Sinusoidal Steady State Analysis (AC Analysis) Part I

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1 Sinusoidal Steady State Analysis (AC Analysis) Part I Amin Electronics and Electrical Communications Engineering Department (EECE) Cairo University

2 OUTLINE Previously on ELCN102 Solution of AC Circuits Simplification Method Loop Analysis Method Node Analysis Method Superposition Method 2

3 Previously on ELCN102 Phasor Relationships for Circuit Elements Resistor Inductor Capacitor v R t = Ri R t v L t = L di L t dt i C t = C dv C t dt V R = R I R V L = ωli L 90 o = jωl I L I C = ωcv C 90o = jωc V C The impedance Z of a circuit is the ratio of the phasor voltage V to the phasor current I, measured in Ω. Z R = R Z L = jωl Z C = 1 jωc = j ωc

4 Previously on ELCN102 Phasor Relationships for Circuit Elements Resistor Inductor Capacitor v R t = Ri R t v L t = L di L t dt i C t = C dv C t dt V R = R I R V L = ωli L 90 o = jωl I L I C = ωcv C 90o = jωc V C Y R = 1 R Y L = 1 jωl = j ωl Y C = jωc The admittance Y of a circuit is the ratio of the phasor current I to the phasor voltage V, measured in Ω 1.

5 Previously on ELCN102 Impedance and Admittance Impedance The impedance Z of a circuit is the ratio of the phasor voltage V to 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 Z L = jωl Z C = 1 jωc = j ωc

6 Previously on ELCN102 Impedance and Admittance 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 Y L = 1 jωl = j ωl Y C = jωc

7 Previously on ELCN102 Impedance Combination Series Combination Z eq = Z 1 + Z Z N

8 Previously on ELCN102 Impedance Combination Parallel Combination 1 = Z eq Z 1 Z 2 Z N

9 Previously on ELCN102 Admittance Combination Series Combination 1 Y eq = 1 Y Y Y N

10 Previously on ELCN102 Admittance Combination Parallel Combination Y eq = Y 1 + Y Y N

11 Previously on ELCN102 Star-Delta Transformation Z A = Z AB Z BC Z B = Z AC + Z BC + Z AB Z AB Z AC Z AC + Z BC + Z AB Z C = Z BC Z AC Z AC + Z BC + Z AB Z AB = Z A + Z B + Z AZ B Z C Z AC = Z A + Z C + Z AZ C Z B Z BC = Z B + Z C + Z BZ C Z A

12 Definition Solution of AC Circuits A 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).

13 Solution of AC Circuits Methods of Solution of AC Circuits To 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

14 Solution of AC Circuits 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

15 Simplification Method Source Transformation A voltage source V AC with a series impedance Z can be transformed into a current source I AC = V AC /Z and a parallel impedance Z A current source I AC with a parallel impedance Z can be transformed into a voltage source V AC = I AC Z and a series impedance Z

16 Example (1) Simplification Method Use simplification method to find V x for the circuit shown.

17 Example (2) Simplification Method Use simplification method to find I x for the circuit shown.

18 Definition Loop Analysis Method 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.

19 Matrix Form Loop Analysis Method Z 11 Z 12 Z 1N Z 21 Z 22 Z 2N Z N1 Z N2 Z NN I 1 I 2 I N = V 1 V 2 V N Z ii = impedance in loop i Z ij = Common impedance between loops i and j = Z ji V i = voltage sources in loop i V is +ve if it supplies current in the direction of the loop current

20 Example (3) Loop Analysis Method Use loop analysis to find I x for the circuit shown.

21 Example (4) Loop Analysis Method Use loop analysis to find I x for the circuit shown.

22 Example (5) Loop Analysis Method Use loop analysis to find V x for the circuit shown.

23 Definition Node Analysis Method 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. 23

24 Matrix Form Node Analysis Method Y 11 Y 12 Y 1N Y 21 Y 22 Y 2N Y N1 Y N2 Y NN V 1 V 2 V N = I 1 I 2 I N Y ii = admittance of node i Y ij = common admittance between node i and j = Y ji I i = current sources at node i I is +ve if it supply current into the node

25 Example (6) Node Analysis Method Use node analysis to find V 1 & V 2 for the circuit shown.

26 Definition Superposition Theorem For 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 o V 5 0 o A Total I a I b I = I a + I b

27 Example (7) Superposition Theorem Use superposition theorem to find I x for the circuit shown.

28 Definition Thevenin s Theorem A linear two-terminal circuit, can be replaced by an equivalent circuit consisting of a voltage source V th in series with a impedance Z th.

29 Solution Steps Thevenin s Theorem 1) 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 V th 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 Z th of the Thevenin equivalent circuit.

30 Example (8) Thevenin s Theorem Obtain the Thevenin equivalent at terminals a and b of the circuit shown.

31 Definition Norton s Theorem A linear two-terminal circuit can be replaced by equivalent circuit consisting of a current source I N in parallel with a impedance Z N

32 Solution Steps Norton s Theorem 1) 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 Z N 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 I N of the Norton equivalent circuit.

33 Norton s Theorem Thevenin and Norton equivalent circuits Thevenin equivalent circuit must be equivalent to Norton equivalent circuit Z N = Z th, V th = I N Z N, I N = V th Z Z th = V th th I N

34 Example (9) Thevenin s Theorem Obtain the Norton equivalent at terminals a and b of the circuit shown.

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