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 - 6 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
 In the previous chapter we considered circuits with a
single storage element (a capacitor or an inductor).
 Such circuits are first-order because the differential
equations describing them are first-order.
 In this chapter we will consider circuits containing
two storage elements.
 These are known as second-order circuits because
their responses are described by differential
equations that contain second derivatives.
EE201 - Circuit Theory I
A second-order circuit is characterized by a second-order
differential equation. It consists of resistors and the equivalent
of two energy storage elements.
EE201 - Circuit Theory I
 Our analysis of second-order circuits will be similar to that
used for first-order.
 Natural response (source-free circuits)
 Step response
 Finding Initial & Final Values
 We begin by learning how to obtain the initial conditions
for the circuit variables and their derivatives, as this is
crucial to analyzing second-order circuits.
 Unless otherwise stated in this chapter, ‘v’ denotes
capacitor voltage, while ‘i’ is the inductor current.
EE201 - Circuit Theory I
 Finding Initial & Final Values
 There are two key points to keep in mind in determining the initial
conditions.
 First—as always in circuit analysis—we must carefully handle the polarity of
voltage v(t) across the capacitor and the direction of the current i(t) through
the inductor. Keep in mind that v and i are defined strictly according to the
passive sign convention.
 Second, keep in mind that the capacitor voltage and inductor current are
always continuous so that
EE201 - Circuit Theory I
 Finding Initial & Final Values (Example)
The switch in figure has been closed for a long time. It is open at
t = 0. Find:
(a) i(0+), v(0+),
(b) di(0+)dt , dv(0+)/dt ,
(c) i(∞), v(∞).
Answers :
a) 2A, 4V
b) 20V/s, 0 A/s
c) 0A, 12V
EE201 - Circuit Theory I
 Finding Initial & Final Values (Example)
The switch in figure was open for a long time but it is closed at
t = 0. Find:
(a) i(0+), v(0+),
(b) di(0+)dt , dv(0+)/dt ,
(c) i(), v().
EE201 - Circuit Theory I
 Finding Initial & Final Values (Example)
Answers :
a) 0A, -20V, 4V
b) 0A/s, 2V/s, 2/3 V/s
c) 1A, -20V, 4V
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 A source-free series RLC circuit occurs when its dc source is
suddenly disconnected.
 The circuit is being excited by the energy initially stored in the
capacitor and inductor. The energy is represented by the
initial capacitor voltage 𝑉0 and initial inductor current 𝐼0 .
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 at t=0
 Applying KVL
 This is a second-order differential equation and is the reason for calling
the RLC circuits in this chapter second-order circuits. Our goal is to
solve this equation and to find v(t) and i(t).
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 To solve such a second-order differential equation requires that we
have two initial conditions, such as the initial value of “i” (i(0))and its
first derivative (di(0)/dt) or initial values of some “i” and “v” (i(0) & v(0))
or
 Now the equation below can be solved with two initial conditions.
 The solution is of exponential form. So we let,
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 By inserting this solution into the general equation,
or
 If this equation is zero, it is only possible when ???
 This quadratic equation is known as the characteristic equation of
the differential equation, since the roots of the equation dictate the
character of “i”. The two roots of equation above are:
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 A more compact way of expressing the roots is;
 where
 The roots s1 and s2 are called natural frequencies, measured in
nepers per second (Np/s), because they are associated with the natural
response of the circuit; w0 is known as the resonant frequency or
strictly as the undamped natural frequency, expressed in radians per
second (rad/s); and α is the neper frequency or the damping factor,
expressed in nepers per second.
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 By inserting these parameters we can obtain;
 As solution has two roots, it must be in same exponential form as;
 where A1 and A2 are constants and must be determined by the
initial values i(0) and di(0)/dt.
We can infer that there are three types of solutions:
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 Overdamped Case (  0)
 Possible when C  4 L / R 2 , s1 and s2 are real and negative
decays and approaches zero as t  
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 Critically Damped Case ( = 0)
 Possible when C  4L / R2 , s1  s2     R 2L
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 Underdamped Case ( < 0)
s1     (02   2 )    jd
 Possible when C  4 L / R 2 ,
s2     (02   2 )    jd
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 Underdamped Case ( < 0)
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 Underdamped Case ( < 0)
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 Example:
R=40 ohm, L=4H, C=0.25F. Calculate the characteristic roots of the
circuit. Is the natural response overdamped, underdamped or critically
damped?
s1  0.101, s2  9.899
EE201 - Circuit Theory I
The Natural Response of a Series RLC Circuit (Source-Free Circuit)
 Example:
Find i(t) in the circuit. Assume that the circuit has reached steady state at
t  0
i(t )  e 9t (cos 4.359  0.6882 sin 4.359 t ) A
EE201 - Circuit Theory I
The Natural Response of a Parallel RLC Circuit (Source-Free Circuit)
 at t=0
 Applying KCL
We obtain the characteristic equation by replacing the first derivative
by s and the second derivative by 𝑠 2 . By following the same reasoning
used in series RLC circuit, the characteristic equation is obtained as;
EE201 - Circuit Theory I
The Natural Response of a Parallel RLC Circuit (Source-Free Circuit)
where
EE201 - Circuit Theory I
The Natural Response of a Parallel RLC Circuit (Source-Free Circuit)
 Overdamped Case (  0)
 Possible when L  4 R 2C , s1 and s2 are real and negative
 Critically Damped Case ( = 0)
 Possible when L  4R2C, s1  s2     1
2RC
 Underdamped Case ( < 0)
 Possible when L  4 R 2C , s1, 2    jd , d  02   2
EE201 - Circuit Theory I
The Natural Response of a Parallel RLC Circuit (Source-Free Circuit)
 Example:
Find v(t) for t>0. Assume v(0)=5V, i(0)=0A, L=1H and C=10mF. Consider
these cases R=1.923 , R=5  and R=6.25 .
v(t )  10 .625 e 2t  5.625 e 50t V
v(t )  (5  150 t )e 10tV
v(t )  (5 cos 6t  20 sin 6t )e 8tV
EE201 - Circuit Theory I
The Natural Response of a Parallel RLC Circuit (Source-Free Circuit)
 Example:
v(t )  10 .625 e 2t  5.625 e 50t V
v(t )  (5  150 t )e 10tV
v(t )  (5 cos 6t  20 sin 6t )e 8tV
EE201 - Circuit Theory I
The Step Response of a Series RLC Circuit (Forced Response)
 Applying KVL around the loop for t>0
𝑑𝑖
L𝑑𝑡 + 𝑅𝑖 + 𝑣 = 𝑉𝑠
𝑑𝑣
i= 𝐶 𝑑𝑡
𝑑 2 𝑣 𝑅 𝑑𝑣
𝑣
𝑉𝑠
+
+
=
𝑑𝑡 2 𝐿 𝑑𝑡 𝐿𝐶 𝐿𝐶
 The solution of equation above has two components: the natural
response 𝑣𝑛 (𝑡) and the forced response 𝑣𝑓 (𝑡) ; that is,
𝑣 𝑡 = 𝑣𝑛 𝑡 + 𝑣𝑓 (𝑡)
Natural response
as already obtained
𝑣𝑓 (𝑡)=𝑣𝑓 () =𝑉𝑠
EE201 - Circuit Theory I
The Complete Responses of a Series and Parallel RLC Circuits
 Thus, the complete solutions for the overdamped, underdamped,
and critically damped cases are:
 For Series RLC Circuit
 For Parallel RLC Circuit
Example:
Find v(t) and i(t) for t>0. Consider these cases R=5 , R=4  and R=1 .
EE201 - Circuit Theory I
Example:
EE201 - Circuit Theory I
Example:
Having been in position a for a long time, the switch is moved to
position b at t=0. Find 𝑣(𝑡) and 𝑣𝑅 (𝑡) for t>0.
EE201 - Circuit Theory I
Example:
Find 𝑖(𝑡) and 𝑖𝑅 (𝑡) for t>0.
EE201 - Circuit Theory I
Example:
Find 𝑖(𝑡) and v(𝑡) for t>0.
EE201 - Circuit Theory I
END OF CHAPTER 7
(END OF TERM)
Dr. Yılmaz KALKAN
EE201 - Circuit Theory I