Chapter 24
Time-Varying Currents and
Fields
AC Circuit


An AC circuit consists of a combination of
circuit elements and an AC generator or
source
The output of an AC generator is
sinusoidal and varies with time according
to the following equation
• v = v0 sin 2ƒt



v is the instantaneous voltage
V0 is the maximum voltage of the generator
ƒ is the frequency at which the voltage changes, in
Hz
Root Mean Square (rms)
You have a variable x. Square, take average, and
put square-root.
xrms 
xrms 
x  xo sin( 2ft)
x2
AV
x
2
AV
xrms 
xo
 0.707 xo
2
Use rms values of v and i for AC to evaluate DC equivalent
Power.
170 V
-170 V
Period = 1/60 = 16.7 ms
From the wall outlet, you will see voltage signal like this with
an amplitude of 170 V. However, when you estimate average
Power, use vrms = 170x0.707 = 120 V.
Case1: Seinfeld used a heater connected to a wall-outlet
for 30 min.
Case2: Kramer somehow found a road-kill DC voltage source
which produces 120 V. Kramer sneaked in Jerry’s place
and took his heater and run for 30 mins.
Used the same amount of electric power and $$!!!
R
C
L
http://www.educatorscorner.com/
Resistor in an AC Circuit




Consider a circuit
consisting of an AC
source and a resistor
The graph shows the
current through and the
voltage across the
resistor
The current and the
voltage reach their
maximum values at the
same time
The current and the
voltage are said to be in
phase
More About Resistors in an AC
Circuit


The direction of the current has no effect
on the behavior of the resistor
The rate at which electrical energy is
dissipated in the circuit is given by
• P = i2 R



where i is the instantaneous current
the heating effect produced by an AC current with a
maximum value of Imax is not the same as that of a
DC current of the same value
The maximum current occurs for a small amount of
time
rms Current and Voltage

The rms current is the direct current
that would dissipate the same
amount of energy in a resistor as is
actually dissipated by the AC current
I  I rms

i0

 0.707 i0
2
Alternating voltages can also be
discussed in terms of rms values
v0
V  Vrms 
 0.707 v0
2
Ohm’s Law in an AC Circuit

rms values will be used when discussing
AC currents and voltages
• AC ammeters and voltmeters are designed
to read rms values
• Many of the equations will be in the same
form as in DC circuits

Ohm’s Law for a resistor, R, in an AC
circuit
•V=IR

Also applies to the maximum values of v and i
Capacitors in an AC Circuit


Consider a circuit containing a capacitor
and an AC source
The current starts out at a large value and
charges the plates of the capacitor
• There is initially no resistance to hinder the
flow of the current while the plates are not
charged

As the charge on the plates increases, the
voltage across the plates increases and
the current flowing in the circuit decreases
More About Capacitors in an AC
Circuit



The current reverses
direction
The voltage across
the plates decreases
as the plates lose
the charge they had
accumulated
The voltage across
the capacitor lags
behind the current
by 90°(or T/4)
v
V
A
AC
i
v= vosin(wt)
Q=CV
i= C w vo
cos(wt)
io
io = Cwvo
vo = (1/wC)io
V lags 90 deg behind i.
Capacitive Reactance and
Ohm’s Law

The impeding effect of a capacitor on the
current in an AC circuit is called the
capacitive reactance and is given by
1
XC 
2 ƒC
• When ƒ is in Hz and C is in F, XC will be in
ohms

Ohm’s Law for a capacitor in an AC circuit
• V = I XC
RC Circuits




A DC circuit may contain capacitors and
resistors, the current will vary with time
When the circuit is completed, the
capacitor starts to charge
The capacitor continues to charge until it
reaches its maximum charge (Q = Cε)
Once the capacitor is fully charged, the
current in the circuit is zero
Charging Capacitor in an RC
Circuit

The charge on the capacitor varies with
time
• q = Q(1 – e-t/RC)
• i=i0 e-t/RC
• The time constant, =RC


The time constant represents the time
required for the charge to increase from
zero to 63.2% of its maximum
i=i0 e-1 = 0.368 i0
Discharging Capacitor in an RC
Circuit

When a charged
capacitor is placed in
the circuit, it can be
discharged
• q = Qe-t/RC


The charge decreases
exponentially
At t =  = RC, the
charge decreases to
0.368 Qmax
• In other words, in one
time constant, the
capacitor loses 63.2%
of its initial charge
RC Circuit
-
Q=CV
dQ/dt = C (dV/dt)
dV/dt = (1/C) (dQ/dt)
V
I
Constant I-source
Q
V
+
+
+
+
+
+
slope
Vc
Vc
V
V=0
t = 0:
Vc = 0
I0 = (V – Vc)/R = V/R
t = t1:
Vc = V1 (>0)
I1 = (V –V1)/R (< I0)
t = t2:
Vc = V2 (> V1 >0)
t
I2 = (V – V2)/R (< I1 < I2)
Level
Time
Vc
Vc = V (1 – e-t/RC)
V
0.63V
t = RC
Vc = V (1 – e-1)
= 0.63 V
RC: time constant
t
Vc
[RC] = [(V/I)(Q/V)]
= [Q/I]
= C/(C/s)
=s
V
Vc = V e-t/RC
0.37V
RC
t
Q. A 100 microF capacitor is fully charged with 5 V DC source.
This capacitor is discharged through 10 K resistor for 1 s.
How much of charge is left in the capacitor in C?
Total amount of charge: Q = C V = (100 x 10-6 F)(5 V)
= 5 x 10-4 C
Time constant: t = RC = (100 x 10-6 F)(10 x 103 Ohm)
=1s
Since Q is proportional to V, after one time constant
(100 – 63)% of initial charge is left: Q = 0.37 (5 x 10-4) C
Self-inductance

Self-inductance occurs when the changing
flux through a circuit arises from the
circuit itself
• As the current increases, the magnetic flux
through a loop due to this current also
increases
• The increasing flux induces an emf that
opposes the current
• As the magnitude of the current increases, the
rate of increase lessens and the induced emf
decreases
• This opposing emf results in a gradual increase
of the current
Self-inductance cont

The self-induced emf must be proportional
to the time rate of change of the current
di
  L
dt
• L is a proportionality constant called the
inductance of the device
• The negative sign indicates that a changing
current induces an emf in opposition to that
change
Self-inductance, final


The inductance of a coil depends on
geometric factors
The SI unit of self-inductance is the
Henry
• 1 H = 1 (V · s) / A
For infinitely long solenoid
B = monI
n: number of turns/m
http://www.bugman123.com/Physics/Physics.html