1. Methods of Analysis and Network Theorems of A.C. Circuits
Electrical Engineering (2)
Mechanical Engineering/ first class
College of Engineering
University of Al-Qadisiya
Lecturer: Yousif M. H.

2. Methods of Analysis and Network Theorems of A.C. Circuits
Mesh Analysis
Superposition Theorem
Source Transformation
Thevenin and Norton Equivalent Circuits
Resonance

3. EXAMPLE: Using mesh analysis, find the current I1 in Fig.
Solution:
Z1 = j XL = j 2 Ω
E1 = 2 V ∠0°
Z2 = R = 4 Ω
E2 = 6 V ∠0°
Z3 = j XC = _ j 1 Ω

4. Mesh Analysis

5. Mesh Analysis
EXAMPLE : Write the mesh currents for the network of Fig. having an independent current source.

6. HOMEWORK: Using the mesh analysis, find the current I2 in Fig.

8. SUPERPOSITION THEOREM
EXAMPLE: Using the superposition theorem, find the current I through the 4-Ω reactance (XL2) of Fig. Considering the effects of the voltage source E1

10. Considering the effects of the voltage source E2

11. SUPERPOSITION THEOREM
EXAMPLE : Using superposition, find the current I through the 6Ωresistor of Fig.
Solution:
Z1 = j 6, Z2 = 6 - j 8
Consider the effects of the current source. Applying CDR

12. SUPERPOSITION THEOREM
Consider the effects of the voltage source. Applying Ohm’s law
The total current through the 6Ω resistor is

13. THEVENIN’S THEOREM
Thévenin’s theorem, for sinusoidal ac circuits, is changed impedance instead of resistance; that is,
any two-terminal linear ac network can be replaced with an equivalent circuit consisting of a voltage source and an impedance in series.

14. THEVENIN’S THEOREM Independent Sources
1. Remove that portion of the network across which the Thévenin equivalent circuit is to be found.
2. Calculate ZTh by first setting all voltage and current sources to zero (short circuit and open circuit, respectively) and then finding the resulting impedance between the two marked terminals.
3. Calculate ETh by first replacing the voltage and current sources and then finding the open-circuit voltage between the marked terminals.
4. Draw the Thévenin equivalent circuit.

15. THEVENIN’S THEOREM
EXAMPLE : Find the Thévenin equivalent circuit for the network external to resistor R in Fig.
Solution:

16. THEVENIN’S THEOREM
Calculate Eth
The Thévenin equivalent circuit is

17. NORTON’S THEOREM

18. NORTON’S THEOREM Independent Sources
1. Remove that portion of the network across which the Norton equivalent circuit is to be found.
2. Calculate ZN by first setting all voltage and current sources to zero (short circuit and open circuit, respectively) and then finding the resulting impedance between the two marked terminals.
3. Calculate IN by first replacing the voltage and current sources and then finding the short-circuit current between the marked terminals.
4. Draw the Norton equivalent circuit.

19. NORTON’S THEOREM
EXAMPLE : Determine the Norton equivalent circuit for the network external to the 6Ωresistor of Fig.
Solution:
Step 1: Remove the resistance 6Ω
2- Replace the voltage source E with
Short circuit and then calculate ZN

20. NORTON’S THEOREM
Step 3: Calculate IN by finding the short-circuit current between the marked terminals. Step 4: The Norton equivalent circuit is shown in Fig
Determining IN for the network

21. NORTON’S THEOREM
HOMEWORK: Find the Norton equivalent circuit for the network external to the 7Ω capacitive reactance in Fig.

22. Resonance The resonant condition
SERIES RESONANT CIRCUIT
The total impedance of this network at any frequency is determined by
The resonant condition
The total impedance at resonance is then simply

23. Resonance
Power factor at
resonant is

24. THE QUALITY FACTOR (Q )
quality factor (Q)of a series resonant circuit is defined as the ratio of the reactive power of either the inductor or the capacitor to the average power of the resistor at resonance; that is,

25. THE QUALITY FACTOR (Q )
The bandwidth (BW) is

26. EXAMPLE : For the series resonant circuit of Fig.
find I, VR, VL, and VC at resonance.
What is the Qs of the circuit?
If the resonant frequency is 5000 Hz, find the bandwidth.

27. EXAMPLE : The bandwidth of a series resonant circuit is 400 Hz. a
EXAMPLE : The bandwidth of a series resonant circuit is 400 Hz. a. If the resonant frequency is 4000 Hz, what is the value of Qs? b. If R =10Ω , what is the value of XL at resonance? c. Find the inductance L and capacitance C of the circuit.

28. EXAMPLE : A series R-L-C circuit is designed to resonant at
ws =10^5 rad/s, have a bandwidth of 0.15 fs, and draw 16 W from a 120-V source at resonance.
Determine the value of R.
Find the bandwidth in hertz.