Chap. 11 Energy and Power in

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

C R T

if CV

e CV

dt R e

V V dt

i V E

S

C R

T S

T RC

t S

S T

S

1 1

2

2

0 1

0 (1 ¹ )

1

1 1

1













 

^{ }

2

2 1

S

C CV

E 

2

1 2

1

CVS

E 

During T2

If we assume

Capacitor energy is fully discharged = Energy dissipated by the resister:

- independent of R C R T₂  ₂

2

2 2

1

CVS

E 

Putting Together

 Energy dissipated in each cycle

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

 Average power

2 2

1 E CV_S

E

E   

f T CV

P  E  _S²

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

ON

L

R

R 

L L L

ON L

dynamic static

C R

T R

R R

P P

 2

 

Example II.1







 k R k

R

_L

100 ,

_ON

10

 Worst static power dissipation

RL

VS

Some numbers

L L L

ON L

dynamic static

C R

T R

R R

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

f CV P  _S²

Static Power of CMOS

 Leakage power

 Transient current

vIN

iT

Some Examples

 Example II.6: Processor SA27E

- # of gates = 3M, 25% activity MHz f

V V

fF

C_L  30 , _S 1.5 ,  425

) 2

(#

)

( _{L S}

dynamic Fraction switching gates fC V

P   

Exercise

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

- # of gates = 25M, 25% activity GHz

f V

V fF

C_L 1 , _S 1.5 ,  3

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

(#

)

( _{L S}

dynamic Fraction switching gates fC V

P   

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 V0

) (t v_I

Zero initial condition:

0 ,

0 

 _L

C i

v

Homogeneous Solution

 General Form: vH(t)  Ae^st

j LC s

Ae e

LCAs ^{st st} _{o o} 1

,

2   0      

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

i v v L

W C

peak peak peak

peak

C C L

C

T  ²  ²  

2 2

Example 12.1

) 10

, 5 . 0 (

, 100

,

1 F L H i A v V at some t

C     _L  _C 

RLC Circuits

Series RLC Circuit: second order circuit

) 1 (

) 1 (

) ( )

(

2 2

t LC v

t LC v

dt t dv L R dt

t v d

S C

C

C   

Homogeneous Solution

o

o o

t s t

s C

s

L LC s R

s

e K e

K t

v









































2 2

, 1

2 2

2 1

, 1 , 2

0 2

)

( ^{1 2}

 Overdamped case: real and distinct roots

 Critically damped case: real and repeated roots

 Underdamped case: complex conjugate roots

)

2 (

2 2

,

1 o wo

s  









 



)

2

(

,

1 o o

s      

)

2 (

2 ,

2 ,

1 j d d o o

s          

Natural Response: overdamped case

5 . 1 ,

1 ,

2 1

1  K  _o   

K

Natural Response: Critically damped case

1 ,

1 ,

2 1

1  K  _o   

K

t t

C t st

s

C t K e K te v t e te

v ( )  ₁  ₂  ( )  ^  ^

Natural Response: Underdamped case

2 . 0 ,

1 ,

2 1

1  K  _o   

K

Response to step input voltage

o 1,   0.2 ~ 4  Zeta



Exercise: Parallel RLC Circuit

V V

H L

F C

k

R  5 , 1 , 1 , _S  25 Find i_L(t)

assume zero initial state

Exercise

+

L v(t) 1k

= u(t)

+

- vc(t)

F 3 1

Find the range of L to make v_C(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.

i_L (0) = 4 mA, v_C (0) = 0 (V)

2k

F 1H

1 4 2k

4mA

+

- v_c(t) +-

12V

Exercise

Example 12.4 Ideal Switched Power Supply

+

-

C

vC

iL

S1 S₂

 L

V

S1

S2

T

cycle 1

iL

vC

(1) S1 on, S2 off

L

(2) S1 off, S2 on until inductor VT

current 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 (

t j t

j i I

e V e

t v











Find the particular solution to ^{j t}

i

e

V

^

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

V

C

 V

^C

RC j

V

_i





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

+

2 1H

1F

10cos2t find v^C

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

+

1k 2k

+

-

v(t) 1F

sint 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



_C

L

Z

Z

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

0.2F 600

200mH Z

Exercise

Summary

 Sinusoidal Steady-state Response

 Impedance

 Frequency Response

 Filter