EE212 Passive AC Circuits
Lecture Notes 2a
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Circuit Analysis
When a circuit has more than one element, a circuit analysis is
required to determine circuit parameters (v, i, power, etc.) in different
parts of the circuit.
Circuit Theories:
•Ohm’s Law
•Superposition Theorem
•Kirchhoff’s Voltage and Current
Laws
•Mesh/Nodal Analysis
•Source Transformation
•Thevenin/Norton Theorem
•Wye/Delta Transformation
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Mesh (Loop) Analysis
• Kirchhoff’s voltage law applies to a closed
path in an electric circuit. The close path is
referred to as a loop.
• A mesh is a simple loop. That is, there are no
other loops inside it.
• Mesh analysis applies to planar circuits. That
is, a circuit that can be drawn on a plane with
no crossed wires.
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Mesh (Loop) Analysis (cont)
~
I1
I2
I3
Steps:
-assume mesh current I1 for Mesh 1, I2 for Mesh 2, etc.
-apply KVL for each loop
-obtain ‘n’ equations for ‘n’ meshes
-i.e., the mesh currents are the unknown variables
-solve equations to determine mesh currents
-usually done using matrices
-obtain currents through each circuit element of interest
-apply Ohm’s Law to calculate voltages of interest
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Mesh (Loop) Analysis (continued)
•
•
•
•
•
Single voltage source
Multiple voltage sources
Voltage and current sources
Supermesh
Dependent sources
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Mesh (Loop) Analysis (continued)
• Dependent sources:
– Current controlled voltage source (CCVS)
– Voltage controlled voltage source (VCVS)
– Current controlled current source (CCCS)
– Voltage controlled voltage source (VCCS)
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Example 1: Find the power dissipated in the 50  resistor
using Mesh analysis.
j5 
50 
-j100 
-j5 
+
50/300 V
~
-
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+
-j50 
j50 
~
50/300 V
-
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Mesh Analysis Using Matrix Method
[V]=[Z][I]
For a circuit with ‘n’ loops, the matrix equation is:
V1
V2
..
..
Vn
=
Z11 -Z12 .. .. -Z1n
-Z21 Z22 .. .. -Z2n
… … .. … ..
... .. .. .. ..
-Zn1 -Zn2 .. .. Znn
I1
I2
..
..
In
[ V ] is voltage vector
Vi is the source voltage in Mesh i
Sign convention: +ve voltage if going –ve to +ve (since voltage vector
has been moved to the left hand side of the equation)
[ Z ] is impedance matrix (square matrix)
Diagonal element Zii = sum of all impedances in Mesh i
Zik = impedance between Mesh i and Mesh k
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[ I ] is the unknown current vector
Ii = current in Mesh i
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Example 1: Mesh (Loop) Analysis (cont)
j5 
+
50/300 V
~
-
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I1
-j50 
50 
-j100 
-j5 
+
I2
j50 
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I3
~
50/300 V
-
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Nodal Analysis
• A node is a point in an electric circuit where 2
or more components are connected. (Strictly
speaking, it is the whole conductive surface
connecting those components.)
• Nodal analysis applies to planar and nonplanar circuits.
• Nodal analysis is used to solve for node
voltages.
• Sign convention: current leaving node is +ve
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Nodal Analysis (continued)
V1
V2
~
~
ref. node
Steps:
- identify all the nodes
- select a reference node
(usually the node with the most branches connected to it or the ground
node of a power source)
- assume voltage Vi (w.r.t . reference node) for Node i
- assume current direction in each branch
- apply KCL at each node
- obtain ‘n-1’ equations for ‘n’ nodes (since one node is the reference
node)
-solve the equations to determine node voltages
- apply Ohm’s Law to calculate the currents
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Nodal Analysis (continued)
•
•
•
•
•
Single current source
Multiple current sources
Voltage and current sources
Supernode
Dependent sources
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Example 1 (revisited): Find the power dissipated in the 50 
resistor using Nodal analysis.
j5 
50 
-j100 
-j5 
+
50/300 V
~
~
-
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+
-j50 
j50 
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50/300 V
-
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Solving by Nodal Analysis:
V3
j5 
V1
50 
-j100 
V2
-j5 
V4
+
50/300 V
+
~
~
-
-j50 
j50 
50/300 V
-
ref. node
KCL at Node 1: get Equation 1
KCL at Node 2: get Equation 2
Why no node equations at Nodes 3 and 4?
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Solve Equations 1 and 2 toEEobtain
V1 and V2
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Nodal Analysis Using Matrix Method
[I]=[Y][V]
For a circuit with ‘n+1’ nodes, the matrix equation is:
I1
I2
..
..
In
=
Y11
-Y21
…
...
-Yn1
-Y12 .. ..
Y22 .. ..
… .. …
.. .. ..
-Yn2 .. ..
-Y1n
-Y2n
..
..
Ynn
V1
V2
..
..
Vn
[ I ] is the known current vector
Ii is the source current in Node i
Sign convention: current entering node, +ve (since current vector has been
moved to the left hand side of the equation)
[ Y ] is admittance matrix (square matrix)
Diagonal element Yii = sum of all admittances connected to Node i
Yik = admittance between Node i and Node k
[ V ] is the unknown voltage vector
Vi = voltage at Node i w.r.t. the reference
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Source Transformation
Voltage Source to Current Source
Z
~
A
V
B
Current Source to Voltage Source
A

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I
Z
B
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Example 1 (revisited): Nodal Analysis and
source transformation
V1
V2


1. Current source equivalent for V1
2. Current source equivalent for V2
3. Admittances for all other components
All admittance values in siemens
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Example 1 (revisited): Nodal Analysis and
source transformation
V1
=
V2
V1=
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= 55.56/300
Volts
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Similarly,
V2 = 55.56/300
Volts
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Thevenin’s Theorem
Any linear two terminal network with sources can be replaced by an
equivalent voltage source in series with an equivalent impedance.
Zth
A
Linear
Circuit
~
B
Eth
A
B
Thevenin Voltage, Eth
voltage measured at the terminals A & B with nothing connected to
the external circuit
Thevenin Impedance, Zth
impedance at the terminals A & B with all the sources reduced to
zero
i.e. voltage sources short circuited (0 volts)
current sources open circuited (0 amperes)
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Example: Use a Thevenin equivalent circuit at bus “A-B”
to calculate the short circuit current at A-B.
j5 
-j2 
A
5
+
3
5
110/300 V ~
-
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B
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Maximum Power Transfer Theorem
Maximum power is transferred to a load
when the load impedance is equal to the
conjugate of the Thevenin Impedance.
i.e. ZLoad = Zth*
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