Power Example 1
ÎR
= 200Ω, XC = 150Ω, XL = 80Ω, εrms = 120v, fd = 60 Hz
‹ ωd
= 120π = 377 rad/s
‹
Z = 200 + ( 80 − 150 ) = 211.9Ω
‹
I rms = ε rms / Z = 120 / 211.9 = 0.566 A
‹
2
2
−1 ⎛ 80 − 150 ⎞
φ = tan ⎜
⎟ = −19.3°
⎠
Current leads emf
Capacitive circuit
‹
⎝ 200
cos φ = 0.944
‹
Pave = ε rms I rms cos φ = 120 × 0.566 × 0.944 = 64.1W
‹
Same
2
Pave = I rms
R = 0.5662 × 200 = 64.1W
PHY2049: Chapter 31
42
Power Example 1 (cont)
ÎR
= 200Ω, XC = 150Ω, XL = 80Ω, εrms = 120v, fd = 60 Hz
ÎHow
much capacitance must be added to maximize the
power in the circuit (and thus bring it into resonance)?
‹ Want
XC = XL to minimize Z, so must decrease XC
‹
X C = 150Ω = 1/ ω d C
C = 17.7μF
‹
X C new = X L = 80Ω
Cnew = 33.2μF
‹ So
we must add 15.5μF capacitance to maximize power
‹ Either
change capacitor or put 15.5μF in parallel with 17.7μF
PHY2049: Chapter 31
43
Power Example 2
ÎCircuit
parameters: C = 2.5μF, L = 4mH, εm = 10v
‹ ω0
= (1/LC)1/2 = 10000 rad/s
‹ Plot Power vs ωd / ω0 for different R values (Q = ω0L / R)
R = 2Ω Q = 20
R = 5Ω Q = 8
Pave
Resonance
ω d = ω0
R = 10Ω Q = 4
R = 20Ω Q = 2
PHY2049: Chapter 31
44
Radio Tuner
Set RLC tuner to 103.7 (ugh!)
Other radio stations.
RLC response is less
Circuit response Q = 500
Tune for f = 103.7 MHz
PHY2049: Chapter 31
45
Quiz
ÎA
generator produces current at a frequency of 60 Hz with
peak voltage and current amplitudes of 100V and 10A,
respectively. What is the average power produced?
‹ (1)
‹ (2)
‹ (3)
‹ (4)
‹ (5)
1000 W
707 W
1414 W
500 W
250 W
Pave = 12 ε peak I peak = ε rms I rms
PHY2049: Chapter 31
46
Quiz
ÎThe
figure shows the current and emf of a series RLC
circuit. To increase the rate at which power is delivered to
the resistive load, which option should be taken?
‹ (1)
Increase R
‹ (2) Decrease L
‹ (3) Increase L
‹ (4) Increase C
Current lags applied emf (peak occurs later, φ > 0), thus circuit is
inductive. Max power is at φ = 0, so need to either (1) reduce XL by
decreasing L or (2) cancel XL by increasing XC (decrease C).
PHY2049: Chapter 31
47
RLC Circuit Example
If you wanted to increase the power
delivered to this RLC circuit, which
modification(s) would work?
(a) increase R
(d) decrease R
(b) increase C
(e) decrease C
(c) increase L
(f) decrease L
Again, current lags emf, so inductive.
See previous page for details.
PHY2049: Chapter 31
48
Example (Prob. 31-48)
frequency EMF source with εm=6V connected to a
resistor and inductor. R=80Ω and L=40mH.
ÎVariable
‹ At
what frequency fd does VR = VL?
ω d L = R ⇒ ω d = 2000 rad/s
f d = ω d / 2π = 318Hz
‹ At
εm
VL
VR
Im
that frequency, what is phase angle φ?
80
= 1 ⇒ φ = 45°
tan φ =
80
‹ What
is current amplitude?
I m = ε m / Z = 6 / 802 + 802 = 6 /113 = 0.053A
PHY2049: Chapter 31
49
Transformers
ÎPurpose:
change alternating (AC) voltage to a bigger (or
smaller) value
Input AC voltage
in the primary
produces a flux
Changing flux
in secondary
induces emf
dΦB
Vp = N p
dt
dΦB
Vs = N s
dt
Ns
Vs = V p
Np
PHY2049: Chapter 31
50
Transformers
ÎNothing
comes for free, however!
‹ Increase
in voltage comes at the cost of current.
‹ Output power cannot exceed input power!
‹ power in = power out
i pVp = isVs
is V p N p
=
=
i p Vs N s
PHY2049: Chapter 31
51
Transformers: Sample Problem
ÎA
transformer has 330 primary turns and 1240 secondary
turns. The input voltage is 120 V and the output current
is 15.0 A. What is the output voltage and input current?
Ns
⎛ 1240 ⎞
Vs = V p
= 120 ⎜
⎟ = 451V
Np
⎝ 330 ⎠
i pV p = isVs
“Step-up”
transformer
Vs
⎛ 451 ⎞
i p = is
= 15 ⎜
⎟ = 56.4 A
Vp
⎝ 120 ⎠
PHY2049: Chapter 31
52
Transformers
¾ This is how first experiment by
Faraday was done
¾ He only got a deflection of the
galvanometer when the switch
is opened or closed
¾ Steady current does not make
induced emf.
PHY2049: Chapter 31
53
Applications
Microphone
Tape recorder
PHY2049: Chapter 31
54
ConcepTest: Power lines
ÎAt
large distances, the resistance of power lines becomes
significant. To transmit maximum power, is it better to
transmit (high V, low i) or (high i, low V)?
‹ (1)
high V, low i
‹ (2) low V, high i
‹ (3) makes no difference
Power loss is i2R
PHY2049: Chapter 31
55
Electric Power Transmission
i2R: 20x smaller current ⇒ 400x smaller power loss
PHY2049: Chapter 31
56