Transcript AC Network Theorems

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 opencircuit 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 shortcircuit 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
2
PL 
E Th R L
R Th  RL 2
2
E Th
Pmax 
4R Th
2
IN ZN
Pmax 
4RN
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
RL  RTh   X  X Th 
2
2