CHAPTER 2: SINUSOIDAL STEADYSTATE ANALYSIS

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CHAPTER 2 SINUSOIDAL STEADY-STATE ANALYSIS

• Nodal Analysis and Mesh Analysis
• Source Transformation Superposition Principles
• Thevenins and Norton Theorem

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NODAL ANALYSIS

• Based on KCLAssigns unknown voltages to all its
  essential node

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Step 1 Mark Essential Node
Remember between 2 different nodes, there
should be at least 1 element
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Step 2 Reference Node
Select a reference node. Mark the reference node
with the earth sign - or downward arrow ?.A
reference node is the node from where all the
other node voltages join.
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Step 3 Assign Unknown Node Voltages
V1
V2
V3
Assign node voltages at the marked essential
nodes.
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Step 4 Decide on Number of Equations Required
V1
V2
V3
There are 3 unknowns V1, V2, V3
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Step 5 Perform KCL at the Selected Nodes
KCL is performed with the current going out of
the node as positive (i.e. currents going out are
added, going in are subtracted)
(1)
(2)
and so on until all the simultaneous
equations are performed for all unknowns
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Step 6 Solve the equations
(1)
(2)
V1 ?
V3 ?
V2 ?
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Nodal Analysis
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Nodal Analysis
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Nodal Analysis
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Example 1
Find ix in the circuit below using nodal analysis.
Reference Alexander, Sadiku Chapter 10 - page
414
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Example 2
Compute V1 and V2 in the circuit below using
nodal analysis
Reference Alexander, Sadiku Chapter 10 - page
416
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Exercise 1
Using nodal analysis, find v1 and v2 in the
circuit below
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MESH ANALYSIS

• Based on KVL
• Assigns unknown currents to all the meshes in the
  circuit
• When solving a circuit, we will have to employ
  either Nodal or Mesh Anaysis do not choose both!

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MESH ANALYSIS
STEP 1 Label all meshes in the circuit with a
clockwise mesh current. STEP 2 Decide on how many
equations are required.   STEP 3 Apply
Kirchhoff's Voltage Law for each mesh in the
circuit. Voltage drops added and the voltage rise
subtracted.  STEP 4 Solve the equations with
Cramer's Rule.
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Example 3
Determine I0 for the circuit below using mesh
analysis
Reference Alexander, Sadiku Chapter 10 - page
417
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Example 4
Find V0 for the circuit below.
Reference Alexander, Sadiku Chapter 10 - page
419
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Exercise 2
Find I0 for the circuit below using mesh
analysis.
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Circuit with Dependent Sources
If a dependent source is present in the circuit,
we need to come up with a constraint equation
imposed by the presence of the dependent source.
The constraint equation is an equation describing
the dependent term (of the dependent source) in
terms of node voltages or values.
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Circuit with Dependent Sources
In this case, the dependent source is the 8io
voltage source. The dependent term is io. From
Ohm's Law, we obtain the constraint equation
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Nodal or Mesh Analysis??

• Go for the analysis that will result in lesser
  number of simultaneous equations.
• Compare the number of node-voltage equations to
  the number of mesh-current equations required.
• The one that is less represents the analysis that
  would be the better choice.

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Superposition Theorem

• The superposition principle states that the
  voltage across (or current through) an element in
  a linear circuits is the algebraic sum of the
  voltage across (or current through) that element
  due to each independent source acting alone.

Current Source ? open circuit(0 A) Voltage Source
? short circuit (0 V)
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Superposition Theorem

• Step to apply
• Turn off all independent sources except one
  source. Find the output (voltage or current) due
  to that active source.
• Repeat step 1 for each other independent sources.
• Find the total contribution by adding
  algebraically all the contribution due to the
  independent source.

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Example 5
Use the superposition theorem to find I0 in the
circuit
Reference Alexander, Sadiku Chapter 10 - page
421
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Exercise 3
Calculate V0 for the circuit below by using
superposition theorem
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Source Transformation

• A source transformation is the process of
  replacing a voltage source vs in series with a
  impedance Z by a current source is in parallel
  with a impedance Z , or vice versa.

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Source Transformation
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Example 6
Calculate Vx in the circuit below using source
transformation method
Reference Alexander, Sadiku Chapter 10 - page
425
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Exercise 4
Find I0 in the circuit below using concept of
source transformation.
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Thevenins Theorem

• A linear two terminal circuit can be replaced by
  an equivalent circuit consisting of a voltage VTh
  in series with an impedance ZTh , where VTh is
  the open circuit voltage at the terminals and ZTh
  is the input or equivalent impedance at the
  terminals when the independent source are turned
  off.

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Thevenins Theorem
Original circuit
Z
Thevenin Equivalent circuit
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Thevenins Theorem
Finding VTh and ZTh.
Z
Z
Z
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Thevenins Theorem
Finding ZTh when circuit has dependent sources
Z
Z
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Example 7
Obtain the Thevenin equivalent at terminal a-b
of the circuit in figure below
Reference Alexander, Sadiku Chapter 10 - page
426
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Example 8
Find Thevenin equivalent of the circuit in figure
below as seen on terminals a-b
Reference Alexander, Sadiku Chapter 10 - page
428
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Exercise 5
Find the Thevenin equivalent at
terminals a-b of the circuit below
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Norton Theorem

• A linear two-terminal circuit can be replaced by
  an equivalent circuit consisting of a current
  source IN in parallel with an impendence ZN,
  where IN is the short circuit current through the
  terminals and ZN is the input or equivalent
  impedance at the terminals when the independent
  source are turned off.

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Norton Theorem
Z
(a) Original circuit (b) Norton equivalent
circuit.
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Norton Theorem
Finding Norton current IN.
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Example 9
Obtain current I0 in figure below using Nortons
theorem.
Reference Alexander, Sadiku Chapter 10 - page
429
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Exercise 6
Determine the Norton equivalent of the circuit in
figure below as seen from terminals a-b. Use the
equivalent to find I0.