Alternating Current Circuits

1
Alternating Current Circuits

• Chapter 33
• (continued)

2
Phasor Diagrams

• A phasor is an arrow whose length represents the
  amplitude of an AC voltage or current.
• The phasor rotates counterclockwise about the
  origin with the angular frequency of the AC
  quantity.
• Phasor diagrams are useful in solving complex AC
  circuits.
• The y component is the actual current or
  voltage.

Resistor Capacitor Inductor
VRp
VLp
Ip
Ip
Ip
w t
w t
w t
VCp
3
Impedance in AC Circuits
The impedance Z of a circuit or circuit element
relates peak current to peak voltage
(Units Ohms)
(This is the AC equivalent of Ohms law.)
4
Phasor Diagrams
Circuit element Impedance Amplitude Phase
Resistor R VR IP R I, V in phase
Capacitor Xc1/wC VCIP Xc I leads V by 90
Inductor XLwL VLIP Xc I lags V by 90
Resistor Capacitor Inductor
VRp
VLp
Ip
Ip
Ip
w t
w t
w t
VCp
5
RLC Circuit
Use the loop method V - VR - VC - VL 0 I
is same through all components.
BUT Voltages have different PHASES ? they
add as PHASORS.
6
RLC Circuit
Ip
VRp
VLp
f
VP
(VCp- VLp)
VCp
7
RLC Circuit
Ip
VRp
VLp
f
VP
(VCp- VLp)
VCp
By Pythagorass theorem (VP )2 (VRp )2
(VCp - VLp)2 Ip2 R2 (Ip XC - Ip
XL) 2
8
RLC Circuit
Solve for the current
9
RLC Circuit
Solve for the current
Impedance
10
RLC Circuit
The currents magnitude depends on the driving
frequency. When Z is a minimum, the current is a
maximum. This happens at a resonance frequency
The circuit hits resonance when 1/wC-wL0 w
r1/ When this happens the capacitor and inductor
cancel each other and the circuit behaves purely
resistively IPVP/R.
L1mH C10mF
The current dies away at both low and
high frequencies.
wr
w
11
Phase in an RLC Circuit
We can also find the phase tan f (VCp
- VLp)/ VRp (XC-XL)/R
(1/wC - wL) / R
12
Phase in an RLC Circuit
We can also find the phase tan f (VCp
- VLp)/ VRp (XC-XL)/R
(1/wC - wL) / R
More generally, in terms of impedance cos f
R/Z
At resonance the phase goes to zero (when the
circuit becomes purely resistive, the current and
voltage are in phase).
13
Power in an AC Circuit
The power dissipated in an AC circuit is PIV.
Since both I and V vary in time, so does the
power P is a function of time.
Use V VP sin (wt) and I IP sin (w tf )
P(t) IpVpsin(wt) sin (w tf ) This
wiggles in time, usually very fast. What we
usually care about is the time average of this
(T1/f )
14
Power in an AC Circuit
Now
15
Power in an AC Circuit
Now
16
Power in an AC Circuit
Now
Use and
So
17
Power in an AC Circuit
Now
Use and
So
which we usually write as
18
Power in an AC Circuit
(f goes from -900 to 900, so the average power is
positive)
cos(f) is called the power factor. For a purely
resistive circuit the power factor is 1. When
R0, cos(f)0 (energy is traded but not
dissipated). Usually the power factor depends on
frequency, and usually 0ltcos(f)lt1.
19
Power in a purely resistive circuit
V
f 0
V(t) VP sin (wt)
I
I(t) IP sin (wt)
p
wt
2p
(This is for a purely resistive circuit.)
P
P(t) IV IP VP sin 2(wt) Note this
oscillates twice as fast.
p
wt
2p
20
Power in a purely reactive circuit
The opposite limit is a purely reactive circuit,
with R0.
I
P
This happens with an LC circuit. Then f
900 The time average of P is zero.
V
wt
21
Transformers
Transformers use mutual inductance to change
voltages
Primary (applied voltage)
Secondary (produced voltage)
Faradays law on the left If the flux per turn
is f then VpNp(df/dt).
Faradays law on the right The flux per turn is
also f, so VsNs(df/dt).
?
22
Transformers
Transformers use mutual inductance to change
voltages
Primary (applied voltage)
Secondary (produced voltage)
In the ideal case, no power is dissipated in the
transformer itself.
Then IpVpIsVs ?
23
Transformers Power Transmission
Transformers can be used to step up and
step down voltages for power transmission.
110 turns
20,000 turns
Power I2 V2
V220kV
V1110V
Power I1 V1
We use high voltage (e.g. 365 kV) to transmit
electrical power over long distances. Why do we
want to do this?
24
Transformers Power Transmission
Transformers can be used to step up and step
down voltages, for power transmission and other
applications.
110 turns
20,000 turns
Power I2 V2
V220kV
V1110V
Power I1 V1
We use high voltage (e.g. 365 kV) to transmit
electrical power over long distances. Why do we
want to do this? P I2R (P power
dissipation in the line - I is smaller at high
voltages)