NETWORK ANALYSIS
UNIT – I INTRODUCTION TO ELECTRICAL CIRCUITS:
Circuit concept – R-L-C parameters
Voltage and current sources
Independent and dependent sources
Source transformations
Kirchhoff’s laws
network reduction techniques
series, parallel, series parallel
star – to –delta and or delta – to – star transformation
Mesh Analysis
Nodal analysis
Super mesh
super node concept
• Network: The interconnection of two or more
circuit elements (voltage sources ,resistors ,
inductors and capacitors) is called an electrical
network. If the network contains at least one
closed path is called circuit.
• Every circuit is a network , but all the networks
are not circuit.
Active Components (have directionality)
Voltage and current sources
Passive Components (Have no directionality)
Resistors, capacitors, inductors
(with all the initial conditions are zero)
4
Ohm’s Law
I=V/R
Georg Simon Ohm (1787-1854)
I
= Current (Amperes) (amps)
V
= Voltage (Volts)
R
= Resistance (ohms)
How you should be thinking
about electric circuits:
Voltage: a force that
pushes the current
through the circuit (in
this picture it would be
equivalent to gravity)
How you should be thinking
about electric circuits:
Resistance: friction that
impedes flow of current
through the circuit
(rocks in the river)
How you should be thinking
about electric circuits:
Current: the actual
“substance” that is
flowing through the
wires of the circuit
(electrons!)
Basic Electrical Quantities
• Basic quantities: current, voltage and power
– Current: time rate of change of electric charge
I = dq/dt
1 Amp = 1 Coulomb/sec
– Voltage: electromotive force or potential, V 1 Volt
= 1 Joule/Coulomb = 1 N·m/coulomb
– Power:
P=IV
1 Watt = 1 Volt·Amp = 1 Joule/sec
Lect1
EEE 202
9
Overview of Circuit Theory
• Power is the rate at which energy is being
absorbed or supplied.
• Power is computed as the product of voltage
and current:
pt   vt i t  or P  VI
• Sign convention: positive power means that
energy is being absorbed; negative power
means that power is being supplied.
Active vs. Passive Elements
• Active elements can generate energy
– Voltage and current sources
– Batteries
• Passive elements cannot generate energy
– Resistors
– Capacitors and Inductors (but CAN store energy)
Lect1
EEE 202
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Energy Storage Elements
• Capacitors store energy in an electric field.
• Inductors store energy in a magnetic field.
• Capacitors and inductors are passive
elements:
– Can store energy supplied by circuit
– Can return stored energy to circuit
– Cannot supply more energy to circuit than is
stored.
Types of sources
Independent sources :
1. Voltage source
2. Current source
Dependent sources:
1. Voltage dependent voltage source
2. Voltage dependent current source
3. current dependent voltage source
4. current dependent current source
Ideal voltage source:
•An ideal voltage source has zero internal resistance so that changes in load
resistance will not change the voltage supplied.
•An ideal voltage source gives a constant voltage, whatever the current is.
A simple example is a 10V battery. For example, a 1ohm resistor or a
10ohm resistor could be connected to it; the voltage across both resistors
would be 10V but the currents would be different.
Practical voltage source:
Practical voltage source has an internal resistance (greater than zero),
but we treat this internal resistance as being connected in series with
an ideal voltage source.
An ideal voltage source has zero internal resistance
Ideal current source:
An ideal current source is a circuit element that maintains a prescribed
current through its terminals regardless of the voltage across those
terminals.
A ideal current source gives a constant current whatever the load is.
If you have a 2A current source for example:
-with a 3 ohm resistor it would automatically change the voltage to 6V
-with a 30 ohm resistor it would automatically change the voltage to 60V
but the current would be 2A whichever resistor was connected.
Practical current source:
Practical current source has an internal resistance, but we treat this
internal resistance as being connected in parallel with an ideal current
source.
An ideal current source has infinite internal resistance.
Dependent sources :
Dependent sources behave just like independent voltage and current
sources, except their values are dependent in some way on another
voltage or current in the circuit.
A dependent source has a value that depends on another
voltage or current in the circuit.
Source transformation
Another circuit simplifying technique
It is the process of replacing a voltage source vS in series with a
resistor R by a current source iS in parallel with a resistor R, or vice
versa
R
vs
a
a
is
+

b
Terminal a-b sees:
Open circuit voltage: vs
Short circuit current: vs/R
R
b
For this circuit to be equivalent, it
must have the same terminal
charateristics
Source Transformations
A method called Source Transformations will allow the transformations of a voltage
source in series with a resistor to a current source in parallel with resistor.
R
a
vs
+

a
is
b
R
b
The double arrow indicate that the transformation is bilateral , that we can start with either
configuration and drive the other
R
a
a
vs
+

iL
is
RL
iRL
b
b
iL 
RL
vs
R + RL
R
iL 
is
R + RL
Equating we have ,
vs
R

is
R + RL R + RL
 is 
vs
R
OR v s  Ri s
Simple Circuits
• Series circuit
– All in a row
– 1 path for electricity
– 1 light goes out and the
circuit is broken
• Parallel circuit
– Many paths for electricity
– 1 light goes out and the
others stay on
Resistors in Series
• A single loop circuit is one which has only a
single loop.
• The same current flows through each element
of the circuit - the elements are in series.
Resistors in Series
Two elements are in series if the current that
flows through one must also flow through the
other.
Series
R1
R2
Resistors in Series
Consider two resistors in series with a voltage
v(t) across them:
Voltage division:
i(t)
+
R1
v2(t)
R2
v2 (t )  v(t )
R1 + R2
+
v(t)
R2
-
v1(t)
R1
v1 (t )  v(t )
R1 + R2
+
-
Resistors in Series
• If we wish to replace the two series resistors
with a single equivalent resistor whose voltagecurrent relationship is the same, the equivalent
resistor has a value given by
Req  R1 + R2
Resistors in Series
• For N resistors in series, the equivalent resistor
has a value given by
R1
R2
R3
Req  R1 + R2 + R3 +  + RN
Req
Resistors in Parallel
• When the terminals of two or more circuit
elements are connected to the same two
nodes, the circuit elements are said to be in
parallel.
Resistors in Parallel
Consider two resistors in parallel with a voltage
v(t) across them:
Current division:
i(t)
+
v(t)
-
i1(t)
i2(t)
R1
R2
R2
i1 (t )  i (t )
R1 + R2
R1
i2 (t )  i (t )
R1 + R2
Resistors in Parallel
• If we wish to replace the two parallel resistors
with a single equivalent resistor whose voltagecurrent relationship is the same, the equivalent
resistor has a value given by
R1 R2
Req 
R1 + R2
Resistors in Parallel
• For N resistors in parallel, the equivalent
resistor has a value given by
R1
R2
R3
1
Req 
1
1
1
1
+
+
++
R1 R2 R3
RN
Req
Parallel
Two elements are in parallel if they are
connected between (share) the same two
(distinct) end nodes.
R1
R1
R2
R2
Parallel
Lect1
Not Parallel
EEE 202
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Series-Parallel Combinations of Inductance and
Capacitance
• Inductors in Series
– All have the same current
di
v L
dt
1
1
di
v L
dt
2
2
v v +v +v
1
2
ECE 201 Circuit Theory I
di
v L
dt
3
3
3
34
To Determine the Equivalent Inductance
v v +v +v
1
2
3
di
di
di
v L
+L
+L
dt
dt
dt
di
v  (L + L + L )
dt
di
v L
dt
L L +L +L
1
2
1
2
3
3
eq
eq
1
2
3
ECE 201 Circuit Theory I
35
The Equivalent Inductance
ECE 201 Circuit Theory I
36
Inductors in Parallel
All Inductors have the same voltage across
their terminals.
ECE 201 Circuit Theory I
37
1
i 
L
1
i 
L
1
i 
L
1
 vd
t
+ i (t )
1
t0
0
1
2
 vd
t
+ i (t )
2
t0
0
2
3
 vd
t
+ i (t )
t0
3
0
3
ECE 201 Circuit Theory I
38
i i +i +i
1
2
3
1 1 1
i   + +   vd + i (t ) + i (t ) + i (t )
L L L 
1
i
 vd + i (t )
L
1
1 1 1
 + +
L
L L L
t
1
t0
1
2
0
2
0
3
0
3
t
0
t0
eq
eq
1
2
3
i (t )  i (t ) + i (t ) + i (t )
0
1
0
2
0
3
ECE 201 Circuit Theory I
0
39
Summary for Inductors in Parallel
ECE 201 Circuit Theory I
40
Capacitors in Series
Problem # 6.30
ECE 201 Circuit Theory I
41
Capacitors in Parallel
Problem # 6.31
ECE 201 Circuit Theory I
42
6.3 Series and Parallel Capacitors
i  i1 + i2 + i3 + ... + iN
dv
dv
dv
dv
i  C1 + C2 + C3 + ... + C N
dt
dt
dt
dt
N
dv
dv


   CK   Ceq
dt
 k 1  dt
Ceq  C1 + C2 + C3 + .... + C N
• The equivalent capacitance of N parallelconnected capacitors is the sum of the
individual capacitance.
Ch06 Capacitors and Inductors
43
Fig 6.15
1
1
1
1
1
 +
+
+ ... +
Ceq C1 C2 C3
CN
Ch06 Capacitors and Inductors
44
Series Capacitors
v(t )  v1 (t ) + v2 (t ) + ... + vN (t )
1
Ceq
1
1
1
1 t
id  ( C1 + C2 + C3 + ... + CN )id
t
q(t ) q(t ) q(t )
q(t )

+
++
Ceq
C1
C2
CN
• The equivalent capacitance of seriesconnected capacitors is the reciprocal of the
sum of the reciprocals of the individual
capacitances.
C1C2
1
1 1
Ceq 
 +
C1 + C2
Ceq C1 C2
Ch06 Capacitors and Inductors
45
Table 6.1
Ch06 Capacitors and Inductors
46
Y
Star
 transformation
delta transformation
Delta -> Star
Star -> Delta
Rb Rc
R1 
( Ra + Rb + Rc )
Ra 
R1 R2 + R2 R3 + R3 R1
R1
Rc Ra
R2 
( Ra + Rb + Rc )
Rb 
R1 R2 + R2 R3 + R3 R1
R2
Ra Rb
R3 
( Ra + Rb + Rc )
Rc 
R1 R2 + R2 R3 + R3 R1
R3
Kirchhoff’s Laws
• Kirchhoff’s Current Law (KCL)
– sum of all currents entering a node is zero
– sum of currents entering node is equal to sum of
currents leaving node
• Kirchhoff’s Voltage Law (KVL)
– sum of voltages around any loop in a circuit is zero
Lect1
EEE 202
49
KCL (Kirchhoff’s Current Law)
i1(t)
i5(t)
i2(t)
i4(t)
i3(t)
The sum of currents entering the node is zero:
n
 i (t )  0
j 1
j
Analogy: mass flow at pipe junction
Lect1
EEE 202
50
Open Circuit
• What if R =  ?
i(t)=0
+
The Rest
of the
Circuit
v(t)
–
i(t)=0
• i(t) = v(t)/R = 0
Lect1
EEE 202
51
Short Circuit
• What if R = 0 ?
i(t)
The Rest
of the
Circuit
+
v(t)=0
–
• v(t) = R i(t) = 0
Lect1
EEE 202
52
Resistors
• A resistor is a circuit element that dissipates
electrical energy (usually as heat)
• Real-world devices that are modeled by
resistors: incandescent light bulbs, heating
elements (stoves, heaters, etc.), long wires
• Resistance is measured in Ohms (Ω)
Lect1
EEE 202
53
Overview of Circuit Theory
• Basic quantities are voltage, current, and
power.
• The sign convention is important in
computing power supplied by or absorbed by
a circuit element.
• Circuit elements can be active or passive;
active elements are sources.
KCL and KVL
• Kirchhoff’s Current Law (KCL) and Kirchhoff’s
Voltage Law (KVL) are the fundamental laws of
circuit analysis.
• KCL is the basis of nodal analysis – in which
the unknowns are the voltages at each of the
nodes of the circuit.
• KVL is the basis of mesh analysis – in which
the unknowns are the currents flowing in each
of the meshes of the circuit.
KCL and KVL
• KCL
– The sum of all currents
entering a node is
zero, or
– The sum of currents
entering node is equal
to sum of currents
leaving node.
i1(t)
i5(t)
i2(t)
i4(t)
i3(t)
n
 i (t )  0
j 1
j
KCL and KVL
• KVL
– The sum of voltages
around any loop in a
circuit is zero.
-
+
v2(t)
n
v
j 1
-
+
v1(t)
+
v3(t)
-
j
(t )  0
KCL and KVL
• In KVL:
– A voltage encountered + to - is positive.
– A voltage encountered - to + is negative.
• Arrows are sometimes used to represent
voltage differences; they point from low to
high voltage.
+
v(t)
-
≡
v(t)