9/17/2004 EE 42 fall 2004 lecture 8 1

Lecture #8 Circuits with Capacitors

•Circuits with Capacitors

•Next week, we will start exploring semiconductor materials (chapter 2).

Reading: Malvino chapter 2 (semiconductors)

9/17/2004 EE 42 fall 2004 lecture 8 2

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

9/17/2004 EE 42 fall 2004 lecture 8 3

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”.

9/17/2004 EE 42 fall 2004 lecture 8 4

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.

dt

dVCi

9/17/2004 EE 42 fall 2004 lecture 8 5

CAPACITORS

C

i

dt

dV So

capacitance is defined by

Ci(t)

| (V

+

dt

dVCi

9/17/2004 EE 42 fall 2004 lecture 8 6

CAPACITORS IN PARALLEL

C1i(t) C2

| ( | (

+

V

dtdV

CdtdV

C)t(i 21

Equivalent capacitance defined by

dt

dVCi eq

Ceqi(t)

| ( +

V(t)

Clearly, PARALLELIN CAPACITORS CCC 21eq

9/17/2004 EE 42 fall 2004 lecture 8 7

CAPACITORS IN SERIES

, C

i

dt

dV ,

C

i

dt

dV So

2

2

1

1

dtdV

Cdt

dVCi 2

21

1

Equivalent capacitance defined by

dt

)Vd(VC

dt

dVCi and VVV 21

eqeq

eq21eq

Clearly, SERIES IN CAPACITORS CC

CC

C1

C1

1C

21

21

21

eq

C1

V1

i(t)C2

| ( | (V2+ +

Ceqi(t)

| (Veq

+

Equivalent to

eq21

eq

C

i)

C

1

C

1i(

dt

dV so

9/17/2004 EE 42 fall 2004 lecture 8 8

Charging a Capacitor with a constant current

C

i

dt

dV(t)

Ci

| (V(t) +

C

i

dt

dV(t) t

0

t

0 dtdt

C

ti

C

iV(t)

t

0

dt

time

voltage

9/17/2004 EE 42 fall 2004 lecture 8 9

Discharging a Capacitor through a resistor

RC

V(t)

C

i(t)

dt

dV(t)

Ci

V(t) +

R

This is an elementary differential equation, whose solution is the exponential:

// 1

dt

d tt ee /0)( teVtV Since:

i

9/17/2004 EE 42 fall 2004 lecture 8 10

Voltage vs time for an RC discharge

0

0.2

0.4

0.6

0.8

1

1.2

0 1 2 3 4

Voltage

Time

9/17/2004 EE 42 fall 2004 lecture 8 11

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.

The capacitor cannot

change its voltage instantly,

as charges can’t jump instantly

to the other plate, they must go through the circuit!

Vout

R

CVin Vout

+_

+

_

9/17/2004 EE 42 fall 2004 lecture 8 12

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.

The value of R and C

for a digital circuit

determine how long it will

take the capacitor to change its

voltage—the gate delay.

Vout

R

CVin Vout

+_

+

_

9/17/2004 EE 42 fall 2004 lecture 8 13

RC Circuit Model

With the digital context in mind, Vin will usually be a time-varying voltage that switches instantaneously between logic 1 voltage and logic 0 voltage.

We often represent this switching voltage with a switch in the circuit diagram.

Vout

R

CVin Vout

+_

+

_

t = 0

i +

Vout

–

Vs = 5 V+

9/17/2004 EE 42 fall 2004 lecture 8 14

Analysis of RC Circuit

• By KVL,

• Using the capacitor I-V relationship,

• We have a first-order linear differential equation for the output voltage

Vout

R

CVin Vout

+_

+

_I

0outVRIinV

0outVdtoutdV

RCinV

9/17/2004 EE 42 fall 2004 lecture 8 15

Analysis of RC Circuit

• What does that mean?

• One could solve the

differential equation using

Math 54 techniques to get

Vout

R

CVin Vout

+_

+

_I

)RC/(teinV)0(outVinV)t(outV

9/17/2004 EE 42 fall 2004 lecture 8 16

Insight

• 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.

)RC/(teinV)0(outVinV)t(outV

time

Vout

00

Vout

0

Vin

time0

Vout(0) Vin

Vout(0)

bigger RC

9/17/2004 EE 42 fall 2004 lecture 8 17

Time Constant

• The value RC is called the time constant.• After 1 time constant has passed (t = RC), the above works out to:

• 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.

)RC/(teinV)0(outVinV)t(outV

)0(outV37.0inV63.0)t(outV

time

Vout

00

Vin

.63 V1

Vout

Vout(0)

time00

.37 Vout(0)

9/17/2004 EE 42 fall 2004 lecture 8 18

Transient vs.Steady-State

• 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.

Vout

R

CVin Vout

+_

+

_I

9/17/2004 EE 42 fall 2004 lecture 8 19

General RC Solution• Every current or voltage (except the source voltage) in an RC circuit has the following form:

• 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 solve for any current or voltage in an RC circuit.

)RC/()0tt(efx)0t(xfx)t(x

9/17/2004 EE 42 fall 2004 lecture 8 20

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.

9/17/2004 EE 42 fall 2004 lecture 8 21

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.

9/17/2004 EE 42 fall 2004 lecture 8 22

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.

9/17/2004 EE 42 fall 2004 lecture 8 23

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.

We will discuss how to combine capacitors in series and in parallel in the next lecture.

9/17/2004 EE 42 fall 2004 lecture 8 24

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)

9/17/2004 EE 42 fall 2004 lecture 8 25

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.

9/17/2004 EE 42 fall 2004 lecture 8 26

RC Circuits

Every node in a circuit has natural capacitance, and it is the charging of these capacitances that limits real circuit performance (speed)

We compute with pulses

We send beautiful pulses in

But we receive lousy-looking pulses at the output

Capacitor charging effects are responsible!

time

volt

age

time

volt

age