Download Thevenin and Max Power

EE301
AC Thèvenin and
Max Power Transfer
Learning Objectives



Apply Thèvenin’s Theorem to AC circuits
Explain under what conditions a source transfers maximum
power to a load
Determine the value of load impedance for which maximum
power is transferred from the circuit
Thévenin’s theorem for AC


ETh is the open circuit voltage at the terminals,
ZTh is the input or equivalent resistance at the
terminals when the independent sources are
turned off.
Review
Determining ETh

Remove the load (open-circuit) and measure the
resulting voltage.
Eth = open ckt voltage
Determining ZTh

With the load disconnected, turn off all
independent sources.
sources – 0 V is equivalent to a short-circuit.
 Current sources – 0 A is equivalent to a open-circuit.
 Voltage

ZTh is the equivalent resistance looking into the
“dead” circuit through terminals a-b.
Zth
Applying Thévenin equivalent

Once ETh and ZTh have been found, the original
circuit is replaced by its equivalent and solving
for ILD and VLD becomes trivial.
I LD
ETh
=
ZTh + Z LD
VLD
Z LD
=
ETh
ZTh + Z LD
Example Problem 1
Convert the source below into a Thévenin equivalent and
determine the current through load Zab.
Example Problem 2
Convert the source below into a Thévenin equivalent.
ZLOAD
Example Problem 3
Convert the source below into a Thévenin equivalent and
determine the power dissipated by the load.
Conjugates

The conjugate of C is written
as C*, which has the same
real value but the opposite
imaginary part:
C  a  jb  C
C  a  jb  C  
Review
Review
Maximum power transfer theorem

Maximum power is transferred to the load
when RLD = RTh.
PMAX 
E
2
Th
4 RTh
Max Power Transfer in AC Circuits

In AC circuits, max power transfer occurs when load
impedance (ZL) is the complex conjugate of the Thévinin
equivalent impedance (ZTh).
ZTH  RTH  jX TH  Z LD  RLD  jX LD


This means the load has a capacitor if the Thèvenin
impedance includes an inductor
XLD cancels out XTH!
Max Power Transfer in AC Circuits

Since RLD=RTH, and the reactances cancel out, the
resulting PMAX equation is the same as with DC!
Z LD  RLD  jX LD
IL 

ZTH  RTH  jX TH
ETh
ETh

 ZTh  Z LD  ( RTh  RLD )  j ( X Th  X LD )
ETh
ETh
E

 Th since RTh  RLD
 RTh  RLD   RTh  RTh  2 RTh
2
PL  I L2 RLD
ETh2

4 RTh
 ETh 
ETh2

RTh
 RLD 
2
4 RTh
 2 RTh 
Max Power Transfer in AC Circuits


#1 Mistake with AC Max Power is not using the REAL
value of resistance when calculating Pmax
Transform ZTH into rectangular form to determine RTH
ZTH  ZXTH   RTH  jX TH
Pmax
ETh2

when Z LD  ZTh
4 RTh
Example Problem 4
Determine the load ZLOAD that will allow maximum power to
be delivered to the load the circuit below. Find the power
dissipated by the load.
Example Problem 5
Determine the load ZLD that will allow maximum power to
be delivered to the load the circuit below. Frequency is
191.15 Hz. Find the maximum power. What will happen to
power if the frequency is changed to 95.575?