Chapter 28: Alternating Current

1
Chapter 28 Alternating Current

• Phasors and Alternating Currents

• Alternating current (AC current)

• Current which varies sinusoidally in time is
  called alternating current (AC) as opposed to
  direct current (DC). One example of AC current
  source is a coil of wire rotating with constant
  angular velocity in a magnetic field.

The symbol

is used to denote an AC source. In general a
source
means either a source of alternating current or
voltage.

• In the U.S. and Canada, commercial
  electric-power distribution system
• uses a frequency of f 60 Hz, corresponding
  to w 377 rad/s. In much
• of the rest of the world uses f 50 Hz. In
  Japan, however, the country is
• divided in two regions with f 50 Hz and 60
  Hz.

2
Phasors and Alternating Currents

• Phasors

w
phasor

• A convenient way to express a quantity
• varying sinusoidally with time is by
• a phasor in phasor diagram as shown.

IP
IIP sin wt
wt

• Rectifier and rectified current

O
-


-
3
Phasors and Alternating Currents

• Rectifier and rectified current (contd)

4
Phasors and Alternating Currents

• Root-mean-square current and voltage

• Root-mean-square current of a sinusoidal current

time averaged

• Root-mean-square voltage of a sinusoidal voltage

For 120-volt AC, V170 V.
5
Reluctance

• Resistance, inductance, capacitance and reactance

• Resistor in an AC circuit

Given
Þ
Voltage across R in phase with current through R
I
IR
VR
wt
uR
At time t
Iem/R
6
Reluctance

• Resistance, inductance, capacitance and reactance

• Inductor in an AC circuit

Given
I
Þ
L
Þ
Voltage across L leads current through L by
one-quarter cycle (90).
I
IL
e
VL
em
wt
Iem/(wL)
At time t
7
Reluctance

• Resistance, inductance, capacitance and reactance

• Capacitor in an AC circuit

Given
Þ
Þ
Voltage across C lags current through C by
one-quarter cycle (90).
I
IC
wt
em
e
wt
VC
IwCem
At time t
8
Reluctance

• LRC series circuit and reluctance

LRC circuit summary
Given
Assume the solution for current
(See derivation later)
amplitude
9
Reluctance

• LRC series circuit and reluctance (contd)

What is reactance?
fw/2p
You can think of it as a frequency-dependent
resistance.
For high ?, ?C0 - Capacitor looks like a wire
(short) For low ?, ?C?8 - Capacitor looks like
a break
For low ?, ?L0 - Inductor looks like a wire
(short) For high ?, ?L?8 - Inductor looks like
a break (inductors resist change in current)
10
LRC Circuits

• LRC series circuit (contd)

amplitude
This picture corresponds to a snapshot at
t0. The projections of these phasors along
the vertical axis are the actual values of the
voltages at the given time.
Im
11
LRC Circuits

• LRC series circuit (contd)

Problem Given Vdrive em sin ?t, find VR,
VL, VC, IR, IL, IC

• Strategy
• Draw Vdrive phasor at t0
• Guess iR phasor
• Since VR iR R, this is also the direction for
  the VR phasor.

-f
f
(No L or C ? f 0)

1. Realize that due to Kirchhoffs current law, iL
   iC iR (i.e., the same current flows through
   each).

12
LRC Circuits

• LRC series circuit (contd)

1. The inductor current IL always lags VL ? draw VL
   90 further counterclockwise.
2. The capacitor voltage VC always lags IC ? draw VC
   90 further clockwise.

-f
VC I XC
The lengths of the phasors depend on R, L, C, and
?. The relative orientation of the VR, VL, and
VC phasors is always the way we have drawn it.
f is determined such that VR VL VC e
(Kirchhoffs voltage rule) These are added like
vectors.
13
LRC Circuits

• Phasor diagrams for LRC circuits Example

amplitude of current
14
LRC Circuits

• Filters Example

Ex. C 1 µF, R 1O
15
LRC Circuits

• Filters

High- pass filter
?0 No current Vout 0 ?8 Capacitor
wire Vout e
? 8 No current Vout 0 ? 0 Inductor
wire Vout e
Low- pass filter
? 0 No current because of capacitor ? 8 No
current because of inductor
(Conceptual sketch only)
Band-pass filter
16
LRC Circuits

• Phasor diagrams for LRC circuits Example 2

Reluctance for inductor
Reluctance for capacitor
amplitude
ß
Impedance Z
17
LRC Circuits

• Phasor diagrams for LRC circuits Tips

• This phasor diagram was drawn as a snapshot of
  time t0 with the voltages being given as the
  projections along the y-axis.

f

• Sometimes, in working problems, it is easier to
  draw the diagram at a time when the current is
  along the x-axis (when I0).

f
f
From this diagram, we can also create a triangle
which allows us to calculate the impedance Z
f
f
18
Resonance in Alternating Current Circuits

• Resonance

For fixed R, C, L the current Im will be a
maximum at the resonant frequency w0 which makes
the impedance Z purely resistive.
i.e.
reaches a maximum when
X
X

L
C
This condition is obtained when
resonance frequency
Þ

• Note that this resonant frequency is identical to
  the natural frequency of the LC circuit by
  itself!
• At this frequency, the current and the driving
  voltage are in phase!

19
Resonance in Alternating Current Circuits

• Resonance (contd)

e

cos
f
I
m
m
R
Plot the current versus w, the frequency of the
voltage source ?
20
Resonance in Alternating Current Circuits

• Resonance (contd)

On Resonance
f0 and ZR
On resonance, the voltage across the reactive
elements is amplified by Q! Necessary to pick up
weak radio signals, cell phone transmissions, etc.
21
Power in Alternating Current Circuits

• Power

• The instantaneous power (for some frequency, w)
  delivered at time t is given by

• The most useful quantity to consider here is not
  the instantaneous power but rather the average
  power delivered in a cycle.

• To evaluate the average on the right, we first
  expand the sin(wt-f) term.

22
Power in Alternating Current Circuits

• Power

• Expanding,

• Taking the averages,

• Generally

• Putting it all back together again,

23
Power in Alternating Current Circuits

• Power

This result is often rewritten in terms of rms
values
24
Power in Alternating Current Circuits

• Power

Power, as well as current, peaks at w w0. The
sharpness of the resonance depends on the values
of the components. Recall
We can write this in the following manner (which
we wont try to prove)
introducing the curious factors Q and x...
25
Resonance in Alternating Current Circuits

• Power and resonance

where Umax is max energy stored in the system and
DU is the energy dissipated in one cycle
For RLC circuit, Umax is
Losses only come from R
period
This gives
And for completeness, note
26
Resonance in Alternating Current Circuits

• Power and resonance

For Q gt few,
FWHM Full Width at Half Maximum Q Quality of
the peak Higher Q sharper peak better quality
27
Transformers

• Transformers

• AC voltages can be stepped up or stepped down by
  the use of transformers.

The AC current in the primary circuit creates a
time-varying magnetic field in the iron
This induces an emf on the secondary windings due
to the mutual inductance of the two sets of
coils.

• The iron is used to maximize the mutual
  inductance. We assume that the entire flux
  produced by each turn of the primary is trapped
  in the iron.

28
Transformers

• Ideal transformer without a load

Nothing connected on secondary
No resistance losses
All flux contained in iron
The primary circuit is just an AC voltage source
in series with an inductor. The change in flux
produced in each turn is given by

• The change in flux per turn in the secondary
• coil is the same as the change in flux per
  turn in the primary coil (ideal case).
• The induced voltage appearing across the
  secondary coil is given by

• Therefore,
• N2 gt N1 ? secondary V2 is larger than primary V1
  (step-up)
• N1 gt N2 ? secondary V2 is smaller than primary
  V1 (step-down)
• Note no load means no current in secondary.
  The primary current,
• termed the magnetizing current is small!

29
Transformers

• Ideal transformer with a load

What happens when we connect a resistive load to
the secondary coil? Changing flux produced by
primary coil induces an emf in secondary which
produces current I2
iron
R
V
e
V

1
2
N
N
1
2
(secondary)
(primary)
This current produces a flux in the secondary
coil µ N2I2, which opposes the change in the
original flux -- Lenzs law
This induced changing flux appears in the
primary circuit as well the sense of it is to
reduce the emf in the primary, to fight the
voltage source. However, V1 is assumed to be a
voltage source. Therefore, there must be an
increased current I1 (supplied by the
voltage source) in the primary which produces a
flux µ N1I1 which exactly cancels the flux
produced by I2.
30
Transformers

• Ideal transformer with a load (contd)

Power is dissipated only in the load resistor R.
Where did this power come from? It could come
only from the voltage source in the primary
31
Exercises

• Exercise 1

Suppose em 100 volts, f1000 Hz, R10 Ohms,
L4.22 mH, Find XL, Z, I, VR, and Vl.
32
Exercises

• Exercise 2 Calculate power lost in R in
  Exercise 1

To calculate power produced by the generator you
need to take account of the phase difference
between the voltage and the current. In general
you can write
For an inductor P 0 because the phase
difference between current through the inductor
and voltage across the inductor is 90 degrees