Chapter 7
AC Power
Jaesung Jang
Average Power & Power Factor Correction
Ideal Transformer
Impedance Matching & Maximum Power Transfer
1
Instantaneous & Average Power
• When a linear electric circuit is excited by a sinusoidal
source, all voltages and currents in the circuit are also
sinusoids of the same frequency as that of the excitation
source (voltage or current source).
v (t ) = V cos (ω t − θ V
)
and i (t ) = I cos (ω t − θ I )
→ General expression for the voltage and current delivered to an arbitrary load
Assume θ V = 0 and θ I = θ : phase difference between vo ltage and current
Instantaen ous power p (t ) = v (t )i (t ) = VI cos (ω t ) cos (ω t − θ )
=
VI
[cos (θ ) + cos (2ω t − θ )] where VI cos (θ ) : Average power
2
2
2
Instantaneous & Average Power (Cont.)
1
Average Power Pav =
T
1
p(t )dt =
0
T
T
∫
T VI
∫  2
0
[cos(θ ) + cos(2ωt −θ )]dt = VI cos(θ )

2
It is customary in AC power analysis to use the rms value of the AC analysis.
V( jω ) Ve j 0
V − jθ
− jθ
j0
jθ
V( jω ) = Ve
and Z( jω ) = Z e → I( jω ) =
=
=
=
e
Ie
Z( j ω ) Z e j θ
Z
I
~
~
= V and I rms =
= I : rms voltage and current values
2
2
~
~~
V2
~
→ Pav = VI cos θ =
cos θ = I 2 Z cos θ ; where power factor (pf) = cos θ
Z
~
~
~
~
~
~
V = Vrms e jθV = V e jθV = V ∠θV and I = I rms e jθ I = I e jθ I = I ∠θ I : rms voltage and current phasors
Vrms =
V
~
VS = 110∠0; RS = 2Ω; RL = 16Ω; C = 100 µF
(
)
(
)
Z T ( jω ) = Z RS + Z RL || Z C , where Z RL || Z C =
~
VL =
(Z R || Z C )
Z R + (Z R || Z C )
L
S
L
1
 1


+ jωC 
 RL

~
VL2
~
VS : voltage divider → Pav =
cos θ
ZL
=
RL
(1 + jωRL C )
3
Complex Power
Im
~~
S = V I ∗ : Complex Power
~ ~
~ ~ ~
~
V = V e j 0 = V , I = I e − jθ = I ∠(− θ ) and Z( jω ) = Z e jθ from Impedance Triangle
R = Z cos θ , X = Z sin θ from Impedance Triangle
z
Z
X
θ
R
Re
~ ~
~ ~
~ ~
V = I Z( jω ) → V = I e − j θ × Z e j θ → V = I Z
(
)
Complex power triangle
~~
~~
~~
~~
~~ R
~~ X
+ jV I
S = V I ∗ = V I e jθ = V I cos θ + jV I sin θ = VI
Z
Z
~
~
~
S = I 2 R + jI 2 X = Pav + jQ = I 2 Z
where Pav : real power (unit : Watts)
Q : reactive power (unit : Volt Amperes Reactive, VAR)
S = Apparent Power (Unit : Volt Amperes VA)
4
Power Factor Correction
• Although the reactive power does not contribute to any average
power dissipation in the load, it may have an adverse effect on
power consumption because it increases the overall rms
current flowing in the circuit. -> desirable to minimize the
reactive power in a load.
• Example 7.6
– Resistor + Inductor: percent real power transfer: 40%
– Resistor only: percent real power transfer: 71%
– A power factor close to unity means efficient energy transfer
from the AC source to the load.
– Power factor correction: making the power factor close to unity
(zero reactive power) for more efficient energy transfer.
5
Transformers
AC circuits are very commonly connected to each other by means of transformers.
• Mutual inductance is a key phenomena of the transformers that occurs when
current flowing through one conductor, creating a magnetic field and inducing a
voltage in a nearby conductor.
The primary winding is connected to a source of ac power.
The secondary winding is connected to the load and induced currents flow.
•The transformer transfers power from the primary (source) to the secondary (load).
Same power on both sides
for ideal transformers
Transformer steps up voltage (to 100V) and steps current down (to 1A)
Iron-core transformer with 1:10 turns ratio. Primary current IP induces secondary voltage VS,
which produces current in secondary load RL.
6
Ideal Transformers
• Turns ratio N =nS/nP, where nP (= n1) equals the number of turns in the primary and
nS (= n2) equals the number of turns in the secondary.
• The voltage ratio (of an ideal transformer) is the same as the turns ratio:
VP / VS = nP / nS = IS / IP
– VP = primary voltage, VS = secondary voltage,
– NP = number of turns of wire in the primary
– NS = number of turns of wire in the secondary
• When transformer efficiency is 100%, the power at the primary equals the power at
the secondary. The ideal transformer has 100% efficiency. VP IP = VS IS
~
~
~
~
V2 n2
I1
V2 n2
~ ~ ~~
~∗ ~
~ ∗ V2 ~ ∗ ~
= N , V2 I2 = V1 I1 → ~ =
= N = ~ → S1 = I1 V1 = N I2
= I2 V2 = S 2
~ =
N
V1 n1
V1 n1
I2
• Step-up transformer: N > 1 and the output voltage is
greater than the input voltage.
• Step-down transformer: N < 1 and the output voltage is
smaller than the input voltage.
• Isolation transformer: N=1.
• No DC at the primary will appear at the secondary coil.
• % Efficiency = Pout/Pin x 100
7
Impedance Reflection & Power Transfer
~
V2 n2
~ ~ ~~
=
=
N
,
V
~
2 I2 = V1 I1 → Z ′ =
n
V1
1
•
•
~
~
V1 V2 1
1
=
=
~
~
N N I2 N 2
I1
~
V2
1
=
Z
~
2 L
I2
N
The secondary load impedance is said to be reflected back into the
primary and is called a reflected impedance.
Reflected impedance (Z’) is impedance seen by the primary:
8
Impedance Matching & Maximum
Power Transfer
•
(
~~
~ ~
PL = VL I L cos θ = Re VL IL∗
When the load impedance is equal to
the complex conjugate of the source
impedance, the load and source
impedance are matched and
maximum power is transferred to
the load.
)
~
ZL
~
~ ∗  VS
VS and IL =
Z +Z
ZL + ZS
S
 L
~ 2
~∗
V
S ZL
ZL
VS
~ ~
~
VL IL∗ =
VS
=
ZL + ZS
(Z L + Z S )∗ Z L + Z S 2
~
where VL =
∗
~

VS∗
 =

(Z L + Z S )∗

~ 2
 ~ 2


V
Z
V


S
L 
S
~~
~ ~
=
PL = VL I L cos θ = Re VL IL∗ = Re
Re

2 
2
+
Z
Z
 L
 ZL + ZS
S 



~
VS2
=
R is greatest when RL = RS and
2
2 L
(R L + R S ) + ( X L + X S )
(
)
Maximum power transfer occurs
when
ZL
N2
= Z S∗ → RL = N 2 RS and X L = − N 2 X S
~ 2


V


S
=
Re
(
Z
)
Re


L

 ZL + ZS


X L = − X S → Z L = Z S∗
2


 RL


Ex. 7.14: Use of
transformers to
increase power
efficiency 9
Impedance Matching & Maximum
Power Transfer (cont.)
VG
)2 × RL
ri + R L
100V
=(
)2 × 8Ω
200 Ω + 8 Ω
= 1.85W
PL = (
Internal resistance (ri)
is 200 Ω and RL is 8
Ω.
R L = N 2 R S → 8 = N 2 200
8
1 n
=N = = S
200
5 nP
PL=12.5W
Maximum power transfer occurs
when
ZL
N2
= Z S∗ → RL = N 2 RS and X L = − N 2 X S
Transferring power from an amplifier to a load RL. (a) Amplifier has ri = 200 Ω and RL = 8 Ω.
10