Chap. 11 Energy and Power in
Digital Circuits
Average Power in an RC Circuit
Power Dissipation in Logic Gates
CMOS Logic
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
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
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
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
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
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
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
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 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
Find the resonance frequency and the equivalent impedance at resonance.
2k
0.2F 600
200mH
Z
Exercise