Transcript Lecture 8

Lecture #8 Circuits with Capacitors
•Circuits with Capacitors
•Next week, we will start exploring semiconductor
materials (chapter 2).
Reading:
Malvino chapter 2 (semiconductors)
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1
Applications of capacitors
Capacitors are used to store energy:
– Power supplies
Capacitors are used to filter:
• Block steady voltages or currents
– Passing only rapid oscillations
• Block fast variations
– Remove ripple on power supplies
Capacitance exists when we don’t want it!
– Parasitic capacitances
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Capacitors and Stored Charge
• So far, we have assumed that electrons keep on moving around and
around a circuit.
• Current doesn’t really “flow through” a capacitor. No electrons can go
through the insulator.
• But, we say that current flows through a capacitor. What we mean is that
positive charge collects on one plate and leaves the other.
• A capacitor stores charge. Theoretically, if we did a KCL surface around
one plate, KCL could fail. But we don’t do that.
• When a capacitor stores charge, it has nonzero voltage. In this case, we
say the capacitor is “charged”. A capacitor with zero voltage has no
charge differential, and we say it is “discharged”.
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Capacitors in circuits
• If you have a circuit with capacitors, you
can use KVL and KCL, nodal analysis, etc.
• The voltage across the capacitor is related
to the current by a differential equation
instead of Ohms law.
dV
iC
dt
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CAPACITORS
+V 
|(
C
i(t)
capacitance is defined by
dV
iC
dt
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dV i
So

dt C
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CAPACITORS IN PARALLEL
+
C2
|(
C1
|(
i(t)
i( t )  C1
V

Clearly, Ceq  C1  C2
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i(t)
Ceq
|(
Equivalent capacitance defined by
dV
i  C eq
dt
dV
dV
 C2
dt
dt
+
V(t)

CAPACITORS IN PARALLEL
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CAPACITORS IN SERIES
+ V 1  + V2 
i(t)
i  C1
|(
|(
C1
C2
dV1
dV
 C2 2
dt
dt
dV1
i
So

,
dt C1
+ Veq 
|(
Equivalent to
i(t)
Ce
q
Equivalent capacitance defined by
dVeq
d(V1  V2 )
Veq  V1  V2 and i  C eq
 C eq
dt
dt
dV2
i

,
dt
C2
so
dVeq
dt
 i(
1
1
i
 )
C1 C 2
C eq
1
CC
 1 2
CAPACITORS IN SERIES
Clearly, Ceq 
1
1 C1  C2

C1 C2
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Charging a Capacitor with a constant current
+ V(t)

|(
C
i
dV(t) i

dt
C
t
t
dV(t)
i
0 dt dt  0 C dt
voltage
i
i t
V(t)   dt 
C
C
0
t
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time
8
Discharging a Capacitor through a resistor
 V(t)
+
i
C
i
R
dV(t)
i(t)
V(t)


dt
C
RC
This is an elementary differential equation, whose
solution is the exponential:
V (t )  V0 e
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 t /
Since:
d t / 
1 t / 
e
 e
dt

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Voltage vs time for an RC
discharge
1.2
Voltage
1
0.8
0.6
0.4
0.2
0
0
1
2
3
4
Time
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RC Circuit Model
The capacitor is used to model the response of a digital circuit
to a new voltage input:
The digital circuit is modeled by
a resistor in series with a capacitor.
+
_
R
Vout
Vin
The capacitor cannot
change its voltage instantly,
as charges can’t jump instantly
to the other plate, they must go through the circuit!
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+
C Vout
_
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RC Circuit Model
Every digital circuit has natural resistance and capacitance. In
real life, the resistance and capacitance can be estimated
using characteristics of the materials used and the layout of the
physical device.
R
Vout
The value of R and C
+
+
for a digital circuit
Vout
Vin
C
_
determine how long it will
_
take the capacitor to change its
voltage—the gate delay.
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RC Circuit Model
With the digital context in
mind, Vin will usually be a
time-varying voltage that
Vin
switches instantaneously
between logic 1 voltage and
logic 0 voltage.
R
Vout
+
C Vout
_
+
_
t=0
We often represent this
switching voltage with a
switch in the circuit
diagram.
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i
+ V =5V
s

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+
Vout
–
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Analysis of RC Circuit
R
• By KVL,
 Vin  RI  Vout  0
Vin
+
_
Vout
I
+
C Vout
_
• Using the capacitor I-V relationship,
dVout
 Vin  RC
 Vout  0
dt
• We have a first-order linear differential equation for the
output voltage
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Analysis of RC Circuit
R
• What does that mean?
Vin
• One could solve the
differential equation using
Math 54 techniques to get
+
_
Vout
I
+
C Vout
_

t
/(
RC
)
Vout (t )  Vin  Vout (0)  Vin e
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Insight
Vout (t )  Vin  Vout (0)  Vin et /(RC)
• Vout(t) starts at Vout(0) and goes to Vin asymptotically.
• The difference between the two values decays exponentially.
• The rate of convergence depends on RC. The bigger RC is,
the slower the convergence.
Vout
Vout(0)
Vout
Vin
bigger RC
Vin
Vout(0)
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0
0
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time
0
16
0
time
Time Constant
Vout (t )  Vin  Vout (0)  Vin et /(RC)
• The value RC is called the time constant.
• After 1 time constant has passed (t = RC), the above works out to:
Vout (t )  0.63 Vin  0.37 Vout (0)
• So after 1 time constant, Vout(t) has completed 63% of its transition, with
37% left to go.
• After 2 time constants, only 0.372 left to go.
Vout
Vout
Vout(0)
Vin
.63 V1
.37 Vout(0)
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0
0

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time
0
0

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time
Transient vs.
Steady-State
R
Vin
+
_
Vout
I
+
C Vout
_
• When Vin does not match up with Vout , due to an abrupt
change in Vin for example, Vout will begin its transient
period where it exponentially decays to the value of Vin.
• After a while, Vout will be close to Vin and be nearly
constant. We call this steady-state.
• In steady state, the current through the capacitor is
(approx) zero. The capacitor behaves like an open
circuit in steady-state.
• Why? I = C dVout/dt, and Vout is constant in steady-state.
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General RC Solution
• Every current or voltage (except the source voltage)
in an RC circuit has the following form:

(
t

t
)
/(
RC
)



0
x( t )  x f   x( t0 )  x f e


• x represents any current or voltage
• t0 is the time when the source voltage switches
• xf is the final (asymptotic) value of the current or
voltage
All we need to do is find these values and plug in to
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solve
for any current orEEvoltage
in 8an RC circuit.
19
Solving the RC Circuit
We need the following three ingredients to fill in our equation for
any current or voltage:
• x(t0+) This is the current or voltage of interest just after the
voltage source switches. It is the starting point of our transition,
the initial value.
• xf This is the value that the current or voltage approaches as t
goes to infinity. It is called the final value.
• RC This is the time constant. It determines how fast the current
or voltage transitions between initial and final value.
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Finding the Initial Condition
To find x(t0+), the current or voltage just after the switch, we use the
following essential fact:
Capacitor voltage is continuous; it cannot jump when a switch occurs.
So we can find the capacitor voltage VC(t0+) by finding VC(t0-), the voltage
before switching.
We can assume the capacitor was in steady-state before switching. The
capacitor acts like an open circuit in this case, and it’s not too hard to find
the voltage over this open circuit.
We can then find x(t0+) using VC(t0+) using KVL or the capacitor I-V
relationship. These laws hold for every instant in time.
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Finding the Final Value
To find xf , the asymptotic final value, we assume that the circuit will be in
steady-state as t goes to infinity.
So we assume that the capacitor is acting like an open circuit. We then find the
value of current or voltage we are looking for using this open-circuit
assumption.
Here, we use the circuit after switching along with the open-circuit
assumption.
When we found the initial value, we applied the open-circuit assumption to the
circuit before switching, and found the capacitor voltage which would be
preserved through the switch.
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Finding the Time Constant
It seems easy to find the time constant: it equals RC.
But what if there is more than one resistor or
capacitor?
R is the Thevenin equivalent resistance with respect
to the capacitor terminals.
Remove the capacitor and find RTH. It might help to
turn off the voltage source. Use the circuit after
switching.
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Alternative Method
Instead of finding these three ingredients for
the generic current or voltage x, we can
• Find the initial and final capacitor voltage
(it’s easier)
• Find the time constant (it’s the same for
everything)
• Form the capacitor voltage equation VC(t)
• Use KVL or the I-V relationship to find x(t)
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Computation with Voltage
When we perform a sequence of computations using
a digital circuit, we switch the input voltages between
logic 0 and logic 1.
The output of the digital circuit fluctuates between logic 0 and logic 1
as computations are performed.
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We compute with pulses
voltage
RC Circuits
We send beautiful pulses in
voltage
But we receive lousy-looking
pulses at the output
time
time
Capacitor charging effects are responsible!
Every node in a circuit has natural capacitance, and it is the charging of
these capacitances that limits real circuit performance (speed)
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