Circuits with inductors
and alternating currents
Chapter 20 #45, 46, 47, 49
RL circuits Ch. 20 (last section)
Symbol for inductor looks like a spring.
An inductor is a circuit element that has a large
inductance. (usually a coil of wire).
Because of the inductance in a RL circuit, the
current changes gradually. When you close a
switch, the current takes time to go from zero
to its maximal value.
• The time it takes for the current to rise
depends on the time constant.
• In a RC circuit, time constant ( ) depended on the
resistance and the capacitance. = RC
• In a RL circuit, = L/R
• See graph of current vs. time on pg. 684
• After each time constant, the current rises to
63.2% of the remainder to the max value.
I
R
(1 e
t/
)
I
R
(1 e
Rt / L
)
RL circuit
The larger the time constant the longer it takes
for the current to reach its maximum value.
= L/R
increase inductance, increase
increase resistance, decrease
Note the difference between RL and RC
circuits.
Behavior of the inductor
Take a RL circuit with the switch open. (fig. 20.27)
When the switch is closed, the current is
changing the most rapidly. The back emf from
the inductor is largest at this time.
As the current approaches the steady state
value, the back emf is reduced.
After a long time, the inductor behaves like a
ordinary wire.
Do example 20.8.
Ch. 21 AC circuits
concept # 2, 8, 13, 15
problems# 1, 5, 7, 9, 11, 13, 14, 18,
19, 29, 37, 39, 43, 45, 47
AC circuit consists of an AC generator or other
AC source hooked up to electrical devices:
capacitors, resistors, inductors
The voltage alternates between a maximum
and minimum value.
The output is a sinusoidal wave
v = Vmaxsin 2 ft
v = instantaneous voltage
f = frequency at which the voltage changes
t = time
The current spends just as much time going
current in one direction as it does with the
current going the opposite direction. The
average current is zero.
RMS current (Irms)
The square root of the mean of the square
Irms= I m ax 0.707 I
2
m ax
For the AC circuit with the resistor. The current
and the voltage peak at the same time.
See fig. on pg 697 for a AC circuit with a resistor.
They are said to be ‘in phase’.
applying Ohm’s law
Vrms = Vm ax 0.707 Vm ax
2
Vrms = IrmsR
Average Power: Pave
and
I
Vmax = ImaxR
2
rms
R
Capacitors in AC circuit
Circuit with AC source and capacitor:
At initial time there is no charge on the capacitor. The
current is maximum then because there is no charge to
fight the new charges from moving onto the plates.
The current decreases as the charge build up. (the voltage
across capacitor increases)
When the current reverses direction, the voltage drops
because the plates are losing their charge.
This process repeats over an over again.
See the figures on page 700.
Capacitors in AC circuit
The current peaks before the voltage.
The current and voltage are not in phase.
The voltage lags behind the current by 900.
Impeding effect of a capacitor is called the
capacitance reactance Xc.
Xc = 1/(2 fC)
Xc depends on the fequency
Vc,rms = Irms Xc
Inductors in AC circuit
Inductor is a coil and due to Faraday’s law the
inductor impedes a changing current.
The effective resistance of an inductor is called
the inductive reactance, XL.
XL = 2 fL
depends on frequency.
Because of the inductance, the current lags
behind the voltage. They are out of phase.
See fig. 21.7 on pg 701.
The voltage peaks before the current.
RLC circuit
Now a resistor, capacitor, and inductor are all
in an AC circuit.
Because the voltages across all the
components are not in phase, we cannot just
add them together. They need to be added
like vectors. See figures 21.9 and 21.11.
The total impedance, Z, of the circuit is
defined as: Z R 2 ( X L X C )2
Vmax = ImaxZ
see table 21.2
Phase shift between potential difference and
current is .
tan
see figure 21.11
XL
XC
R
Power in AC circuit
Energy stored in capacitor is PEc = ½ C( Vmax)2
During half of a cycle the capacitor is charged.
During the other half the charge returns to the
voltage source.
Average power supplied by the source is zero.
No power losses occur in capacitor in AC circuit.
Energy stored in inductor is PEL = ½ L(Imax)2
The current source does work against the back emf
of the inductor. When the current decreases the
stored energy is returned to the source as the
inductor attempts to maintain the current in the
circuit.
There is no power loss through the inductor.
All the power in RLC circuit is converted to internal
energy in the resistor.
2
In RLC circuit
Pav I rms R
power
Pav
I
2
rms
R
VR
Pav
Pav
I rms VR
Vrms cos
I rms Vrms cos
cos = power factor
The power depends on the phase shift between the
current and voltage.
If a large generator has a large inductance,
capacitors can be used to shift the phase.
Resonance in series RLC circuit.
I rms
Vrms
Z
Vrms
R
2
(XL
XC )
2
When the impedance has its minimum value the
current is maximized. This happens when:
XL = XC
Then Z = R
This happens when the frequency of the circuit is
just right. Called the resonance frequency.
Resonance frequency.
To find the resonance frequency, f0.
Set the XL = XC
2 f0L = 1/(2 f0C)
f0
1
2
LC
See fig 21.13. At f = f0, the current spikes. If the
resistance was zero, the current would become
infinite.
Transformer
Used to change the magnitude of an AC voltage.
Works on basis of Faraday’s Law.
Simple transformer is made up of 2 coils
wrapped around a soft iron core.
When the current through the primary coil
changes, the flux through the secondary coil
changes producing an induced emf.
The voltages in each coil is proportional to the
number of turns in the coil.
Transformer
V1 = -N1
B/
t
and
V2 = -N2
B/
t
Assuming no flux is lost in the iron core, the
term
B/ t is common to both coils.
V2 = (N2/N1) V1
When N2 >N1 the voltage is increased, step up
transformer.
When N2 < N1 the voltage is decreased, step
down transformer.
The power input to the primary coil equals the
power output at the secondary coil for an
ideal transformer. (No power loss)
I1 V1 = I2 V2
So for a step up transformer the voltage is
increased, but the current is decreased.
In a real transformer, there is some power loss
due to eddy currents in the iron core. Real
transformers have efficiencies ranging from
90% - 99%.
Power lost due to the currents in transmission
lines is lost as resistive heating.
P = I2R
Want the current to be as small as possible to
reduce power loss.
More economical to transmit electric power
using high voltage and low current.