Circuit Analysis Techniques

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LECTURE 2
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Circuit Analysis Techniques
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Circuit Analysis Techniques

• Voltage Division Principle
• Current Division Principle
• Nodal Analysis with KCL
• Mesh Analysis with KVL
• Superposition
• Thévenin Equivalence
• Nortons Equivalence

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Circuit Analysis using Series/Parallel Equivalents

1. Begin by locating a combination of resistances
   that are in series or parallel. Often the place
   to start is farthest from the source.
2. Redraw the circuit with the equivalent resistance
   for the combination found in step 1.
3. Repeat steps 1 and 2 until the circuit is reduced
   as far as possible. Often (but not always) we
   end up with a single source and a single
   resistance.
4. Solve for the currents and voltages in the final
   equivalent circuit.

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Working Backward
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Voltage Division
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Application of the Voltage-Division Principle
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Current Division
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Application of the Current-Division Principle
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• Voltage division and
• current division

• Voltage division

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• Current division

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Although they are veryimportant
concepts,series/parallel equivalents andthe
current/voltage divisionprinciples are not
sufficient to solve all circuits.
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Node Voltage (Nodal) Analysis
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Writing KCL Equations in Terms of the Node
Voltages for Figure 2.16
node 1
node 2
node 3
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node 1
node 2
node 3
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No. of unknown v1, v2, v3 No. of linear
equation 3 Setting up nodal equation with KCL
at Node 1, Node 2, Node 3
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No. of unknown v1, v2, v3 No. of linear
equation 3 Setting up nodal equation with KCL
at Node 1, Node 2, Node 3
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Problem with node 3, it is rather hard to set the
nodal equation at node 3, but still solvable.
Why? As there is no way to determinethe current
through the voltage source, but v3Vs
v3
Problem with node 3, it is rather hard to set the
nodal equation at node 3 but still solvable.
Same as before.
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May Not Be That Simple
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Circuits with Voltage Sources

• We obtain dependent equations if we use all of
  the nodes in a network to write KCL equations.
• Any branch with a voltage source
• define SUPERNODE, sum all current either in or
  out at the supernode with KCL
• use KVL to set up dependent equation involving
  the voltage source.

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At node 1
At the supernode
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A
B
Summing all the current out from the supernode A
Why? As the current via the 10V source is equal
to the current via R4 plus the current via R3
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B
Summing all the current into the supernode B
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-v1 -10 v2 0
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• Any branch with a voltage source
• define supernode, sum all current either in or
  out at the
• supernode with KCL
• use KVL to set up dependent equation involving
  the
• voltage source.

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Node-Voltage Analysis with a Dependent Source

• First, we write KCL equations at each node,
  including the current of the controlled source
  just as if it were an ordinary current source.
• Next, we find an expression for the controlling
  variable ix in terms of the node voltages.

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Substitution yields
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Node-Voltage Analysis
1. Select a reference node and assign variables
for the unknown node voltages. If the reference
node is chosen at one end of an independent
voltage source, one node voltage is known at the
start, and fewer need to be computed.
2. Write network equations. First, use KCL to
write current equations for nodes and supernodes.
Write as many current equations as you can
without using all of the nodes. Then if you do
not have enough equations because of voltage
sources connected between nodes, use KVL to write
additional equations.
3. If the circuit contains dependent sources,
find expressions for the controlling variables in
terms of the node voltages. Substitute into the
network equations, and obtain equations having
only the node voltages as unknowns.
4. Put the equations into standard form and solve
for the node voltages. 5. Use the values found
for the node voltages to calculate any other
currents or voltages of interest.
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Step 1.Reference node
Step 1. v1
Step 2.
Step 1 v2
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Step 1
Step 1
v2
v1
Step 2
v3
node 1
supernode
Step 1
ref
supernode
Step 3
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Mesh Current Analysis
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Definition of a loop
Definition of a mesh
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Choosing the Mesh Currents
When several mesh currents flow through one
element, we consider the current in that element
to be the algebraic sum of the mesh currents.
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Writing Equations to Solve for Mesh Currents
If a network contains only resistors and
independent voltage sources, we can write the
required equations by following each current
around its mesh and applying KVL.
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For mesh 1, we have
For mesh 2, we obtain
For mesh 3, we have
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Mesh Currents in Circuits Containing Current
Sources
A common mistake is to assume the voltages
across current sources are zero. Therefore, loop
equation cannot be set up at mesh one due to the
voltage across the current source is unknown
Anyway, the problem is still solvable.
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As the current source common to two mesh,
combine meshes 1 and 2 into a supermesh. In other
words, we write a KVL equation around the
periphery of meshes 1 and 2 combined.
It is the supermesh.
Mesh 3
Three linear equations and three unknown
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Creating a supermesh from meshes 1 and 3
-7 1 ( i1 - i2 ) 3 ( i3 - i2 ) 1 i3
0 1
Around mesh 2 1 ( i2 - i1 ) 2 i2 3
( i2 - i3 ) 0 2
Finally, we relate the currents in meshes 1 and
3 i1 - i3 7 3
Rearranging, i1 - 4 i2 4 i3
7 1 -i1 6 i2 - 3 i3 0 2 i1
- i3 7 3
Solving, i1 9 A, i2 2.5 A, and i3 2
A.
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supermesh of mesh1 and mesh2
branch current
current source
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Three equations and three unknown.
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Mesh-Current Analysis
1. If necessary, redraw the network without
crossing conductors or elements. Then define the
mesh currents flowing around each of the open
areas defined by the network. For consistency, we
usually select a clockwise direction for each of
the mesh currents, but this is not a requirement.
2. Write network equations, stopping after the
number of equations is equal to the number of
mesh currents. First, use KVL to write voltage
equations for meshes that do not contain current
sources. Next, if any current sources are
present, write expressions for their currents in
terms of the mesh currents. Finally, if a current
source is common to two meshes, write a KVL
equation for the supermesh.
3. If the circuit contains dependent sources,
find expressions for the controlling variables in
terms of the mesh currents. Substitute into the
network equations, and obtain equations having
only the mesh currents as unknowns.
4. Put the equations into standard form. Solve
for the mesh currents by use of determinants or
other means. 5. Use the values found for the mesh
currents to calculate any other currents or
voltages of interest.
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Superposition

• Superposition Theorem the response of a circuit
  to more than one source can be determined by
  analyzing the circuits response to each source
  (alone) and then combining the results

Insert Figure 7.2
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Superposition
Insert Figure 7.3
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Superposition

• Analyze Separately, then Combine Results

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Use superposition to find the current ix.
Current source is zero open circuit as I 0
and solve iXv Voltage source is zero short
circuit as V 0 and solve iXv
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Use superposition to find the current ix.
The controlled voltage source is included in all
cases as it is controlled by the current ix.
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Voltage and Current Sources
Insert Figure 7.7
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Voltage and Current Sources
Insert Figure 7.8
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Voltage and Current Sources
Insert Figure 7.9
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Source Transformation
Under what condition, the voltage and current of
the load is the same whenoperating at the two
practical sources?For voltage source
For current source
,
We have,
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Voltage and Current Sources

• Equivalent Voltage and Current Sources for
  every voltage source, there exists an equivalent
  current source, and vice versa

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Thevenins Theorem

• Thevenins Theorem any resistive circuit or
  network, no matter how complex, can be
  represented as a voltage source in series with a
  source resistance

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Thevenins Theorem

• Thevenin Voltage (VTH) the voltage present at
  the output terminals of the circuit when the load
  is removed

Insert Figure 7.18
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Thevenins Theorem

• Thevenin Resistance (RTH) the resistance
  measured across the output terminals with the
  load removed

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Thévenin Equivalent Circuits
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Thévenin Equivalent Circuits
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Thévenin Equivalent Circuits
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Finding the Thévenin Resistance Directly
When zeroing a voltage source, it becomes a short
circuit. When zeroing a current source, it
becomes an open circuit. We can find the
Thévenin resistance by zeroing the sources in the
original network and then computing the
resistance between the terminals.
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Computation of Thévenin resistance
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Equivalence of open-circuit and Thévenin voltage
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A circuit and its Thévenin equivalent
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Superposition
As the voltage source does not contribute any
output voltage, Only the current source has the
effect.
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Determine the Thévenin and Norton Equivalents of
Network A in (a).
Source transformation
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Find the Thévenin equivalent of the circuit shown
in (a).
v
As i -1, therefore, the controlled voltage
source is -1.5V.Use nodal analysis at node v,

Thus, Rth v/I 0.6/1 0.6 ohms
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Applications of Thevenins Theorem

• Load Voltage Ranges Thevenins theorem is most
  commonly used to predict the change in load
  voltage that will result from a change in load
  resistance

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Applications of Thevenins Theorem

• Maximum Power Transfer
• Maximum power transfer from a circuit to a
  variable load occurs when the load resistance
  equals the source resistance
• For a series-parallel circuit, maximum power
  occurs when RL RTH

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Applications of Thevenins Theorem

• Multiload Circuits

Insert Figure 7.30
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Nortons Theorem

• Nortons Theorem any resistive circuit or
  network, no matter how complex, can be
  represented as a current source in parallel with
  a source resistance

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Nortons Theorem

• Norton Current (IN) the current through the
  shorted load terminals

Insert Figure 7.35
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Computation of Norton current
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Nortons Theorem

• Norton Resistance (RN) the resistance measured
  across the open load terminals (measured and
  calculated exactly like RTH)

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Nortons Theorem

• Norton-to-Thevenin and Thevenin-to-Norton
  Conversions

Insert Figure 7.39
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Step-by-step Thévenin/Norton-Equivalent-Circuit
Analysis
1. Perform two of these a. Determine the
open-circuit voltage Vt voc. b. Determine
the short-circuit current In isc. c. Zero
the sources and find the Thévenin resistance Rt
looking back into the terminals.
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2. Use the equation Vt Rt In to compute the
remaining value. 3. The Thévenin equivalent
consists of a voltage source Vt in series with Rt
. 4. The Norton equivalent consists of a current
source In in parallel with Rt .
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Maximum Power Transfer
The load resistance that absorbs the maximum
power from a two-terminal circuit is equal to the
Thévenin resistance.
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Graphical representation of maximum power transfer
Power transfer between source and load
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