Sinusoidal Steady State Analysis
(AC Analysis)
Part I
Dr. Mohamed Refky Amin
Electronics and Electrical Communications Engineering Department (EECE)
Cairo University
http://scholar.cu.edu.eg/refky/
OUTLINE
• Previously on ELCN102
• Solution of AC Circuits
Simplification Method
Loop Analysis Method
Node Analysis Method
Superposition Method
Dr. Mohamed Refky 2
Previously on ELCN102
Dr. Mohamed Refky
Phasor Relationships for Circuit Elements
The impedance 𝑍 of a circuit is the ratio of the phasor voltage 𝑉to the phasor current 𝐼, measured in Ω.
Resistor Inductor Capacitor
𝑣𝑅 𝑡 = 𝑅𝑖𝑅 𝑡 𝑣𝐿 𝑡 = 𝐿𝑑𝑖𝐿 𝑡
𝑑𝑡𝑖𝐶 𝑡 = 𝐶
𝑑𝑣𝐶 𝑡
𝑑𝑡
𝑉𝑅 = 𝑅 × 𝐼𝑅𝑉𝐿 = 𝜔𝐿𝐼𝐿∠90
𝑜
= 𝑗𝜔𝐿 × 𝐼𝐿
𝐼𝐶 = 𝜔𝐶𝑉𝐶∠90𝑜
= 𝑗𝜔𝐶 × 𝑉𝐶
𝑍𝑅 = 𝑅 𝑍𝐿 = 𝑗𝜔L 𝑍𝐶 =1
𝑗𝜔𝐶= −
𝑗
𝜔𝐶
Previously on ELCN102
Dr. Mohamed Refky
Phasor Relationships for Circuit Elements
The admittance 𝑌 of a circuit is the ratio of the phasor current 𝐼to the phasor voltage 𝑉, measured in Ω−1.
Resistor Inductor Capacitor
𝑣𝑅 𝑡 = 𝑅𝑖𝑅 𝑡 𝑣𝐿 𝑡 = 𝐿𝑑𝑖𝐿 𝑡
𝑑𝑡𝑖𝐶 𝑡 = 𝐶
𝑑𝑣𝐶 𝑡
𝑑𝑡
𝑉𝑅 = 𝑅 × 𝐼𝑅𝑉𝐿 = 𝜔𝐿𝐼𝐿∠90
𝑜
= 𝑗𝜔𝐿 × 𝐼𝐿
𝐼𝐶 = 𝜔𝐶𝑉𝐶∠90𝑜
= 𝑗𝜔𝐶 × 𝑉𝐶
𝑌𝑅 =1
𝑅𝑌𝐿 =
1
𝑗𝜔L= −
𝑗
𝜔L𝑌𝐶 = 𝑗𝜔𝐶
Previously on ELCN102
Dr. Mohamed Refky
Impedance and Admittance
The impedance 𝑍 of a circuit is the ratio of the phasor voltage 𝑉to the phasor current 𝐼, measured in Ω.
𝑍 = 𝑅 + 𝑗𝑋
𝑅 is the resistance & 𝑋 is the reactance
𝑍 is inductive if 𝑋 is +𝑣𝑒.
𝑍 is capacitive if 𝑋 is −𝑣𝑒.
𝑍, 𝑅, and 𝑋 are in units of Ω
Impedance
𝑍𝐿 = 𝑗𝜔L
𝑍𝐶 =1
𝑗𝜔𝐶= −
𝑗
𝜔𝐶
Previously on ELCN102
Dr. Mohamed Refky
Impedance and Admittance
The admittance 𝑌 of a circuit is the ratio of the phasor current 𝐼 to
the phasor voltage 𝑉, measured in Ω−1.
𝑌 = 𝐺 + 𝑗𝐵
𝐺 is the conductance & 𝐵 is the susceptance.
𝑌 is inductive if 𝐵 is −𝑣𝑒.
𝑌 is capacitive if 𝐵 is +𝑣𝑒.
𝑌, 𝐺, and 𝐵 are in units of Ω−1
Admittance
𝑌𝐿 =1
𝑗𝜔L= −
𝑗
𝜔L
𝑌𝐶 = 𝑗𝜔𝐶
Previously on ELCN102
Dr. Mohamed Refky
Impedance Combination
𝑍𝑒𝑞 = 𝑍1 + 𝑍2 +⋯+ 𝑍𝑁
Series Combination
Previously on ELCN102
Dr. Mohamed Refky
Impedance Combination
1
𝑍𝑒𝑞=
1
𝑍1+
1
𝑍2+⋯+
1
𝑍𝑁
Parallel Combination
Previously on ELCN102
Dr. Mohamed Refky
Admittance Combination
1
𝑌𝑒𝑞=
1
𝑌1+1
𝑌2+⋯+
1
𝑌𝑁
Series Combination
Previously on ELCN102
Dr. Mohamed Refky
Admittance Combination
𝑌𝑒𝑞 = 𝑌1 + 𝑌2 +⋯+ 𝑌𝑁
Parallel Combination
Previously on ELCN102
Dr. Mohamed Refky
Star-Delta Transformation
𝑍𝐴𝐵 = 𝑍𝐴 + 𝑍𝐵 +𝑍𝐴𝑍𝐵𝑍𝐶
𝑍𝐴𝐶 = 𝑍𝐴 + 𝑍𝐶 +𝑍𝐴𝑍𝐶𝑍𝐵
𝑍𝐵𝐶 = 𝑍𝐵 + 𝑍𝐶 +𝑍𝐵𝑍𝐶𝑍𝐴
𝑍𝐴 =𝑍𝐴𝐵𝑍𝐴𝐶
𝑍𝐴𝐶 + 𝑍𝐵𝐶 + 𝑍𝐴𝐵𝑍𝐶 =
𝑍𝐵𝐶𝑍𝐴𝐶𝑍𝐴𝐶 + 𝑍𝐵𝐶 + 𝑍𝐴𝐵
𝑍𝐵 =𝑍𝐴𝐵𝑍𝐵𝐶
𝑍𝐴𝐶 + 𝑍𝐵𝐶 + 𝑍𝐴𝐵
Solution of AC Circuits
Dr. Mohamed Refky
DefinitionA circuit is said to be solved when the voltage across and the
current in every element have been determined due to input
excitation (voltage and/or current sources).
Solution of AC Circuits
Dr. Mohamed Refky
Methods of Solution of AC CircuitsTo solve a AC circuit you can use one or more of the following
methods:
• Simplification Method
• Loop Analysis Method
• Node Analysis Method
• Superposition Method
• Thevenin equivalent circuit
• Norton equivalent circuit
Solution of AC Circuits
Dr. Mohamed Refky
Simplification Method In step by step simplification we can use:
• Source transformation
• Combination of active elements
• Combination of series and parallel elements
• Star-delta & delta-star transformation
Simplification Method
Dr. Mohamed Refky
Source Transformation“A voltage source 𝑉𝐴𝐶 with a series impedance 𝑍 can be
transformed into a current source 𝐼𝐴𝐶 = 𝑉𝐴𝐶/𝑍 and a parallel
impedance 𝑍”
“ A current source 𝐼𝐴𝐶 with a parallel impedance 𝑍 can be
transformed into a voltage source 𝑉𝐴𝐶 = 𝐼𝐴𝐶 × 𝑍 and a series
impedance 𝑍”
Simplification Method
Dr. Mohamed Refky
Example (1)Use simplification method to find 𝑉𝑥 for the circuit shown.
Simplification Method
Dr. Mohamed Refky
Example (2)Use simplification method to find 𝐼𝑥 for the circuit shown.
Loop Analysis Method
Dr. Mohamed Refky
Definition
The Loop Analysis Method (Mesh Method) uses KVL to generate
a set of simultaneous equations.
1) Convert the independent current sources into equivalent
voltage sources
2) Identify the number of independent loop (𝐿) on the circuit
3) Label a loop current on each loop.
4) Write an expression for the KVL around each loop.
5) Solve the simultaneous equations to get the loop currents.
Loop Analysis Method
Dr. Mohamed Refky
Matrix Form
𝑍11 −𝑍12 ⋯ −𝑍1𝑁−𝑍21 𝑍22 −𝑍2𝑁⋮
−𝑍𝑁1
⋮−𝑍𝑁2
⋱ ⋮⋯ 𝑍𝑁𝑁
𝐼1𝐼2⋮𝐼𝑁
=
𝑉1𝑉2⋮𝑉𝑁
𝑍𝑖𝑖 =𝑖𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒 𝑖𝑛 𝑙𝑜𝑜𝑝 𝑖
𝑍𝑖𝑗 =𝐶𝑜𝑚𝑚𝑜𝑛 𝑖𝑚𝑝𝑒𝑑𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑙𝑜𝑜𝑝𝑠 𝑖 𝑎𝑛𝑑 𝑗 = 𝑍𝑗𝑖
𝑉𝑖 =𝑣𝑜𝑙𝑡𝑎𝑔𝑒 𝑠𝑜𝑢𝑟𝑐𝑒𝑠 𝑖𝑛 𝑙𝑜𝑜𝑝 𝑖𝑉 is +ve if it supplies
current in the direction
of the loop current
Loop Analysis Method
Dr. Mohamed Refky
Example (3)Use loop analysis to find 𝐼𝑥 for the circuit shown.
Loop Analysis Method
Dr. Mohamed Refky
Example (4)Use loop analysis to find 𝐼𝑥 for the circuit shown.
Loop Analysis Method
Dr. Mohamed Refky
Example (5)Use loop analysis to find 𝑉𝑥 for the circuit shown.
Node Analysis Method
Dr. Mohamed Refky 23
Definition
The Node Analysis Method (Nodal Analysis) uses KCL to
generate a set of simultaneous equations.
1) Convert independent voltage sources into equivalent current
sources.
2) Identify the number of non simple nodes (𝑁) of the circuit.
3) Write an expression for the KCL at each 𝑁 − 1 Node
(exclude the ground node).
4) Solve the resultant simultaneous equations to get the voltages.
Node Analysis Method
Dr. Mohamed Refky
Matrix Form
𝑌11 −𝑌12 ⋯ −𝑌1𝑁−𝑌21 𝑌22 −𝑌2𝑁⋮
−𝑌𝑁1
⋮−𝑌𝑁2
⋱ ⋮⋯ 𝑌𝑁𝑁
𝑉1𝑉2⋮𝑉𝑁
=
𝐼1𝐼2⋮𝐼𝑁
𝑌𝑖𝑖 =𝑎𝑑𝑚𝑖𝑡𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑛𝑜𝑑𝑒 𝑖
𝑌𝑖𝑗 =𝑐𝑜𝑚𝑚𝑜𝑛 𝑎𝑑𝑚𝑖𝑡𝑡𝑎𝑛𝑐𝑒 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑛𝑜𝑑𝑒 𝑖 𝑎𝑛𝑑 𝑗 = 𝑌𝑗𝑖
𝐼𝑖 =𝑐𝑢𝑟𝑟𝑒𝑛𝑡 𝑠𝑜𝑢𝑟𝑐𝑒𝑠 𝑎𝑡 𝑛𝑜𝑑𝑒 𝑖 𝐼 is +ve if it supply
current into the node
Node Analysis Method
Dr. Mohamed Refky
Example (6)Use node analysis to find 𝑉1 & 𝑉2 for the circuit shown.
Superposition Theorem
Dr. Mohamed Refky
DefinitionFor a linear circuit containing multiple independent sources, the
voltage across (or current through) any of its elements is the
algebraic sum of the voltages across (or currents through) that
element due to each independent source acting alone.
10∠30𝑜V 𝐼𝑎
5∠0𝑜A 𝐼𝑏
Total 𝐼 = 𝐼𝑎 + 𝐼𝑏
Superposition Theorem
Dr. Mohamed Refky
Example (7)Use superposition theorem to find 𝐼𝑥 for the circuit shown.
Thevenin’s Theorem
Dr. Mohamed Refky
DefinitionA linear two-terminal circuit, can be replaced by an equivalent
circuit consisting of a voltage source 𝑉𝑡ℎ in series with a
impedance 𝑍𝑡ℎ.
Thevenin’s Theorem
Dr. Mohamed Refky
Solution Steps1) Identify the load impedance and introduce two nodes 𝑎 and 𝑏
2) Remove the load impedance between node 𝑎 and 𝑏
3) Calculate the open circuit voltage between nodes 𝑎 and 𝑏.This voltage is 𝑉𝑡ℎ of the Thevenin equivalent circuit.
4) Set all the independent sources to zero (voltage sources are
SC and current sources are OC) and calculate the impedance
seen between nodes 𝑎 and 𝑏. This impedance is 𝑍𝑡ℎ of the
Thevenin equivalent circuit.
Thevenin’s Theorem
Dr. Mohamed Refky
Example (8)Obtain the Thevenin equivalent at terminals 𝑎 and 𝑏 of the circuit
shown.
Norton’s Theorem
Dr. Mohamed Refky
DefinitionA linear two-terminal circuit can be replaced by equivalent circuit
consisting of a current source 𝐼𝑁 in parallel with a impedance 𝑍𝑁
Norton’s Theorem
Dr. Mohamed Refky
Solution Steps1) Identify the load impedance and introduce two nodes 𝑎 and 𝑏
2) Remove the load impedance between node 𝑎 and 𝑏 and set all
the independent sources to zero (voltage sources are SC and
current sources are OC) and calculate the impedance seen
between nodes 𝑎 and 𝑏. This resistance is 𝑍𝑁 of the Norton
equivalent circuit.
3) Replace the load impedance with a short circuit and calculate
the short circuit current between nodes 𝑎 and 𝑏. This current
is 𝐼𝑁 of the Norton equivalent circuit.
Norton’s Theorem
Dr. Mohamed Refky
Thevenin and Norton equivalent circuits
Thevenin equivalent circuit must be equivalent to Norton
equivalent circuit
𝑍𝑁 = 𝑍𝑡ℎ, 𝑉𝑡ℎ = 𝐼𝑁𝑍𝑁, 𝐼𝑁 =𝑉𝑡ℎ𝑍𝑡ℎ
→ 𝑍𝑡ℎ =𝑉𝑡ℎ𝐼𝑁
Thevenin’s Theorem
Dr. Mohamed Refky
Example (9)Obtain the Norton equivalent at terminals 𝑎 and 𝑏 of the circuit
shown.
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