ENGI 251/ELEC 241
AC Power
ENGI 241
Power in AC Circuits
Chapter 10
Net Impedance in an AC Circuit
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ENGI 251/ELEC 241
AC Power
Voltage, Current, and Average Power Waveforms
Purely Resistive Circuit
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Voltage, Current, and Average Power Waveforms
Purely Inductive Circuit
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Purely Capacitive Circuit
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ENGI 251/ELEC 241
AC Power
Types of Power Measurement
• Real Power
–
–
–
–
–
the power consumed by the resistive elements in the circuit
Symbol is P
Unit of measurement is W (watt)
P = |Vrms | |Irms | cos(θ)
P = | Irms |2 Re(Z) ( the Real part of the impedance
• Reactive Power
–
–
–
–
–
the power consumed by the net reactive elements in the circuit
Symbol is Q
Unit of measurement is VAR (volt-ampere reactive)
Q = |Vrms| |Irms | sin(θ)
Q = |Irms|2 Im(Z) ( the imaginary part of the impedance)
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Types of Power Measurement
• Apparent Power
–
–
–
–
–
the total power delivered by the source or consumed by the components
Symbol is S
Unit of measurement is the VA (volt-ampere)
Is the algebraic sum of the real and net reactive power
S = P + jQ
• pf = cos θ
• rf = sin θ
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where θ = θv - θi
where θ = θv - θi
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ENGI 251/ELEC 241
AC Power
Phasor Diagrams
Impedance Phasor Diagram
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Power Phasor Diagram
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Power Phasor Diagram for a Net RC
When
-90° ≥ θ > 0, the load looks like an RC circuit
⎛
1 ⎞
⎜ ωL <
⎟
ωC
⎝
⎠
and pf is leading.
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ENGI 251/ELEC 241
AC Power
Power Phasor Diagram for a net RL
When
0 > θ ≥ + 90°, the load looks like an RL circuit
⎛
1 ⎞
⎜ ωL >
⎟
ωC
⎝
⎠
and pf is lagging.
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Power Phasor Diagram
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ENGI 251/ELEC 241
AC Power
Possible Power Cases for Net Circuit
Case 1
If Z = R (purely resistive load), then:
P = | Irms |2 R, and Q = 0
Case 2
If Z = jωL (purely inductive load), then:
P = 0, and Q = | Irms |2 ωL
Case 3
If (purely capacitive load), then:
⎛ -1 ⎞
P = 0, and Q = | Irms |2 ⎜
⎟
⎝ ωC ⎠
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Possible Power Cases for Net Circuit
Case 4
1
For a series RLC load, Z = R + jωL - j
ωC
2
then P = Irms R
1 ⎞
⎛
Q = | Irms |2 ⎜ ωL ⎟
ωC ⎠
⎝
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ENGI 251/ELEC 241
AC Power
Example 1
Calculate all Component Powers
Verify S delivered by source = S consumed by all components
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Example 2
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ENGI 251/ELEC 241
AC Power
Maximum Power Transferred to Load
• It is sometimes important to be able to transfer as much real
power as possible from a source to a load
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Thevenin Equivalent Circuits
• Consider the Thevenin equivalent circuit of the network
• We wish to determine the value of the load impedance ZL for
which the real power PL absorbed by the load is a maximum.
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ENGI 251/ELEC 241
AC Power
Maximum Power Transfer
• For a complex load impedance, the maximum power is
transferred to a load when the load impedance equal to the
complex conjugate of the Thevenin impedance
ZL = ZTH*
• For a purely resistance load, the maximum power is
transferred to the load when the load resistance equal to the
Thevenin resistance
RL = RTH
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Example 3
Find ZL for maximum power to load
Find PL
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