(L/R)
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power removed from the circuit (Natural Response)
a
R'
V
b
t=0
+
-
-
L
+
+
R
i
The current is:
(V/R') e-t/(L/R)
-
Some things to note:
The time constant τ of this circuit is L/R, meaning that the current
reaches 36.8% of its initial value after τ seconds, or
the voltage decays by 63.2% after τ seconds.
The current begins at the initial value of the inductor current (V/R') and
decays to zero (in the Natural Response)
DC Circuits: Transient Analysis
In this lecture, we will look at the following circuits:
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power removed from the circuit (Natural Response)
3.  Those circuits responding to power delivered/applied to the circuit (Step Response)
a
R'
V
b
t=0
+
-
R
L
This circuit is an RL circuit
which goes into its Step Response at t > 0
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power delivered/applied to the circuit (Step Response)
a
R'
V
b
t=0
+
R
-
L
Right before t = 0 (at t = 0-):
The circuit is presumed to have had the switch in
position B for "a long time" and is in DC Steady State;
the inductor is acting like a short circuit
R'
L
R
Voltage
0
0
0
Current
0
0
0
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power delivered/applied to the circuit (Step Response)
a
R'
V
+
b
t=0
i
R
-
L
Right after t = 0 (at t = 0+):
The switch moves to position A. Only the inductor current cannot
change instantaneously from t = 0- to t = 0+. All other parameters
are calculated based on the inductor current.
R'
L
R
Voltage
0
V
0
Current
0
0
0
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power delivered/applied to the circuit (Step Response)
a
R'
V
b
t=0
+
-
R
L
i
At t = infinity:
The switch has been in position A for a long time. The inductor is
again acting as a short circuit.
R'
L
R
Voltage
V
0
0
Current
V/R'
V/R'
0
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power delivered/applied to the circuit (Step Response)
a
R'
V
b
t=0
+
-
For t > 0 in general,
KVL gives:
R
i
L
V –VR' + VL = 0
V – iR' - Ldi/dt = 0
1/(i - V/R' )di = -R'/L dt
Solving this KVL equation as a first order differential equation:
1/(i - V/R' )di = -R'/L dt
Taking the integral of both sides gives:
ln (i – V/R') di = -t/(L/R)
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power delivered/applied to the circuit (Step Response)
a
b
R'
V
t=0
+
+
-
The limits of integration
are:
R
i
L
i
-
1.  t goes from 0 to t
2.  i goes from i(t = 0+)
to i(t), where i(t = 0+)
is 0
ln (i – V/R') di = -t/(L/R)
Evaluating this equation across the limits of integration gives:
ln (i-V/R') – ln (0-V/R') = -t / (L/R)
ln [ (i – V/R')/ (-V/R')] = -t / (L/R)
i = V/R' - V/R' e-t/(L/R)
and
(i – V/R')/ (-V/R') = e-t/(L/R)
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and inductors (RL circuits)
2.  Those circuits responding to power delivered/applied to the circuit (Step Response)
a
R'
V
b
t=0
+
+
-
R
i
L
i
-
Knowing the inductor
current as a function
of time, we can
calculate any
remaining
parameters in the
circuit as a function
of time.
The inductor current is i(t) = V/R' - V/R' e-t/(L/R)
The resistor (R') current is also i(t) = V/R' - V/R' e-t/(L/R)
The resistor (R') voltage is v(t) = V - V e-t/(L/R)
The inductor voltage is v(t) = V e-t/(L/R)
DC Circuits: Transient Analysis
1.  Those circuits that contain only resistors and capacitors (RC circuits)
2.  Those circuits responding to power delivered/applied to the circuit (Step Response)
a
R'
V
Using the Shortcut
Method below:
t=0
+
-
b
R
i
L
i(t) = V/R'
+
(0- V/R') e
A general solution for all RL circuits (Natural, Step) is:
Unknown variable as a function of time =
The final variable as a function of time
+
(Initial – Final value of variable) exp [- (t – tswitching)/time constant τ)
–t/(L/R)
DC Circuits: Transient Analysis
Our discussion of transient response has been divided into several sections:
1.  Those circuits that contain only resistors and inductors (RL Circuits)
2.  Those circuits responding to power removed from the circuit (Natural Response)
3.  Those circuits responding to power delivered/applied to the circuit (Step Response)
a
R'
V
t=0
+
-
b
R
i
L
We have found a general solution to finding the inductor current in circuits
containing resistors and inductors, whether the inductor is experiencing a
natural or a step response. Once the inductor current is known, it can be used to
find/solve for any other current or voltage in the circuit.