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.

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

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

Mesh Analysis

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

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

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

Considering the effects of the voltage source E2

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

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

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.

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.

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

THEVENIN’S THEOREM Calculate Eth The Thévenin equivalent circuit is

NORTON’S THEOREM

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.

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

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

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

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

Resonance Power factor at resonant is

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,

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

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.

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.

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.