Sinusoidal Steady State Analysis

(AC Analysis)

Part I

Dr. Mohamed Refky Amin

Electronics and Electrical Communications Engineering Department (EECE)

Cairo University

elc.n102.eng@gmail.com

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.