Chap. 11 Energy and Power in

Digital Circuits

Average Power in an RC Circuit

Power Dissipation in Logic Gates

CMOS Logic

RC Circuit with a Switch

Find energy dissipated in each cycle

Find the average power.

During T1

Energy Computation

Total energy provided by the source during T1

Energy stored in the capacitor:

Energy dissipated in the resistor:

– Independent of R

CRTifCV

eCVdteR

VVdtiVE

S

CR

T

S

TCR

t

SS

T

S

11

2

2

01

0)1( 1

1

11

1

2

2

1SC CVE

2

12

1SCVE

During T2

If we assume

Capacitor energy is fully discharged =

Energy dissipated by the resister:

- independent of R

CRT 22

2

22

1SCVE

Putting Together

Energy dissipated in each cycle

– assuming that Capacitor is fully charged and discharged during the period.

Average power

2

21 SCVEEE

fCVT

EP S

2

Power Dissipation of Inverter

During input is low

During input is high

Easy way to solve

Average power = static power + dynamic power

– Static power: independent of the frequency

– Dynamic power: charging-and-discharging of the capacitor

Average Power

Static power + dynamic power

Static power: Independent of the switching frequency.

Dynamic power: related to switching capacitor

ONL RR

LLL

ONL

dynamic

static

CR

T

R

RR

P

P

2

Example II.1

kRkR ONL 10,100

Worst static power dissipation

LR

SV

Some numbers

LLL

ONL

dynamic

static

CR

T

R

RR

P

P

2

Power Minimization

Dynamic power

– Reduce the supply voltage (5V -> 1.5V)• Change the voltage on need

– Reduce the capacitance

– Reduce the frequency• Turn off clock when not in use

Static power

– CMOS logic

PMOS

CMOS Logic

CMOS Inverter

No static power!

- No direct path from the supply to ground

Dynamic Power of CMOS

fCVP S

2

Static Power of CMOS

Leakage power

Transient current

INv

Ti

Some Examples

Example II.6: Processor SA27E

- # of gates = 3M, 25% activity

MHzfVVfFC SL 425,5.1,30

2)(#)( SLdynamic VfCgatesswitchingFractionP

Exercise

Estimate the power of the following processor (~ P-IV)

- # of gates = 25M, 25% activity

GHzfVVfFC SL 3,5.1,1

Summary

Total power = static power + dynamic power

CMOS does not exhibit static loss

Dynamic power

– Reduce the supply voltage

– Reduce the switching activities

– Turn off the logic not in use (clock gating)

2)(#)( SLdynamic VfCgatesswitchingFractionP

Chap. 12 Transients in Second-

order Circuits

LC Circuit

RLC Circuits

LC Circuit

Solutions

Find the particular solution

Find the form of homogeneous solution

Use initial condition to obtain the total solution

< example >

t

0V

)(tvI

Zero initial condition:

0,0 LC iv

Homogeneous Solution

General Form: st

H Aetv )(

LCjsAeeLCAs oo

stst 1,02

Find A1 and A2 from initial conditions

Total Solution

Total Solution: Plot

Undriven LC Circuit

Energy

Observations from undriven LC Circuit

1. LC circuits are capable of oscillation

2. Important time constant:

3. Characteristic impedance

LC

C

L

C

L

i

vi

Lv

CW

peak

peak

peakpeak

C

C

LCT 22

22

Example 12.1

)10,5.0(,100,1 tsomeatVvAiHLFC CL

RLC Circuits

Series RLC Circuit: second order circuit

)(1

)(1)()(

2

2

tvLC

tvLCdt

tdv

L

R

dt

tvdSC

CC

Homogeneous Solution

o

oo

tsts

C

s

LCL

Rss

eKeKtv

2

2,1

22

21

1,

2,02

)( 21

Overdamped case: real and distinct roots

Critically damped case: real and repeated roots

Underdamped case: complex conjugate roots

)(22

2,1 oo ws

)(2,1 oos

)(22

,2,1 ooddjs

Natural Response: overdamped case

5.1,1,121 oKK

Natural Response: Critically damped case

1,1,121 oKK

tt

C

stts

C teetvteKeKtv )()( 21

Natural Response: Underdamped case

2.0,1,121 oKK

Response to step input voltage

Zetao 4~2.0,1

Exercise: Parallel RLC Circuit

VVHLFCkR S 25,1,1,5

Find iL(t)assume zero initial state

Exercise

+

L

1kv(t)

= u(t)

+

-

vc(t)F

3

1

Find the range of L to make vC(t) to be overdamped

1

0 t

vs(t) = u(t)

Find the differential equation for capacitor voltage. Find the total response with the following initial condition. iL (0) = 4 mA, vC (0) = 0 (V)

2k

F

1H

1

4

2k

4mA

+

-

vc(t)

+-

12V

Exercise

Example 12.4 Ideal Switched Power Supply

+

-

CCv

Li

1S 2S

LV

1S

2S

Tcycle1

Li

Cv

(1) S1 on, S2 off

L

VT(2) S1 off, S2 on until inductorcurrent becomes zero

(3) S1 off, S2 off

Summary

LC Circuit

RLC Circuit

– Three kinds of transient responses• Over-damped

• Critically damped

• Under-damped

Chap. 13 Sinusoidal Steady State:

Imdepance and Frequency

Response

Sinusoidal Steady-state Response

Impedance

Frequency Response

Filter

Sinusoidal Response of RC Circuit

2)(

tjtj

iI

eeVtv

Find the particular solution to tj

ieV

Total solution

Sinusoidal Steady State (SSS)

Particular solution for sinusoidal input after transients have died

All information of SSS is contained in the complex amplitude

CV CV

RCj

Vi

1

Magnitude Plot

Phase Plot

Impedance Model

Impedance Model

Circuit corresponding to a complex exponential input

I

C

V

V

Series RLC Circuit

Exercise 1

+

1H2

1F10cos2t Cvfind

Find v(t) at steady state (time unit: msec)

+

1k 2k

+

-

v(t) 1Fsint cos2t

Exercise 2

What we are doing? The Big Picture

Frequency Response

Frequency Response of Basic Elements

RC and RL Circuits

RLC Circuits

Resonance

Equivalent impedance = Resistor

Circuit response is maximum

0 CL ZZ

Another Example

AM Radio Receiver

Selectivity

Quality Factor: Q

Quality Factor

Lower R, higher Q

Another interpretation

Find the resonance frequency and the equivalent impedance at resonance.

2k

0.2F 600

200mH

Z

Exercise

Summary

Sinusoidal Steady-state Response

Impedance

Frequency Response

Filter