Capacitance
• Capacitance occurs whenever electrical
conductors are separated by a dielectric,
or insulating material.
• Applying a voltage to the conductors can
displace the charge within the dielectric.
• Current does not actually flow through the
dielectric.
ECE 201 Circuit Theory I
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Capacitance
• Capacitor
– Structured as two parallel plates
ECE 201 Circuit Theory I
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Capacitor
• Circuit Symbol (a)
• Component
designation (C)
• Units – Farads
– Usually μF or pF
• Reference directions
for voltage and
current (b)
ECE 201 Circuit Theory I
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Voltage-Current Relationship
dv
i C
dt
ECE 201 Circuit Theory I
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Observations
dv
i C
dt
The voltage across a capacitor cannot change
instantaneously (the current would be infinite).
If the voltage across the terminals is constant, the
current will be zero. (looks like an open circuit).
Only a time-varying voltage can produce a
displacement current.
ECE 201 Circuit Theory I
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Express the voltage across the capacitor as
a function of the current
dv
i C
dt
1
idt  Cdv  dv  idt
C
1
 dx   id
C
1
v(t )   id  v(t )
C
1
v(t )   id  v(0)
C
v (t )
t
v ( t0 )
t0
t
t0
0
t
0
ECE 201 Circuit Theory I
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Power and Energy for the Capacitor
dv
p  vi  Cv
dt
1

p i
 id  v (t
C
dw
dv
 Cv
dt
dt
dw  Cvdv
t
t
w
 C  ydy
0
0
 dx
0
)

v
1
w  Cv
2
2
ECE 201 Circuit Theory I
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Example 6.4
• A voltage pulse described as follows is
applied across the terminals of a 0.5μF
capacitor:
v(t )  0V olts, t  0s
v(t )  4tV olts, 0s  t  1s
v(t )  4e V olts, t  1s
 ( t 1)
ECE 201 Circuit Theory I
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Derive the expressions for the capacitor
current, power, and energy.
dv
i C
dt
i  (0.5  10 )(0)  0, t  0s
6
i  (0.5  10 )(4)  2 A, 0s  t  1s
6
i  (0.5  10 )( 4e
6
 ( t 1)
)  2e
ECE 201 Circuit Theory I
 ( t 1)
A, t  1s
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Power
p  vi
p  0, t  0s
p  (4t )(2 A )  8t W , 0s  t  1s
p  (4e
 ( t 1)
)( 2e
 ( t 1)
A )  8e
ECE 201 Circuit Theory I
2 ( t 1)
W , t  1s
10
Energy
1
w  Cv
2
2
w  0, t  0s
1
w  (0.5  10 )(16t )  4t J , 0s  t  1s
2
1
w  (0.5  10 )(16e )  4e J , t  1s
2
6
2
6
2 ( t 1)
ECE 201 Circuit Theory I
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2 ( t 1)
11
Energy is being stored whenever the
power is positive.
Energy is being delivered by the
capacitor whenever the power is
negative.
ECE 201 Circuit Theory I
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