Lec. (4)
Chapter (2) AC- circuits
Capacitors and transient current
1
2.1 – Introduction
Capacitor displays its true
characteristics only when a
change in voltage is made in the
network.
A capacitor is constructed of two
parallel conducting plates separated
by an insulator.
Capacitance is a measure of a
capacitor’s ability to store charge on
its plates.
Transient voltages and currents
result when circuit is switched
2
2.2 – Capacitance
o Capacitance is a measure of a capacitor’s ability to
store charge on its plates.
o A capacitor has a capacitance of 1 farad (F) if 1
coulomb (C) of charge is deposited on the plates
by a potential difference of 1 volt across its plates.
o The farad is generally too large a measure of
capacitance for most practical applications, so the
microfarad (106 ) or picofarad (1012 ) is more
commonly used.
3
Q = C V
where C = eo A / d
for a parallel plate capacitor,
where eo is the permittivity of
the insulating material
(dielectric) between plates.
A charged parallel
plate capacitor.
Recall that we used Gauss's Law
to calculate the electric field (E)
between the plates of a charged
capacitor:
E = s / eo where there is a
vacuum between the plates.
Vab = E d, so E = Vab /d
The unit of capacitance is called the Farad (F).
4
2.2 – Capacitance
 Fringing – At the edge of the capacitor
plates the flux lines extend outside the
common surface area of the plates.
2.3 Types of Capacitors:
1.
2.
Fixed:
mica, ceramic, electrolytic, tantalum and polyester-film
Variable Capacitors:
The capacitance is changed by turning the shaft at one end to vary
the common area of the movable and fixed plates. The greater the
common area the larger the capacitance.
6
The potential energy stored in the system of
positive charges that are separated from the
negative charges is like a stretched spring that
has potential energy associated with it.
Capacitors can store charge and ENERGY
DU = q DV
and the potential V increases as the charge is
placed on the plates
7
(V = Q / C)
Since the V changes as the Q is increased, we have to integrate over
all the little charges “dq” being added to a plate:
DU = q DV
Q
u   Vdq , V 
C
q dq

C
Q
1
  qdq
C0
Q2

,
2C
Q  CV
1
Q
2
u  CV , C 
2
V
1
u  QV
2
8
Energy density:
1
2
u  eE
2
This is an important result because it tells us
that empty space contains energy if there is
an electric field (E) in the "empty" space.
If we can get an electric field to travel
(or propagate) we can send or transmit
energy and information through
empty space!!!
9
The charges induced on the surface of the
dielectric (insulator) reduce the electric field.
10
Q1. 2
You slide a slab of dielectric between the plates of a parallel-plate
capacitor. As you do this, the charges on the plates remain
constant.
What effect does adding the dielectric have on the energy stored
in the capacitor?
U α E2
A. The stored energy increases.
B. The stored energy remains the same.
C. The stored energy decreases.
D. not enough information given to decide
11
2.5 Capacitors are in Series:
When capacitors are in series,
the charge is the same on each capacitor.
Vt  V1  V2  V3
Qt Q1 Q2 Q3
 

Ct C1 C2 C3
Qt  Q1  Q2  Q3
1
1
1
1
 

Ct C1 C2 C3
Q
Q  CV  V 
C
2.5 Capacitors are in Parallel
When capacitors are in parallel ,
the total charge is the sum of that on each capacitor.
Qt  Q1  Q2  Q3
CtVt  C1V1  C2V2  C3V3
Vt V1  V2  V3
Ct  C1  C2  C3
Q  CV
• Charging a capacitor that is discharged
– When switch is closed, the current instantaneously
jumps to E/R
– Exponentially decays to zero
• When switching, the capacitor looks like a short circuit
• Voltage begins at zero and exponentially increases to E
volts
• Capacitor voltage
• cannot change instantaneously
2.5 – Initial Conditions
The voltage across a capacitor at the instant of the
start of the charging phase is called the initial value.
Once the voltage is applied the transient phase will
commence until a leveling off occurs after five time
constants called steady-state as shown in the figure.
RC Circuits
RC-Circuit: Resistance R and capacitance C in series with a source of emf V.
R
a
- -
Start charging capacitor. . .
Applying KVL
C
b
V
i
+
+
V
+
+
b
R
a
- -
 e   iR
q
V   IR
C
q
C
C
RC Circuit: Charging Capacitor
q
V
 IR
C
R
a
V
i
+
+
b
q
V
 IR
C
q
dq
V
R
C
dt
- -
Rearrange terms to place in differential form:
Multiply by C dt :
dt
dq


RC (CV  q)
t

0
q
dt
dq

RC 0 (CV  q)
q
C
C
RC Circuit: Charging Capacitor
t

0
q
dt
dq

RC 0 (CV  q)
t
RC
t
RC
t
RC
t
RC
  ln( CV  q ) 0
q
  ln( CV  q )  ln CV
 ln CV  ln( CV  q )
CV
 ln
CV  q
CV  q  CVe
t
RC
Instantaneous charge q on a
charging capacitor:
q  CV (1  e
t
RC
)
RC Circuit: Charging Capacitor
q  CV (1  e
t
RC
)
At time t = 0: q = CV(1 - 1); q = 0
At time t = : q = CV(1 - 0); qmax = CV
The charge q rises from zero initially to its
maximum value qmax = CV
Example 2.1. What is the charge on a 4mF capacitor
charged by 12V for a time t = RC?
Qmax
Capacitor
q
Rise in Charge
R = 1400 W
b
V
i
+
+
0.63 Q
a
- -
t
Time, t
The time t = RC is known as
the time constant.
q  CV (1  e
4 mF
t
RC
1
)
q  CV (1  e )
e = 2.718
q  (4 10 6 F )(12V )(1  e 1 )
 48 10 6 (1  e 1 )
 30.3  10
6
C  30.3mC
Example 2.1 (Cont.) What is the time constant t?
Qmax
b
V
i
+
+
0.63 Q
Rise in Charge
R = 1400 W
a
Capacitor
q
- -
t
4 mF
Time, t
The time t = RC
is known as the time constant.
t = (1400 W)(4 mF)
t = 5.60 ms
In one time constant
(5.60 ms in this
example), the charge
rises to 63% of its
maximum value (CV).
RC Circuit: Decay of Current
As charge q rises, the current i will decay.
q  CV (1  e
t
RC
Current decay as a capacitor is
charged:
)
dq
i
dt
t
d
 (CV  CVe RC )
dt
t
CV RC

e
RC
V
i e
R
when t = 0
when t = 
imax 
i=0
t
RC
V
R
Current Decay
when t = 0
imax
when t = 
V
i
+
+
b
t
RC
i=0
R
a
V
i e
R
V

R
- -
q
C
C
I
i
Capacitor
Current Decay
0.37 I
t
Time, t
•
The placement of charge on the plates of a capacitor does not
occur instantaneously.
• Transient Period – A period of time where the voltage or
current changes from one steady-state level to another.
• The current ( ic ) through a capacitive network is essentially
zero after five time constants of the capacitor charging phase.
24
Steady State Conditions

Circuit is at steady state



When voltage and current reach their final values and
stop changing
Capacitor has voltage across it, but no current
flows through the circuit
Capacitor looks like an open circuit
25
Example 2.2. What is the current i after one time
constant (t  RC)? Given R=1400W and C=4mF.
V
i e
R
t
RC
The time t = RC is known as the time constant.
i.e t = t = RC
V 1
i e
R
i  0.37 imax
Charge and Current During the Charging
of a Capacitor.
Qmax
Capacitor
q
0.63 I
I
i
Capacitor
Current Decay
Rise in Charge
0.37 I
t
Time, t
t
Time, t
In a time t of one time constant, the charge q rises
to 63% of its maximum, while the current i decays
to 37% of its maximum value.
Capacitor Discharging



Assume capacitor has E volts across it when it begins
to discharge
Current will instantly jump to –E/R
Both voltage and current will decay exponentially to
zero
28
RC Circuit: Discharge
R
a
- -
b
C
V
i
+
+
V
+
+
b
R
a
- -
q
C
C
After C is fully charged, we turn switch to b, allowing it to discharge.
Discharging capacitor. . . loop rule gives:
 E   iR;
q
 iR
C
Negative because of
decreasing I.
29
Discharging From q0 to q:
Instantaneous charge q on discharging capacitor:
dq
q   RCi; q   RC
dt
t
t
ln q q 
o
RC
q
t
ln

qo
RC
b
V
i
q
C
+
+
dq
 dt
q q  0 RC
o
q
R
a
- -
q
q  qo e
t
RC
C
Discharging Capacitor
q  qo e
t
RC
dq
i
dt
t
d

(CVe RC )
dt
t
 CV RC

e
RC
Current i for a discharging capacitor.
b
V
i
q
C
+
+
Note qo = CV
and the instantaneous current is: dq/dt.
R
a
- -
V
i
e
R
t
RC
C
Prob. 2.3 How many time constants are needed
for a capacitor to reach 99% of final charge?
q  qmax (1  e
q
qmax
 1 e
0.99  1  e
t
RC
t
RC
)
q  CV (1  e
t
RC
t
RC
t
RC
e  0.01
t
 ln 0.01
RC
t
 4.61
RC
4.61 time
constants
)
Prob. 2.4. Find time constant, qmax, and time to
reach a charge of 16 mC if V = 12 V and C = 4 mF.
t  ? qmax  ? t  ?
q  16mF V  12V C  4mF
t  RC
qmax  CV
 (16 10 6 F )(12V )
 21.6m C
q  qmax (1  e
t
RC
)
b
V
12 V
R
i
1.8 mF
+
+
 (1.4 106 W)(1.8 106 F )
 2.52 S
1.4 MW
a
- -C
Prob. 2.4. continued
q  qmax (1  e
q
qmax
e
e
t
RC
t
RC
 1 e
 1
t
RC
)
t
q
 ln( 1 /(1 
))
RC
qmax
t
RC
t
16 mC
 ln( 1 /(1 
))
2.52
21.6 mC
t  2.52 ln( 1 / 0.259)
q
qmax
 1 /(1 
q
qmax
)
t
q
 ln( 1 /(1 
))
RC
qmax
t  3.4 S
Time to reach 16 mC: t=3.4S
2.12 – Energy Stored by a Capacitor
o The ideal capacitor does not
dissipate any energy supplied to
it. It stores the energy in the
form of an electric field between
the conducting surfaces.
o The power curve can be
obtained by finding the product
of the voltage and current at
selected instants of time and
connecting the points obtained.
o WC is the area under the curve.
2.14 – Applications
Capacitors find applications in:
o
o
o
o
Electronic flash lamps for cameras
Line conditioners
Timing circuits
Electronic power supplies
36
An RC Timing Application

RC circuits


Used to create delays for alarm, motor control, and
timing applications
Alarm unit shown contains a threshold detector

When input to this detector exceeds a preset value,
the alarm is turned on
37
An RC Timing Application

Pulses have a rise
and fall time


Because they do not
rise and fall
instantaneously
Rise and fall times
are measured
between the 10%
and 90% points
38
The Effect of Pulse Width

Width of pulse relative to a circuit’s time constant


If pulse width >> 5t



Capacitor charges and discharges fully
With the output taken across the resistor, this is a differentiator
circuit
If pulse width = 5t


Determines how it is affected by an RC circuit
Capacitor fully charges and discharges during each pulse
If the pulse width << 5t


Capacitor cannot fully charge and discharge
This is an integrator circuit
39
Simple Wave-shaping Circuits
Circuit (a) provides approximate
integration if 5t >>T
Circuit (b) provides approximate
differentiation if T >> 5t
40
Capacitive Loading(stray capacitance):

Stray Capacitance
Occurs when conductors are
separated by insulating material
 Leads to stray capacitance
 In high-speed circuits this can cause
problems

41