Lecture #8 Circuits with Capacitors

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

9/17/2004

Reading: Malvino chapter 2 (semiconductors)

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

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 3

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.

i  C dV dt 9/17/2004 EE 42 fall 2004 lecture 8 4

9/17/2004 i(t)

CAPACITORS

+ V | ( C  i  C dV dt capacitance is defined by So dV dt  i C EE 42 fall 2004 lecture 8 5

i(t)

CAPACITORS IN PARALLEL

C 1 C 2 + V  i ( t )  C 1 dV dt  C 2 dV dt Equivalent capacitance defined by dV i  C eq dt i(t) C eq + Clearly, C eq  C 1  C 2 CAPACITORS IN PARALLEL 9/17/2004 EE 42 fall 2004 lecture 8 6

i(t) + V 1  | ( C 1 + V 2 | ( C 2 

CAPACITORS IN SERIES

Equivalent to i(t) + V eq  | ( C e q i  C 1 dV 1 dt  C 2 dV 2 dt Equivalent capacitance defined by V eq  V 1  V 2 and i  C eq dV eq dt  C eq d(V 1  V 2 ) dt So dV dt 1  i C 1 , dV dt 2  i C 2 , so dV eq dt  1 i( C 1  1 C 2 )  i C eq Clearly, C eq 9/17/2004  1 1 C 1  1 C 2  C C 1 C 1  2 C 2 CAPACITORS IN SERIES EE 42 fall 2004 lecture 8 7

Charging a Capacitor with a constant current

+ V(t) | ( C  i 9/17/2004 t 0  dV(t) dt dV(t) dt

dt

  t 0  i C i C

dt

V(t)   t 0 i C

dt

 i  C t EE 42 fall 2004 lecture 8 voltage time 8

Discharging a Capacitor through a resistor

 V(t) + i i C R 9/17/2004 dV(t) dt   i(t)  C  V(t) RC This is an elementary differential equation, whose solution is the exponential:

V

(

t

) 

V

0

e



t

/  Since: d dt

e



t

/     1

e



t

/  EE 42 fall 2004 lecture 8 9

Voltage vs time for an RC discharge

Voltage 1.2

1 0.8

0.6

0.4

0.2

0 0 9/17/2004 1 2 EE 42 fall 2004 lecture 8 Time 3 4 10

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 The capacitor cannot

V in

change its voltage instantly, as charges can’t jump instantly + _ to the other plate, they must go through the circuit!

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

out

C +

V out

_ 11

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 V

out

The value of R and C for a digital circuit determine how long it will

V in

+ _ C

V

+ _

out

take the capacitor to change its voltage —the gate delay.

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

RC Circuit Model

+ _ R With the digital context in mind, V

in

will usually be a time-varying voltage that switches instantaneously between logic 1 voltage and logic 0 voltage.

V in t = 0

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

9/17/2004

+



V s

EE 42 fall 2004 lecture 8

= 5 V i

V

out

C +

V out

_

+ V out

13

–

• By KVL,

Analysis of RC Circuit

R V

out

 V in  RI  V out  0

V in

+ _ I C • Using the capacitor I-V relationship,  V in  RC dV out dt  V out  0 +

V out

_ • We have a first-order linear differential equation for the output voltage 9/17/2004 EE 42 fall 2004 lecture 8 14

Analysis of RC Circuit

R V

out

• What does that mean?

in

• One could solve the + _ I C differential equation using +

V out

_ Math 54 techniques to get V out ( t )  V in   V out ( 0 )  V in  e  t /( RC ) 9/17/2004 EE 42 fall 2004 lecture 8 15

Insight

V out ( t )  V in   V out ( 0 )  V in  e  t /( RC ) • V

out

(t) starts at V

out

(0) and goes to V

in

asymptotically.

• The difference between the two values decays exponentially.

• The rate of convergence depends on RC. The bigger RC is, the slower the convergence.

V in

V out V out

V out (0) bigger RC V out (0) 0

time

EE 42 fall 2004 lecture 8 V in 0 0

time

16

V out ( t

Time Constant

)  V in   V out ( 0 )  V in  e  t • The value RC is called the

time constant

.

/( RC ) • After 1 time constant has passed (t = RC), the above works out to: V out ( t )  0 .

63 V in  0 .

37 V out ( 0 ) • So after 1 time constant, V

out

(t) has completed 63% of its transition, with 37% left to go. • After 2 time constants, only 0.37

2

left to go.

V out

V

in

.63 V 1

V

V

out (0 out )

0 

time

.37 V

out (0 )

EE 42 fall 2004 lecture 8 0 0 

time

17

Transient vs.

Steady-State

V in

+ _ R I V

out

C +

V out

_ • When V

in

does not match up with V

out

change in V

in

for example, V

out

, due to an abrupt will begin its

transient period

where it exponentially decays to the value of V

in

.

• After a while, V

out

will be close to V

in

constant. We call this

steady-state

.

and be nearly • In steady state, the current through the capacitor is (approx) zero.

The capacitor behaves like an open circuit in steady-state

.

• Why? I = C dV

out

/dt, and V

out

9/17/2004 is constant in steady-state.

EE 42 fall 2004 lecture 8 18

General RC Solution

• Every current or voltage (except the source voltage) in an RC circuit has the following form: x ( t )  x f     x ( t 0  )  x f    e  ( t  t 0 ) /( RC ) • x represents any current or voltage • t 0 is the time when the source voltage switches • x f is the final (asymptotic) value of the current or voltage All we need to do is find these values and plug in to EE 42 fall 2004 lecture 8 19

Solving the RC Circuit

We need the following three ingredients to fill in our equation for any current or voltage: • x(t

0 +

) 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.

• x

f

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 20

Finding the Initial Condition

To find x(t 0 + ), 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

V

C

(t

0 +

) by finding V

C

(t

0 -

), 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(t 0 + ) using V

C

(t

0 +

) 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 21

Finding the Final Value

To find x

f

, 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 22

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 R TH . It might help to turn off the voltage source. Use the circuit

after switching

.

9/17/2004 EE 42 fall 2004 lecture 8 23 We will discuss how to combine capacitors in series

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 V

C

(t) • Use KVL or the I-V relationship to find x(t) 9/17/2004 EE 42 fall 2004 lecture 8 24

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 25

RC Circuits

We compute with pulses We send beautiful pulses in

time

But we receive lousy-looking pulses at the output

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