Chapter 20 AC Network Theorems. Superposition Theorem The voltage across (or current through) an element is determined by summing the voltage (or current)

Chapter 20

AC Network Theorems

Superposition Theorem• The voltage across (or current through) an

element is determined by summing the voltage (or current) due to each independent source.

• All sources other than the one being considered are eliminated.

• Replace current sources with opens.• Replace voltage sources with shorts.

Superposition Theorem

• A circuit may operate at more than one frequency at a time.

• Diode and transistor circuits will have both dc and ac sources.

• Superposition can still be applied.

Superposition Theorem

• The superposition theorem can be applied only to voltage and current.

• It cannot be used to solve for the total power dissipated by an element.

• This is because power is not a linear quantity, but instead follows a square-law relationship.

Thévenin’s Theorem• Thévenin’s theorem converts an ac circuit

into a single ac voltage source in series with an equivalent impedance.

• First, remove the element or elements across which the equivalent circuit is to be found.

• Label the two terminals.• Set all sources to zero - replace voltage

sources with shorts, current sources with opens.

Thévenin’s Theorem

• Calculate the Thévenin equivalent impedance.

• Replace the sources and determine the open-circuit voltage.

• If more than one source is involved, use superposition.

• Draw the resulting Thévenin equivalent circuit, including the portion removed.

Norton’s Theorem

• Norton’s theorem converts an ac network into an equivalent circuit consisting of a single current source and a parallel impedance.

• First, remove element or elements across which the Norton circuit is to be found.

• Label the terminals.• Set all sources to zero.

Norton’s Theorem

• Determine the Norton equivalent impedance.• Replace the sources and calculate the short-

circuit current.• Superposition may used for multiple sources.• Draw the resulting Norton circuit with

elements which were removed replaced.

Thévenin and Norton Circuits

• It is possible to find the Norton equivalent circuit from the Thévenin equivalent circuit.

• ZN = ZTh

• IN = ETh/ZTh

Thévenin’s and Norton’s Theorems

• If a circuit contains a dependent source which is controlled by an element outside the area of interest, the previous methods cannot be used to find the Thévenin or Norton circuit.

• If a circuit contains a dependent source controlled by an element in the circuit, other methods must be used.

Thevenin’s and Norton’s Theorems

• If a circuit has a dependent source which is controlled by an element in the circuit, use the following steps to determine the equivalent circuit.

• First, remove the branch across which the equivalent circuit is to be determined.

• Label the terminals.

Thevenin’s and Norton’s Theorems

• Calculate the open-circuit voltage. The dependent source cannot be set to zero.Its effects must be considered.

• Determine the short-circuit current.

• ZN = ZTh = ETh/IN• Draw the equivalent circuit, replacing the

removed branch.

Thevenin’s and Norton’s Theorems

• A circuit may have more than one independent source.

• It is necessary to determine the open-circuit voltage and short-circuit current due to each independent source.

• The effects of the dependent source must be considered simultaneously.

Maximum Power Transfer Theorem

• Maximum power will be delivered to a load when the load impedance is the complex conjugate of the Thévenin or Norton impedance.

• ZTh = 3 + j4 ZL = 3 - j4

• ZTh = 10 30° ZL = 10 -30°

Maximum Power Transfer Theorem

• If the Z is replaced by its complex conjugate, the maximum power will be

N

NNmax

Th

Thmax

Th

Th

R

ZIP

R

EP

RR

REP

L

LL

4

422

2

2

2

Relative Maximum Power

• If it is not possible to adjust the reactance part of a load, then a relative maximum power will be delivered.

• The load resistance has a value determined by

22ThTh XXRRL