EE201 – Circuit Theory I
2015 – Spring
Dr. Yılmaz KALKAN
1. Basic Concepts (Chapter 1 of Nilsson - 3 Hrs.)
Introduction, Current and Voltage, Power and Energy
2. Basic Laws (Chapter 2&3 of Nilsson - 6 Hrs.)
Voltage and Current Sources, Ohm’s Law,
Kirchhoff’s Laws, Resistors in parallel and in series, Voltage and Current Division
3. Techniques of Circuit Analysis (Chapter 4 of Nilsson - 12 Hrs.)
Node Analysis, Node-Voltage Method and Dependent Sources, Mesh Analysis, Mesh-Current
Method and Dependent Sources, Source Transformations, Thevenin and Norton Equivalents,
Maximum Power Transfer, Superposition Theorem
4. Operational Amplifier (Chapter 5 of Nilsson - 6 Hrs.)
Op-Amp Terminals & Ideal Op-Amp, Basic Op-Amp Circuits, Buffer circuit, Inverting and Noninverting Amplifiers, Summing Inverter, Difference Amplifier, Cascade Op-Amp Circuits
5. Capacitors and Inductors (Chapter 6 of Nilsson - 3 Hrs.)
Inductors, Capacitors, Series and Parallel Combinations of them
6. First Order Circuits (Chapter 7 of Nilsson - 9 Hrs.)
The Natural Response of an RL & RC Circuits, The Step Response of RL and RC
Circuits, A General Solution for Step and Natural Responses, Integrating Amplifier
Circuit
7. Second Order Circuits (Chapter 8 of Nilsson - 6 Hrs.)
The Natural Response of a Parallel RLC Circuit, The Forms of Natural Response of a Parallel RLC
Circuit, The Step Response of a Parallel RLC Circuit, Natural and Step Responses of a Series RLC
Circuit
EE201 - Circuit Theory I

Michael Faraday (1791–1867), an English
chemist and physicist, was probably the
greatest experimentalist who ever lived.

Joseph Henry (1797–1878), an American
physicist,
discovered
inductance
and
constructed an electric motor.

He made several contributions in all areas of
physical science and coined such words as
electrolysis, anode, and cathode. His discovery
of electromagnetic induction in 1831 was a
major breakthrough in engineering because it
provided a way of generating electricity. The
electric motor and generator operate on this
principle. The unit of capacitance, the farad,
was named in his honor.

He conducted several experiments on
electromagnetism and developed powerful
electromagnets that could lift objects weighing
thousands of pounds. Interestingly, Joseph
Henry discovered electromagnetic induction
before Faraday but failed to publish his
findings. The unit of inductance, the henry,
was named after him.
EE201 - Circuit Theory I
 Now that we have considered the three passive elements
(resistors, capacitors, and inductors) and one active element
(the op amp) individually.
 We are prepared to consider circuits that contain various
combinations of two or three of the passive elements.
 In this chapter, we shall examine two types of simple
circuits:
a circuit comprising a resistor and capacitor and
a circuit comprising a resistor and an inductor.
These are called RC and RL circuits, respectively.
EE201 - Circuit Theory I
 We carry out the analysis of RC and RL circuits by
applying Kirchhoff’s laws, as we did for resistive circuits.
 The only difference is that applying Kirchhoff’s laws to
purely resistive circuits results in algebraic equations,
while applying the laws to RC and RL circuits produces
differential equations, which are more difficult to solve
than algebraic equations.
 The differential equations resulting from analyzing RC
and RL circuits are of the first order. Hence, the circuits
are collectively known as first-order circuits.
EE201 - Circuit Theory I
A first-order circuit is characterized by a first-order
differential equation.
 In addition to there being two types of first-order circuits (RC
and RL), there are two ways to excite the circuits.
 The first way is by initial conditions of the storage elements in
the circuits. In these so called source-free circuits, we assume
that energy is initially stored in the capacitive or inductive
element. The energy causes current to flow in the circuit and is
gradually dissipated in the resistors. Although source free circuits
are by definition free of independent sources, they may have
dependent sources.
EE201 - Circuit Theory I
A first-order circuit is characterized by a first-order
differential equation.
 The second way of exciting first-order circuits is by
independent sources. In this chapter, the independent sources
we will consider are dc sources. (In EE202, we shall consider
sinusoidal and exponential sources.)
 The two types of first-order circuits and the two ways of
exciting them add up to the four possible situations we will study
in this chapter.
EE201 - Circuit Theory I
 RL and RC circuits are also known as first-order circuits,
because their voltages and currents are described by first-order
differential equations.
 No matter how complex a circuit may appear, if it can be
reduced to a Thevenin or Norton equivalent connected to the
terminals of an equivalent inductor or capacitor, it is a firstorder circuit.
 Note that if multiple inductors or capacitors exist in the
original circuit, they must be interconnected so that they can be
replaced by a single equivalent element.
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 A source-free RL circuit occurs when its dc source is suddenly
disconnected.
 The energy already stored in the inductor/capacitor is
released to the resistors.
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 Assume that the independent
current source generates a constant
current of I s A, and that the switch
has been in a closed position for a
long time.
 Long time means that all currents and voltages have reached a constant
value.
 Thus only constant, or dc, currents can exist in the circuit just prior to
the switch's being opened, and therefore the inductor appears as a short
circuit (Ldi/dt = 0) prior to the release of the stored energy.
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 Assume that the independent
current source generates a constant
current of I s A, and that the switch
has been in a closed position for a
long time.
di
vL
dt
vL  0  iR  iR0  0  iL  I s
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 If at t=0, the switch is opened
 Now the problem
becomes of finding v(t)
and i(t) for t  0
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 by using KVL
di
L  Ri  0
dt
 This equation is known as a first order ordinary differential equation.
The highest order derivative appearing in the equation is 1; hence the
term first-order.
di
R
R
di
R
  i  di   idt    dt  i(t )  i (0)e
dt
L
L
i
L
EE201 - Circuit Theory I
R
(  )t
L
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 We know that an instantaneous change of current cannot occur
in an inductor.


i (0 )  i (0 )  i (0)  I 0
 i (t )  I 0 e
(  R )t
L
,t  0
 The current starts from an
initial value I(0) and decreases
exponentially toward zero as t
increases
 The coefficient of t—namely, R/L—determines the rate at which the current
(or voltage) approaches zero. The reciprocal of this ratio is the time constant
of the circuit denoted by .
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
L

R
 i (t )  I 0e
t
( )

,t  0
(tangent at t=0)
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
L

R
 i (t )  I 0e
t
( )

v(t )  i (t ) R  v(t )  I 0 Re
,t  0
t
( )

, t  0
v(0  )  0 , v(0 )  I 0 R  v(0)  ??
 There will be a jump in voltage at t=0, so v(0) is NOT DEFINED!!
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 Power & Energy
2
v
P  v.i  i R   P  I 02 Re
R
2
(
2t

)
,t  0
The energy deliveredto the resistor during any interval of time
afte the switch has been opened is;
t
t
w   pdx   I Re
2
0
0
0
(
2x

)
t
dx  I R  e
2
0
(
2x

)
dx , t  0
0
2t
(

)

1 2

 ,t  0
w  LI 0 1  e

2


EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 Example: 7.1(from textbook)
The switch in the circuit has been closed for a long time before it is opened at
t = 0. Find
iL (t ), i0 (t ) and v0 (t ) for t  0
Also find the percentage of the total energy stored in the 2 H inductor
that is dissipated in the 10 Ω resistor.
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 Example: 7.1(solution)
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 Example: 7.1(solution)
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 Example: 7.1(solution)
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 Example:
The switch in the circuit has been closed for a long time. At t = 0, the
switch is opened. Calculate i(t) for t > 0.
i(t )  6e 4t A ; t  0
EE201 - Circuit Theory I
The Natural Response of An RL Circuit (Source-Free RL Circuit)
 Example:
Assuming that i(0) = 10 A, calculate i(t) and ix (t) in the circuit.
i (t )  10 e
2 t
3
A;t  0
5 23t
ix (t )   e
A ; t  0
3
EE201 - Circuit Theory I
The Natural Response of An RC Circuit (Source-Free RC Circuit)
The natural response of an RC circuit is
analogous to that of an RL circuit.
 Assume that the switch has been in
position “a“ for a long time, and the
capacitor C to reach a steady-state
condition.
 We know that a capacitor behaves as an open circuit in the
presence of a constant voltage. (Cdv/dt = 0)
 The important point is that when the switch is moved from
position a to position b (at t = 0), the voltage on the capacitor is Vg.
EE201 - Circuit Theory I
The Natural Response of An RC Circuit (Source-Free RC Circuit)
 Therefore the problem
reduces to solving the circuit
shown in following figure
EE201 - Circuit Theory I
The Natural Response of An RC Circuit (Source-Free RC Circuit)
 By using KCL
dv v
ic  i  0  C
 0
dt R
dv
1

v 
dt
RC
dv
1

dt  v(t )  v(0)e
v
RC
EE201 - Circuit Theory I
(
t
)
RC
;t  0
The Natural Response of An RC Circuit (Source-Free RC Circuit)
 We know that an instantaneous change of voltagecannot occur
in a capacitor.
v (0  )  v(0  )  v(0)  Vg  V0
and
  RC
(tangent at t=0)
 The voltage starts from an
initial value v(0) and decreases
exponentially toward zero as t
increases
 The coefficient of t—namely, RC—determines the rate at which the oltage
(or current) approaches zero. The reciprocal of this ratio is the time constant
of the circuit denoted by .
EE201 - Circuit Theory I
The Natural Response of An RC Circuit (Source-Free RC Circuit)
v(t )  V0 e
t
( )

;t  0
t
V0 ( )
i (t )  e ; t  0 
R
2
2
0
V
v
P  v.i  i R   P 
e
R
R
2
(
2t

)
, t  0
2t
( ) 

1
w  CV02 1  e   , t  0
2


EE201 - Circuit Theory I
The Natural Response of An RC Circuit (Source-Free RL Circuit)
 Example: 7.3(from textbook)
The switch in the circuit has been in position x for a long time. At t=0, the
switch moves instantaneously to position b. Find,
vC (t ) for t  0, v0 (t ) and i0 (t ) for t  0
Also find the total energy dissipated in the 60 kΩ resistor.
EE201 - Circuit Theory I
The Natural Response of An RC Circuit (Source-Free RL Circuit)
 Example: 7.3(solution)
EE201 - Circuit Theory I
The Natural Response of An RC Circuit (Source-Free RL Circuit)
 Example: 7.3(solution)
EE201 - Circuit Theory I
The Natural Response of An RC Circuit (Source-Free RL Circuit)
 Example:
Let vc(0) = 15 V. Find vc(t), vx(t) , and ix(t) for t > 0
vc (t )  15e
v x (t )  9e
5 t
2
5 t
2
ix (t )  0.75e
V ; t  0
5 t
2
wc (0)  2.25 J
EE201 - Circuit Theory I
V ; t  0
A ; t  0
The Natural Response of An RC Circuit (Source-Free RL Circuit)
 Example:
If the switch opens at t = 0, find v(t) for t ≥ 0 and wc(0).
v(t )  8e 2t V
wc (0)  5.33 J
EE201 - Circuit Theory I
END OF CHAPTER 6 – Part I
Dr. Yılmaz KALKAN
EE201 - Circuit Theory I