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Passive components and circuits - CCP
Lecture 5
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Content
 Capacity




Properties
DC behavior
AC behavior
Transient regime behavior
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Web addresses
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http://en.wikipedia.org/wiki/RC_circuit
http://www.solarbotics.net/bftgu/starting_elect_pass_cap.html
http://www.standrews.ac.uk/~jcgl/Scots_Guide/info/comp/passive/capacit/capacit.htm
http://www.phy.ntnu.edu.tw/oldjava/rc/rc.htm
http://www.ac.wwu.edu/~vawter/PhysicsNet/Topics/DCCurrent/RCSeries.html
http://academic.evergreen.edu/projects/biophysics/technotes/electron/rc.ht
m#time_const
http://www.sciences.univnantes.fr/physique/perso/charrier/tp/rcrlrlc/index.html
http://zone.ni.com/devzone/cda/ph/p/id/217
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Electric capacitance - circuit element
 The electrical property of a
circuit element to store electric
charge when a voltage is
applied.
 Measurement unit: Farad [F]
 F, nF, pF
positive
charge
negative
charge
+
-
C
Vc
Q
C
V
dQ
IC
dV
dt
C

 IC  C
dV dV
dt
dt
dt
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Electric capacitance - circuit element
 Electronic component capacitor.
 It is characterized by its
capacity.
 For a capacitor, the
capacitance depends on
the geometrical size and
the dielectric properties
Terminal
d
Capacitor plate with
surface A
Dielectric characterized
by relative permitivity
r
Capacitor plate with
surface A
A
C   0 r
d
 0  8,85419 pF/m
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Stored energy
 The capacitor doesn’t dissipate power, but it is storing
energy when it is charging and is releasing the energy
when it is discharging:
T
T
T
dV
1
We   Pdt  VIdt  VC
dt  CV 2
dt
2
0
0
0
 We is energy stored when the capacitance is charging
to VC voltage or is discharged from VC to 0.
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Parallel connection
 The equivalent capacitance is equal
with the sum of capacitances
A
C1
C2
A
Cn
Cech
Qi
Q
Ci  ; Cech 
Vi
V
Q   Qi ; V  Vi

n
B
B
Cech   Ci
i 1
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Series connection
 The equivalent capacitance is given by the following
formula:
A
B
C1
C2
A
Cn
B
Cech
Qi
Q
Ci  ; Cech 
Vi
V
V   Vi ; Q  Qi

n
1
1

Cech i 1 Ci
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DC regime behavior
 In DC, the capacitor corresponds to an open-circuit.
V
R1
R2
C
A
AB
R1
A
B
i
V1
V
AB
B
i
C
DC
V2
dvC
dVAB
iC  C
C
0
dt
dt
R2
C =0
V1
V2
VAB  V1  V 2
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AC regime behavior
 In AC regime, the capacitance is equivalent with ZC
impedance. The sinusoidal voltage applied is considered
through the phasorial representation..
dvC
iC  C
; vC  V  e jt  e j
dt
d V  e j  t  e j
iC  C
 j   C  V  e j t  e j 
dt
vC
1
 j  C  vC   Z C 
iC
jC

1
X C  ZC 
C

Capacitance’s
reactance
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AC regime behavior
 Impedance (reactance) is frequency dependent.
 In alternative current, the imittances of the circuits with
capacitors are dependent of the signals frequency.
 The property of a circuit to pass or reject some
frequencies is called filtering.
Low-pass filter
High-pass filter
Band-pass filter
Band reject filter
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RC low-pass filter
vo
vo  vi  R  iC ; iC 
ZC
vi is a sinusoidal voltage with
frequency f = ω/2π (or ω=2π f)

R
v
i
i
C
C
Z
v
o
C
For R=1,6 K and C=100pF :
1
ZC
j C
vo 
 vi 
 vi
1
R  ZC
R
j C
v ( j)
1
H ( j )  o

vi ( j) 1  jRC
H ( jf ) 
1
1

1  j 2  f 1,6 107 1  j  f 106
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RC low-pass filter - frequency characteristics
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RC low-pass filter - frequency characteristics
 Answer to the following questions:
 What kind of representation is used for modulus-frequency and phasefrequency characteristics?
 What is the modulus-frequency characteristic slope for 100Hz – 100KHz
domain? And in10MHz – 100MHz domain?
 What is the phase difference between input and output voltages at the
1MHz?
 Represent, at a 1MHz frequency, the input voltage phasor and the output
voltage phasor. How does this representation look at 100MHz?
 At what frequency is the phase difference between input and output
φ=30○?
 How are the previous characteristics modified if R=16KΩ and C=100pF?
 How are the previous characteristics modified if R=1KΩ and C=1nF?
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RC high-pass filter
vi is a sinusoidal voltage with
frequency f = ω/2π (or ω=2π f)
vo  R  iC ; iC 
C
Z
v
i
i
C
vC vi  vo

ZC
ZC

C
R
For R=1,6 K and C=100pF:
v
o
R
R
vo 
 vi 
 vi
1
R  ZC
R
jC
v ( j )
jRC
H ( j )  o

vi ( j ) 1  jRC
j 2  f 1,6 107
j  f 106
H ( jf ) 

7
1  j 2  f 1,6 10
1  j  f 106
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High-pass RC filter - frequency characteristics
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High-pass RC filter - frequency characteristics
 Answer to the following questions:
 What kind of representation is used for modulus-frequency and
phase-frequency characteristics?
 What is the modulus-frequency characteristic slope for 100Hz –
100KHz domain? And in 10MHz – 100MHz domain?
 What is the phase difference between input and output voltages at
the 1MHz?
 Represent, at a 1MHz frequency, the input voltage phasor and the output
voltage phasor. How does this representation look at 100MHz?
 At what frequency is the phase difference between input and output
φ= =30○?
 How are the previous characteristics modified if R=16KΩ and
C=100pF?
 How are the previous characteristics modified if R=1KΩ and
C=2.2nF?
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The high frequencies behavior
 At high frequencies, the capacitive reactance is much lower
than resistances from the previous circuits. The capacitance
is equivalent with a short-circuit.
R
R
C
v
i
v VHF v
o
i
v =0
o
C
v
i
R
v
o
VHF
v
i
R
v =v
i
o
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Separation and pass capacitances
 In some circuits, the capacitances are used to
separate the DC components (DC - open-circuit)
between two circuits without affecting the signal
variation (AC short-circuit)
 In these situations, they are called separation
capacitances (separation of DC components). In
other situations, realizing the same functions, they
are called passing capacitances (for high frequency
signals).
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Capacitance behavior in transient regime
 In this case, the transient regime consist in the
modification of a DC circuit steady state in a new DC
steady state.
 During these modifications, the capacitance cannot be
considered open-circuit or short-circuit.
 The transient regime analysis presumes the determination
of the way of charging and discharging of the capacitance.
 In transient regime, the circuit operations are described by
differential equations.
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The capacitance charging from a constant
voltage source
 Considering the K switch on
position 1, the capacitance will
be discharged.
 At the time t=t0, the switch is
moved on position 2.
 After enough time, t, the
capacitance will be charged at
the E voltage.
 The transient regime is taking
place between these two DC
steady states.
2
R
K
1
E
v
R
i
C
C
v
C
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The capacitance charging from a constant
voltage source
TKV : E  vR  vC
2
dvC
E  R  iC  vC ; iC  C
dt
dvC
E  RC
 vC ;   RC
dt
dvC
E 
 vC
dt
vC (t )  vC ()  [vC (0)  vC ()]  e
vC (0)  vC (t  t0 )
  RC
vC ()  v(t  )
R
K
1
E

t

v
R
i
C
C
v
C
The solution of the
differential equation
Circuit time constant
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Voltage variation across the capacitance, vC
vC (0)  0; vC ()  E

t
vC (t )  E  (1  e  )
vR (t )  E  vC (t )  E  e

t

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Current variation through the capacitance,
iC
E  vC (t ) E 
iC (t ) 
 e
R
R
t
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Significance of the time constant
 If the transient process has the same slope like in origins
(initial moment), the final values of voltages and currents
will be obtain after a time equal with circuit time
constant.
 As can be seen in the previous figures, the charging
process continues to infinite.
 Practically, the transient regime is considered to be
finished after 3 (95% from the final values) or 5 (99%
from the final values).
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Example (E=1V, R=1KΩ, C=1nF)
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The capacitance discharging regime on a
resistance
 At the initial time, the switch is
considered on position 2. The
capacitance is charged to E voltage.
 At a reference time moment t=t0, The
K switch is moved on position 1.
 After enough time, t, the
capacitance is totally discharged.
2
R
K
1
E
v
R
i
C
C
v
C
 The transient regime is taking place
between these two DC steady states.
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The capacitance discharging regime on a
resistance
R
2
TKV : 0  vR  vC
dv
0  R  iC  vC ; iC  C C
dt
dv
0  RC C  vC ;   RC
dt
dv
0   C  vC
dt
vC (t )  vC ()  [vC (0)  vC ()]  e
vC (0)  vC (t  t0 )  E
vC ()  v(t  )  0
K
1
v
R
i
C
E

t

C
v
C
The solution of the differential
equation

t
vC (t )  E  e  ; vR (t )  vC (t )
t
vR (t )
E 
iC (t ) 
  e
R
R
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Example (E=1V, R=1KΩ, C=1nF)
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Charging the capacitance from a constant
current source
 The K switch is considered in
position 1. The capacitance is
considered initially charged to the
R1
voltage vC(0).
 At reference time t=t0, the K is
switched in position 2.
R
1
2
v
R
K
i
C
v
C
C
I
 The constant current source will
charge the capacitance with the
current I.
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Charging the capacitance from a constant
current source
t
dv (t )
1
iC  C C  vC (t )   iC (t )dt  vC (t0 ) 
dt
C t0
v (t)
C
t

1
I
I
dt

v
(
t
)

(t  t0 )  vC (t0 )
C 0

C t0
C
R
1
slope = I/C
v (0)
C
2
v
R
K
R1
i
C
C
I
v
C
t
0
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Observations
 The voltage across the capacitance will raise linearly.
 The charging slope (or discharging) is independent by
the value of resistance R (so, the resistor can misform
the circuit).
 Theoretically, the voltage across the capacitance can
raise infinitely . In these situations, we must take some
measures to limit the voltage on the capacitance.
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The RC circuits behavior when pulses are
applied
R
 Consider a pulses signal source
applied to a series RC circuit.
 In analyzing the circuit behavior, we
consider both voltages: the voltage
across the capacitor, vC(t), and the
voltage across the, vR(t).
v
R
i
C
v
I
C
v
C
 Applying this signal source, the
phenomena of charging and
discharging described to transient
regime are repetitive.
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Case A – the time constant is much lower
than pulses duration
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Case B – the time constant is much greater
than pulses duration
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Integrating circuit
 If the output voltage is the voltage across the capacitor,
the effect under the input signal is attenuation of edges,
similarly with the integration operation.
 In this situation,(when vO(t)=vC(t)), the circuit is called
integration circuit.
 The integration effect is higher in case B , when the time
constant is greater then the pulse duration.
 The integration function in transient regime corresponds
to low-pass filtering in AC regime.
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Derivative circuit
 If the output voltage is the voltage across the resistor,
the circuit effect under the input signal is an accentuation
of edges, similarly with derivative mathematical
operation.
 In this situation,(when vO(t)= vR(t)), the circuit is called
derivative circuit.
 The derivative effect is higher in case A , when the time
constant is lower then pulse duration.
 The derivative function in transient regime corresponds
to high-pass filtering in AC regime.
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Answer to the following questions
 How are the waveforms modified if the input source for the
RC series circuit is a pulses current source?
 Can the current source pulses be asymmetrical? (from 0
on the current axis)?
 How does the voltage across the capacitor (or resistor)
vary if the pulses have the same duration with the circuit
time constant? Make the analysis starting with the initial
time moment, when the capacitor is completed
discharged.
 For the homework problem from the end of lecture 2,
consider that the resistor R is replaced by a capacitance
C=10nF. Draw the waveform of the voltage across the
capacitance.
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