1. Chapter 7
In chapter 6, we noted that an important attribute of inductors and capacitors
is their ability to store energy
In this chapter, we are going to determine the currents and voltages that arise when energy
is either released or acquired by an inductor or capacitor in response to an abrupt change
in a dc voltage or current
Energy acquired by a capacitor
Energy released by a capacitor
In this chapter, we will focuses on circuits that consist only of sources, resistors, and
either (but not both) inductors or capacitors
Such configurations are called RL (resistor-inductor) and RC (resistor-capacitor) circuits

2. Our analysis of the RL and RC circuits will be divided into three phases:
First Phase we consider the currents and voltages that arise when stored energy
in an inductor or capacitor is suddenly released to a resistive network
This happens when the inductor or capacitor is abruptly disconnected from its dc source and
allowed to discharge through a resistor
The currents and voltages that arise in this configuration are referred to as the natural response of the circuit to emphasize that the nature of the circuit itself , not external sources excitation determine its behavior
you can think of this as when a tank of water is opened suddenly , will the water
in the tank disappear instantaneously in zero second or will it takes some time no
matter how small to empty the tank

3. Second Phase we consider the currents and voltages that arise when energy is being acquired
by an inductor or capacitor due to the sudden application to a dc voltage or
current source
This response is referred to as the step response
you can think of this as when a tank of water is being filled suddenly , will the water in the tank rise instantaneously in zero second or will it takes some time no
matter how small for the water to rise in the tank

4. Third Phase The process for finding both the natural response ( First Phase ) and
step response ( Second Phase ) is the same thus in the third phase we will
develop a general method that can be used to find the response of RL and RC
circuit to any abrupt change in a dc voltage or current

5. 7.1 The Natural Response of an RL Circuits
Let the circuit shown which contain an inductor is shown
Suppose the switch has been in a closed position for a long time
we will define long time
later
For the time being long time
All currents and voltages have reached a constant value
Only constant or DC currents can exist in the circuit just prior to the
switch’s being opened
The inductor appears as a short circuit

6. Because the voltage across the inductive branch is zero
There can be no current in either R0 or R
All the source current IS appears in the inductive branch
Finding the natural response requires finding the voltages and currents at any branch in the
circuit after the switch has been opened.

7. If we let t = 0 denote the instant when the switch is opened
For t ≥ 0 the circuit become
The problem become one of finding v(t) and i(t) ( or i and v ) for t ≥ 0

8. Deriving the Expression for the current
KVL around the loop
This is a first order differential equation because it contains terms involving the
ordinary derivative of the unknown di/dt.
The highest order derivative appearing in the equation is 1.
Hence the first order
We can still describe the equation further.
Since the coefficients in the equation R and L are constant that is not functions of either the dependent variable i or the independent variable t
Thus the equation can also be described as an ordinary differential equation with constant
coefficients

9. To solve the differential equation we processed as follows:
Integrating both side to obtain explicit expression for i as a function of t
Here t0 = 0
Based on the definition of the natural logarithm

10. Since the current through an indicator can not change abruptly or instanteously 
Were I0 is the initial current on the inductor just before the switch opened
( or in some cases closed)
In the circuit above I0 = Is
Therefore
Which shows that the current start from initial value I0
and decreases exponentially toward zero as t increases

11. We derive the voltage across the resistor form direct application of Ohm’s law
Note the voltage v(t) here is defined for t > 0 because the voltage across the resistor R is
zero for t < 0 ( all the current Is was going through L and zero current through R)
Note the voltage v(t) across the resistor and across the indicator can change instanousley or have a jump

12. We derive the power dissipated in the resistor
Whichever form is used, the resulting expression can be reduce to
Note the current i(t) through the resistor is zero for t < 0

13. We derive the energy delivered to the resistor
The energy deliver to the resistor is after the switch is opened because before that there was no current passing through the resistor and the voltage across it was zero

14. Note that just before the switch is opened the current on the indicator is IS = I0
The initial energy stored in the indictor

15. The significant of the time constant
The coefficient at the exponential term namely R/L determine the rate at witch the exponential term in the current approaches zero
The reciprocal of this ratio (L/R ) is the time constant which has the units of seconds
the time constant for the R,L circuit is
Seconds
It is convenient to think of the time elapsed after switching in terms of integral multiple of t
Example
After one time constant, the maximum of the current I0 has dropped to 37% of its value

16. Table showing the value of exponential e-t/t for integral multiples of t
Note that when the time elapsed after switching exceed five time constants
The current is less than 1% of its initial value I0
Thus sometime we say that five time constant after switching has occurred, the currents and voltages have for most practical purposes reached their final values
Thus with 1% accuracy along time
Five or more time consent

17. The existence of the current in the RL circuit is momentary event and is referred to as transient response of the circuit
The response that exist a long time after the switching has taken place is called
steady-state response
The steady-state response in the RL circuit is zero , that were the i(t) will go to as t goes to infinity

18. Determining the time constant
The time constant can be determined as follows
(1) If the RL circuit can be put as
Were L is an equivalent inductor and R is an equivalent resistor
Seen by the equivalent indictor
Example

19. t is the inverse of this constant
(2) If we know the differential equation
t is the inverse of this constant
Example
Suppose the differential equation is given as

20. (2) If a trace or graph of i(t) is given
Differentiating i(t) we have

26. Since the current through an inductor is given as
We need to Find v (t)

27. 12 A 4H 6W
Finding v (t)
The circuit reduce to

30. =

31. The voltage across the capacitor =

32. For t >= 0
KCL on the upper junction

36. Solving for t

37. then at time t=0 the switch move to position b .
For the circuit shown above, the switch was in position a for a long time,
then at time t=0 the switch move to position b .
Find the voltage v(t) for t > 0 ?
Solution

42. Similar to the previous example 7
Similar to the previous example 7.2 , the voltage across a capacitor is given as
We need to Find i(t)
Question: how to find i(t) ?

43. We need to Find i(t)
Question: how to find i(t) ?
Do we find i(t) through the capacitor voltage
The answer is no of course because we do not know
However we can find i(t) through the resistor 250 kW as
we need to find v(t) as will be shown next

44. The solution of the voltage v(t) is given as
The circuit reduce to
The solution of the voltage v(t) is given as
were

45. Note the direction of the current i (t) and the initial voltages
The circuit reduce to
Note the direction of the current i (t) and the initial voltages
and the polarity of

46. The circuit reduce to

48. (a)
(b)
(c)
d) Show that the total energy delivered to the 250 kW resistor is the difference between
the results in (b) and (c)
Comparing the results obtained in (b)
and in (c)

50. 7.3 Step Response of RL and RC Circuits
We are going to find the currents and voltages in 1st order RL and RC circuits
when a DC voltage or current is suddenly applied.
The Step Response of an RL Circuit
The switch is closed at t = 0 , the task is to find the expressions for the current in the circuit
and for the voltage across the inductor after the switch has been closed

51. After the switch has being closed, we have
We can solve the differential equation similar to what we did previously with the natural response by separating the variables i and t and integrating as follows

52. We know separate the variables i and t
Integrating both side to obtain explicit expression for i as a function of t
Were I0 is the current at t = 0 and i(t) is the current at any t > 0
Performing the Integrating and the substitution of the limits
When no initial current on the inductor I0 = 0

53. When no initial current on the inductor I0 = 0
When initial current on the inductor I0 ≠ 0
The equation for the no initial current (I0 = 0) indicate that the after the switch is closed the current will increase exponentially to its final value of Vs / R

54. When no initial current on the inductor I0 = 0
At one time constant t = L / R the current will be
The current will reached 63% of Its final value

55. The rate of change of i(t) or di(t)/dt current will be
The rate of change of i(t) or di(t)/dt at t = 0 ( The tangent at t=0 ) will be
Which when drawn on the plot of i(t) will be as
Here t = L/R therefore the slop of the tangent at
t =0 is

56. When initial current on the inductor I0 ≠ 0
The voltage across the inductor will be
Before the switch close the voltage across the inductor is zero
Just after the switch close the voltage across the inductor will jump to
because initial current on the inductor I0.This will make the voltage drop across the resistor after the switch was closed at t = 0+ to be RI0 → voltage drop across the inductor is (Vs-RI0)
If the initial current on the inductor I0 = 0, the current will be
and the voltage across the inductor will jump to
Note the inductor voltage can jump however the current is not allowed to jump

57. When initial current on the inductor I0 ≠ 0
When no initial current on the inductor I0 = 0
When initial current on the inductor I0 ≠ 0
When no initial current on the inductor I0 = 0

58. When no initial current on the inductor I0 = 0

59. Solving the differential equation when the initial current on the inductor I0 ≠ 0 , we have
What we want to do next is to find the indictor current i(t) without finding the differential equation or solving it
Let us look at the the indictor current i(t)
reciprocal of the time constant t
The constant part which is the
steady state value of the current
or the value of the current at
t = ∞ ( i(∞) )
initial current on the inductor i(0)

60. The time constant t
You find it after you move the switch and it is
You find it after you move the switch and the inductor in the DC state or is short
You find it before you move the switch and the inductor in the DC state or is short

61. Example 7.5
The switch in the circuit shown has been in position a for a long time
At t = 0, the switch moves from position a to position b.
(a) Find the expression for i(t) for t >0.
For t < 0.
Inductor is short

62. Inductor is short

63. We need t ( the time constant)
We look at the circuit t > 0
We deactivate independent sources ( leave dependent)
We now find the equivalent resistant seen by the indictor

64. The time constant

65. The Step Response of an RC Circuit
The switch is closed at t = 0 , the task is to find the expressions for the voltage across
And the current through the capacitor after the switch has been closed

66. After the switch has being closed, we have
We can solve the differential equation similar to what we did previously with the step response for RL circuit by separating the variables v and t and integrating we obtain
Were V0 is the initial voltage at the capacitor

67. Let us look at the the capacitor voltage current v(t)
The time constant t
The steady state value of the voltage vC(∞)
Initial voltage on the capacitor v(0)

69. When initial voltage on the capacitor V0 ≠ 0
The current through the capacitor will be
Before the switch close the current through the capacitor is zero ( open circuit )
Just after the switch close the current through the capacitor will jump to
because initial voltage on the capacitor V0.This will make the current through the resistor after the switch was closed at t = 0+ to be V0/R → current through the capacitor is (Is-V0/R)
Note the capacitor current can jump however the voltage is not allowed to jump

70. Summary of Step Response of RL and RC Circuits
because initial current on the inductor I0.
This will make the voltage drop across the resistor after the switch was closed at t = 0+ to be
RI0 → voltage drop across the inductor is
(Vs-RI0)
because initial voltage on the capacitor V0.
This will make the current through the
resistor after the switch was closed at t = 0+ to be V0/R → current through the capacitor is
(Is-V0/R)

71. Example 7.11

72. For t ≤ 0
For t < 0 both switches are closed causing the
150 mH inductor to short the 18 W resistor
iL(0-) = 6 A
Using source transformations, we find that iL(0-) = 6 A
For 0 ≤ t ≤ 35 ms
Switch 1 is open ( switch 2 is closed)
The 60 V voltage source and 4 W and 12 W
are disconnected from the circuit

73. For t ≤ 0
For 0 ≤ t ≤ 35 ms
The 60 V voltage source and 4 W and 12 W
are disconnected from the circuit
Switch 1 is open ( switch 2 is closed)
The indictor is no longer behaving as a short
circuit because the DC source is no longer in
The circuit
The 18 W resistor is no longer
short-circuited
iL(0-) = 6 A
For 0 ≤ t ≤ 35 ms

74. For t ≤ 0
iL(0-) = 6 A
For 0 ≤ t ≤ 35 ms
t ≥ 35 ms
We find iL(0.035-) from the circuit before
For 0 ≤ t ≤ 35 ms

75. For 0 ≤ t ≤ 35 ms
t ≥ 35 ms
We find iL(0.035-) from the circuit before
For 0 ≤ t ≤ 35 ms

76. For 0 ≤ t ≤ 35 ms
iL(0-) = 6 A
For 0 ≤ t ≤ 35 ms
t ≥ 35 ms

78. Unbounded Response
Example 7.11

79. The Integrating Amplifier