# Maximum Power Delivered Delivered power Digilent Inc

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```					              Lecture 12
(parts A & B)

•Review:
•Source transformations
•Maximum power transfer
•Derivation of maximum power transfer
•Thévenin theorem examples
•Operational Amplifiers
•Related educational materials:
–Chapters 4.5, 4.6, 5.1-5.4
Using Source Transformations in Circuit Analysis
• Any voltage source in series with a resistance can be modeled as a
current source in parallel with the same resistance and vice-versa
Maximum Power Transfer
RL = RTH

• Why?
Maximum Power Transfer – Derivation
• Load voltage:           • Delivered power:
2
2
V    V 
2
RL      
PL  L
  OC
            
RL   RL  RL  RTH




PL
RL
VL  VOC
RL  RTH

RL
Maximizing power
• Set derivative of power to zero:
PL                2      RL      
0           VOC            2
0
R L     RL           RL  RTH  
• Chain rule:
 ( RL  RTH )2  2 RL ( RTH  RL ) 
0
2
V 
( RL  RTH )
OC                        4
                                   
• Set numerator to zero:
RL  RTH
Maximum Power Delivered
• Delivered power:
2
V2
   RL      
PL   OC

R R       

RL     L    TH   
• Letting RL = RTH:
2
VOC
PL 
4 RTH
Example 1: Maximum power transfer
(a) Determine the load resistance, R, which absorbs the
maximum power from the circuit.
(b) What is the maximum power delivered to the load?
Example 1(b): Power delivered
Example 2
• Determine the Norton equivalent of the circuit of example 1
Operational Amplifiers
• So far, with the exception of our ideal power
sources, all the circuit elements we have examined
have been passive
– Total energy delivered by the circuit to the element is
non-negative
• We now introduce another class of active devices
– Operational Amplifiers (op-amps)
– Note: These require an external power supply!
Operational Amplifiers – overview
• We will analyze op-amps as a “device” or “black
box”, without worrying about their internal circuitry
– This may make it appear as if KVL, KCL do not apply to the
operational amplifier
– Our analysis is based on “rules” for the overall op-amp
operation, and not performing a detailed analysis of the
internal circuitry
• We want to use op-amps to perform operations, not
design and build the op-amps themselves
uA741 op-amp schematic

• Source: RFIC Technologies web site
Ideal Operational Amplifiers
• Typical circuit schematic symbol:

ip +
vp - vn = vin
in -
• Three-terminal device (2 inputs, 1 output)
• Operation characterized by:
– Voltage difference between input terminals (vin)
– Currents into the input terminals (ip and in)
Ideal Operational Amplifier “Rules”
• More complete circuit      • Assumptions:
symbol
• (Power supplies shown)      • ip = 0, in = 0

• vin = 0

• V - < vout < V +
Notes on op-amp operation
1. Output current is generally not known (it is
provided by external power supplies)
2. KCL at input nodes is generally a good starting
point in op-amp circuit analysis
3. vin is multiplied by a large number to get vout

4. vout is limited by the external power supplies
Op-amp circuit – example 1
• Find Vout
Op-amp circuit – example 2
• Find Vout

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