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.
Resistor Capacitor Inductor
Vp
Vp
Ip
Ip
Ip
w t
w t
w t
Vp
3
Reactance - Phasor Diagrams
Resistor Capacitor Inductor
Vp
Vp
Ip
Ip
Ip
w t
w t
w t
Vp
4
Impedance of an AC Circuit
The impedance, Z, of a circuit relates
peak current to peak voltage
(Units OHMS)
5
Impedance of an AC Circuit
The impedance, Z, of a circuit relates
peak current to peak voltage
(Units OHMS)
(This is the AC equivalent of Ohms law.)
6
Impedance of an RLC Circuit
E
As in DC circuits, we can use the loop method
E - VR - VC - VL 0 I is same through
all components.
7
Impedance of an RLC Circuit
E
As in DC circuits, we can use the loop method
E - VR - VC - VL 0 I is same through
all components.
BUT Voltages have different PHASES ? they
add as PHASORS.
8
Phasors for a Series RLC Circuit
Ip
VRp
VLp
f
VP
(VCp- VLp)
VCp
9
Phasors for a Series RLC Circuit
Ip
VRp
VLp
f
VP
(VCp- VLp)
VCp
By Pythagoras theorem (VP )2 (VRp )2
(VCp - VLp)2
10
Phasors for a Series RLC Circuit
Ip
VRp
VLp
f
VP
(VCp- VLp)
VCp
By Pythagoras theorem (VP )2 (VRp )2
(VCp - VLp)2 Ip2 R2 (Ip XC - Ip
XL) 2
11
Impedance of an RLC Circuit
Solve for the current
12
Impedance of an RLC Circuit
Solve for the current
Impedance
13
Impedance of an 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
14
Phase in an RLC Circuit
We can also find the phase tan f (VCp
- VLp)/ VRp or tan f (XC-XL)/R. or
tan f (1/wC - wL) / R
15
Phase in an RLC Circuit
We can also find the phase tan f (VCp
- VLp)/ VRp or tan f (XC-XL)/R. or
tan f (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).
16
Power in an AC 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
17
Power in an AC Circuit
The power 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 )
18
Power in an AC Circuit
Now
19
Power in an AC Circuit
Now
20
Power in an AC Circuit
Now
Use and
So
21
Power in an AC Circuit
Now
Use and
So
which we usually write as
22
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.
23
Power in an AC Circuit
What if f is not zero?
I
P
V
Here I and V are 900 out of phase. (f 900) (It
is purely reactive) The time average of P is
zero.
wt
24
Transformers
Transformers use mutual inductance to change
voltages
Primary
Secondary
Power is conserved, though (if 100 efficient.)
25
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?
26
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)