Electric Circuits (Fall 2015)
Pingqiang Zhou
Discussion 2
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Review
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Node, Branch and Loop
KCL and KVL
Series Resistors and Voltage Division
Parallel Resistors and Current Division
Wye-Delta Transformations
With Ohm’s and Kirchhoff’s law established, they may now be applied
to circuit analysis.
Two techniques:
 Nodal analysis, which is based on Kichhoff current law (KCL)
 Mesh analysis, which is based on Kichhoff voltage law (KVL)
The analysis will result in a set of simultaneous equations which may
be solved by Cramer’s rule or computationally (using MATLAB for
example)
Computational circuit analysis using PSpice will also be introduced
here.
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Terminology: Nodes, Branches and Loops
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Node: A point where two or more circuit elements are
connected
Branch: A path that connects two nodes
Loop: Any closed path in a circuit.
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Network Topology
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A loop is independent if it contains at least one branch not
shared by any other independent loops.
Two or more elements are in series if they share a single
node and thus carry the same current
Two or more elements are in parallel if they are connected
to the same two nodes and thus have the same voltage.
Tree 2
How can we find a set of
independent loop?
Tree 1
Is this loop independent?
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Kirchhoff’s Laws
•
Ohm’s law is not sufficient for circuit
analysis
 Kirchhoff’s laws complete it.
• Kirchhoff’s Current Law (KCL):
Gustav Robert Kirchhoff
1824-1887
▪ The algebraic sum of all the currents entering any node in a circuit
equals zero.
▪ Based on conservation of charge.
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Generalized KCL Examples
50 mA
5mA
i
2mA
i
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Notation: Node and Branch Voltages
• Use one node as the reference (the “common” or
“ground” node) – label it with a symbol
• The voltage drop from node x to the reference node is
called the node voltage vx.
• The voltage across a circuit element is defined as the
difference between the node voltages at its terminals.
Example:
– v1 +
a R1 b
+
va +_ vs
_
c
+
R2 vb
_
 REFERENCE NODE
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Kirchhoff’s Voltage Law (KVL)
•The
algebraic sum of all the voltages around any loop in
a circuit equals zero.
▪ Based on conservation of energy
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Electric Circuits (Fall 2015)
Pingqiang Zhou
KVL Example
Three closed paths:
v2
1
+
va

b
+
vb
-
v3
+
a +
2
c
+
vc

3
Path 1:
Path 2:
Is the equation of path 3 necessary?
Path 3:
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Electric Circuits (Fall 2015)
Pingqiang Zhou
When can the Voltage Divider Formula be Used?
I
I
R1
R1
VSS
+
R2
+
V
– 2
VSS +
R2
R3
R3
R4
R4
+
–V2
R5
Why? What is V2?
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Measuring Voltage (Voltmeter)
To measure the voltage drop across an element in a
real circuit, insert a voltmeter (digital multimeter in
voltage mode) in parallel with the element.
Voltmeters are characterized by their “voltmeter input
resistance” (Rin). Ideally, this should be very high
(typical value 10 MW)
Ideal
Voltmeter
Rin
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Effect of Voltmeter
undisturbed circuit
circuit with voltmeter inserted
R1
VSS
+
_
R1
+
R2
–
V2
 R2 
V2  VSS 

R1  R2 
VSS
+
_
+
R2
Rin
 R2 || Rin 

V2  VSS 

R2 || Rin  R1 
Example: VSS  10 V, R2  100K, R1  900K  V2  1V
Rin  10M , V2  ?
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–
V2′
Electric Circuits (Fall 2015)
Pingqiang Zhou
Measuring Current (Ammeter)
To measure the current flowing through an element in a
real circuit, insert an ammeter (digital multimeter in
current mode) in series with the element.
Ammeters are characterized by their “ammeter input
resistance” (Rin). Ideally, this should be very low
(typical value 1W).
Ideal
Ammeter
Rin
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Effect of Ammeter
Measurement error due to non-zero input resistance:
undisturbed circuit
I
circuit with ammeter inserted
Imeas
ammeter
R1
R1
V1 +
_
V1 +
_
R2
V1
I
R1  R2
Rin
R2
V1
Imeas 
R1  R2  Rin
Example: V1 = 1 V, R1= R2 = 500 W, Rin = 1W
1V
I
 1mA, I meas  ?
500W  500W
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Wye-Delta Transformations
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There are cases where resistors are neither parallel nor
series.
Consider the bridge circuit shown here. This circuit can
be simplified to a three-terminal equivalent
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Delta to Wye
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The conversion formula for a delta to wye transformation
are:
Rb Rc
R1 
Ra  Rb  Rc
R2 
Rc Ra
Ra  Rb  Rc
R3 
Ra Rb
Ra  Rb  Rc
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Wye to Delta
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The conversion formula for a wye to delta transformation
are:
R1 R2  R2 R3  R3 R1
Ra 
R1
R1 R2  R2 R3  R3 R1
Rb 
R2
Rc 
R1 R2  R2 R3  R3 R1
R3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Example
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Find the equivalent resistance
Wye -> Delta
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Summary
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KCL and KVL
N
i
n 1
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n
0
M
v
m 1
m
0
Series Resistors and Voltage Division
Parallel Resistors and Current Division
Wye-Delta Transformations
R1 
Rb Rc
Ra  Rb  Rc
Ra 
R1 R2  R2 R3  R3 R1
R1
R2 
Rc Ra
Ra  Rb  Rc
Rb 
R1 R2  R2 R3  R3 R1
R2
R3 
Ra Rb
Ra  Rb  Rc
Rc 
R1 R2  R2 R3  R3 R1
R3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Nodal Analysis
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Instead of focusing on the voltages of the circuit elements, if
one looks at the voltages at the nodes of the circuit, the number
of simultaneous equations to solve for can be reduced.
Given a circuit with n nodes, without voltage sources, the nodal
analysis is accomplished via three steps:
1. Select a node as the reference (i.e., ground) node. Define the
node voltages (except reference node and the ones set by the
voltage sources). Voltages are relative to the reference node.
2. Apply KCL at nodes with unknown voltage, expressing
current in terms of the node voltages (using the I-V relationships
of branch elements).
Special cases: floating voltage sources.
3. Solve the resulting simultaneous equations to obtain the
unknown node voltages.
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Applying Nodal Analysis
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Let’s apply nodal analysis to this circuit to see
how it works.
This circuit has a node that is designed as
ground. We will use that as the reference node
(node 0)
The remaining two nodes are designed 1 and
2 and assigned voltages v1 and v2.
Now apply KCL to each node:
At node 1
I1  I 2  i1  i2
At node 2
I 2  i2  i3
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Apply Nodal Analysis II
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We can now use Ohm’s law to express the unknown currents i1, i2,
and i3 in terms of node voltages.
In doing so, keep in mind that current flows from high potential to low.
From this we get:
v1  0
or i1  G1v1
R1
v v
i2  1 2 or i2  G2 v1  v2 
R2
v 0
i3  2
or i3  G3v2
R3
i1 
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v1 v1  v2

R1
R2
v v
v
I2  1 2  2
R2
R3
I1  I 2 
Substituting
back into the
node
equations
or
I1  I 2  G1v1  G2 v1  v2 
I 2  G2 v1  v2   G3v2
The last step is to solve the system of equations
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Nodal Analysis: Exercise #1
Find v1 and v2 using nodal analysis.
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Nodal Analysis: Exercise #2
R1
Va
R3
V
1
R2
I1
R4
R5
V2
• How many nodes can you find?
• How many equations are needed?
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Nodal Analysis “Floating Voltage Source”
A “floating” voltage source is one for which neither side is
connected to the reference node, e.g. VLL in the circuit below:
Va
VLL
Vb
- +
I1
R2
R4
I2
Problem: We cannot write KCL at nodes a or b because there is no
way to express the current through the voltage source in terms of VaVb.
Solution: Define a “supernode” – that chunk of the circuit containing
nodes a and b. Express KCL for this supernode. Incorporate voltage
source constraint into KCL equation.
Electric Circuits (Fall 2015)
Pingqiang Zhou
Nodal Analysis: Exercise #3
• Find the node voltages for the circuit
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Mesh Analysis
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Another general procedure for analyzing circuits is to use the
mesh currents as the circuit variables.
Recall:
 A loop is a closed path with no node passed more than once
 A mesh is a loop that does not contain any other loop within it
• Mesh analysis uses KVL to find unknown currents
• Mesh analysis is limited in one aspect: It can only apply to
circuits that can be rendered planar.
▪ A planar circuit can be drawn such that there are no crossing
branches.
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Planar vs Nonplanar
The figure on the left is a nonplanar
circuit: The branch with the 13Ω
resistor prevents the circuit from being
drawn without crossing branches
The figure on the right is a planar
circuit: It can be redrawn to avoid
crossing branches
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Planar vs Nonplanar
Which one is nonplanar? For the planar circuit,
redraw the circuits with no crossing
branches.
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Mesh Analysis Steps
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Mesh analysis follows these steps:
1. Assign mesh currents i1,i2,…in to the n meshes
2. Apply KVL to each of the n mesh currents.
3. Solve the resulting n simultaneous equations to get the
mesh currents
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Mesh Analysis Example
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The above circuit has two paths that are meshes (abefa and bcdeb)
 The outer loop (abcdefa) is a loop, but not a mesh
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First, mesh currents i1 and i2 are assigned to the two meshes.
Applying KVL to the meshes:
 V1  R1i1  R3 i1  i2   0
R2i2  V2  R3 i2  i1   0


R1  R3 i1  R3i2  V1
 R3i1  R2  R3 i2  V2
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Mesh Analysis: Exercise #1
Apply mesh analysis to find i.
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Mesh Analysis with Current Sources
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The presence of a current source makes the mesh
analysis simpler in that it reduces the number of equations.
If the current source is located on only one mesh, the
current for that mesh is defined by the source.
For example:
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Here, the current i2 is equal to -5A
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Supermesh
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In this example, a 6A current source is shared between
mesh 1 and 2.
A supermesh is required because mesh analysis uses KVL, but
the voltage across a current source cannot be known in
advance.
 The supermesh is formed by merging the two meshes.
 The current source and the 2Ω resistor in series with it are removed.
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Supermesh Example
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Apply KVL to the supermesh
 20  6i1  10i2  4i2  0 or 6i1  14i2  20
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We next apply KCL to the node in the branch where the two meshes
intersect.
i2  i1  6
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Note that the supermesh required using both KVL and KCL
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Mesh Analysis: Exercise #2
Apply mesh analysis to find I1, I2 and I3.
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Nodal or Mesh?
Find the maximum power of Ro
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Nodal Analysis by Inspection
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A faster way to construct a matrix for solving a circuit by
nodal analysis
 It requires that all current sources within the circuit be
independent
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The equations for the example
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Electric Circuits (Fall 2015)
Pingqiang Zhou
Exercise
•
By inspection, obtain the node-voltage equations for the
circuit shown below
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Electric Circuits (Fall 2015)
Q&A
Pingqiang Zhou