Chapter 13 Output Stages and Power Amplifiers









13.1
13.2
13.3
13.4
13.5
13.6
13.7
13.8
13.9
General Considerations
Emitter Follower as Power Amplifier
Push-Pull Stage
Improved Push-Pull Stage
Large-Signal Considerations
Short Circuit Protection
Heat Dissipation
Efficiency
Power Amplifier Classes
1
Why Power Amplifiers?
 Drive a load with high power.
 Cell phone needs 1W of power at the antenna.
 Audio system needs tens to hundreds Watts of power.
 Ordinary Voltage/Current amplifiers are not equipped for
such applications
CH 13 Output Stages and Power Amplifiers
2
Chapter Outline
CH 13 Output Stages and Power Amplifiers
3
Power Amplifier Characteristics
 Experiences small load resistance.
 Delivers large current levels.
 Requires large voltage swings.
 Draws a large amount of power from supply.
 Dissipates a large amount of power, therefore gets “hot”.
CH 13 Output Stages and Power Amplifiers
4
Power Amplifier Performance Metrics
 Linearity
 Power Efficiency
 Voltage Rating
CH 13 Output Stages and Power Amplifiers
5
Emitter Follower Large-Signal Behavior I
 As Vin increases Vout also follows and Q1 provides more current.
CH 13 Output Stages and Power Amplifiers
6
Emitter Follower Large-Signal Behavior II
 However as Vin decreases, Vout also decreases, shutting off Q1
and resulting in a constant Vout.
CH 13 Output Stages and Power Amplifiers
7
Example: Emitter Follower
 Vout
 1
Vin  VT ln 
 I1    Vout
 IS 
 RL
Vin  0.5V  Vout  211mV
I C1
Vin  VT ln
  I C1  I1  RL
IS
I C1  0.01I1  Vin  390mV
CH 13 Output Stages and Power Amplifiers
8
Linearity of an Emitter Follower
 As Vin decreases the output waveform will be clipped,
introducing nonlinearity in I/O characteristics.
CH 13 Output Stages and Power Amplifiers
9
Push-Pull Stage
 As Vin increases, Q1 is on and pushes a current into RL.
 As Vin decreases, Q2 is on and pulls a current out of RL.
CH 13 Output Stages and Power Amplifiers
10
I/O Characteristics for Large Vin
Vout=Vin-VBE1 for large +Vin
Vout=Vin+|VBE2| for large -Vin
 For positive Vin, Q1 shifts the output down and for negative Vin,
Q2 shifts the output up.
CH 13 Output Stages and Power Amplifiers
11
Overall I/O Characteristics of Push-Pull Stage
 However, for small Vin, there is a dead zone (both Q1 and Q2 are
off) in the I/O characteristic, resulting in gross nonlinearity.
CH 13 Output Stages and Power Amplifiers
12
Small-Signal Gain of Push-Pull Stage
 The push-pull stage exhibits a gain that tends to unity when
either Q1 or Q2 is on.
 When Vin is very small, the gain drops to zero.
CH 13 Output Stages and Power Amplifiers
13
Sinusoidal Response of Push-Pull Stage
 For large Vin, the output follows the input with a fixed DC
offset, however as Vin becomes small the output drops to zero
and causes “Crossover Distortion.”
CH 13 Output Stages and Power Amplifiers
14
Improved Push-Pull Stage
VB=VBE1+|VBE2|
 With a battery of VB inserted between the bases of Q1 and Q2,
the dead zone is eliminated.
CH 13 Output Stages and Power Amplifiers
15
Implementation of VB
 Since VB=VBE1+|VBE2|, a natural choice would be two diodes in
series.
 I1 in figure (b) is used to bias the diodes and Q1.
CH 13 Output Stages and Power Amplifiers
16
Example: Current Flow I
I in  I1  I B1  I B 2
Iin
CH 13 Output Stages and Power Amplifiers
If Vout=0 & β1=β2>>1
=> IB1=IB2
17
Example: Current Flow II
VD1≈VBE → Vout≈Vin
If I1=I2 & IB1≈IB2
→ Iin=0 when Vout=0
CH 13 Output Stages and Power Amplifiers
18
Addition of CE Stage
 A CE stage (Q4) is added to provide voltage gain from the input
to the bases of Q1 and Q2.
CH 13 Output Stages and Power Amplifiers
19
Bias Point Analysis
VA=0
Vout=0
IC1=[IS,Q1/IS,D1]×[IC3]
 For bias point analysis, the circuit can be simplified to the one
on the right, which resembles a current mirror.
 The relationship of IC1 and IQ3 is shown above.
CH 13 Output Stages and Power Amplifiers
20
Small-Signal Analysis
AV=-gm4(rπ1||r π2)(gm1+gm2)RL
 Assuming 2rD is small and (gm1+gm2)RL is much greater than 1,
the circuit has a voltage gain shown above.
CH 13 Output Stages and Power Amplifiers
21
Output Resistance Analysis
Rout
rO3 || rO 4
1


g m1  g m 2 ( g m1  g m 2 )(r 1 || r 2 )
 If β is low, the second term of the output resistance will rise,
which will be problematic when driving a small resistance.
CH 13 Output Stages and Power Amplifiers
22
Example: Biasing
CE AV=5
Output Stage AV=0.8
RL=8Ω
βnpn= 2βpnp=100
IC1≈IC2
g m1  g m 2
1

2
g m1  g m 2   4 
1
I C1  I C 2  6.5mA
r 1 || r 2  133
CH 13 Output Stages and Power Amplifiers
I C 3  I C 4  195 A
23
Problem of Base Current
 195 µA of base current in Q1 can only support 19.5 mA of
collector current, insufficient for high current operation
(hundreds of mA).
CH 13 Output Stages and Power Amplifiers
24
Modification of the PNP Emitter Follower
Rout
1

  2  1 g m3
 Instead of having a single PNP as the emitter-follower, it is now
combined with an NPN (Q2), providing a lower output
resistance.
CH 13 Output Stages and Power Amplifiers
25
Example: Input Resistance

RL
1 

iin 
vin  vin
1
r 3 
RL 

  2  1 g m3

rin  3 (  2  1) RL  r 3
CH 13 Output Stages and Power Amplifiers






26
Additional Bias Current
 I1 is added to the base of Q2 to provide an additional bias
current to Q3 so the capacitance at the base of Q2 can be
charged/discharged quickly.
CH 13 Output Stages and Power Amplifiers
27
Example: Minimum Vin
Min Vin≈0
Vout≈|VEB2|
CH 13 Output Stages and Power Amplifiers
Min Vin≈VBE2
Vout≈|VEB3|+VBE2
28
HiFi Design
 Using negative feedback, linearity is improved, providing
higher fidelity.
CH 13 Output Stages and Power Amplifiers
29
Short-Circuit Protection
 Qs and r are used to “steal” some base current away from Q1
when the output is accidentally shorted to ground, preventing
short-circuit damage.
CH 13 Output Stages and Power Amplifiers
30
Emitter Follower Power Rating
VP 

Pav  I1  VCC  
2 

Pav ,max  T1VCC
 Maximum power dissipated across Q1 occurs in the absence of
a signal.
CH 13 Output Stages and Power Amplifiers
31
Example: Power Dissipation
Avg Power Dissipated in I1


1 T
PI 1   I1 V p sin t  VEE dt
T 0
PI 1   I1VEE
CH 13 Output Stages and Power Amplifiers
32
Push-Pull Stage Power Rating
VP  VCC VP 
Pav 


RL  
4 
Pav,max
2
VCC
 2
 RL
 Maximum power occurs between Vp=0 and 4Vcc/π.
CH 13 Output Stages and Power Amplifiers
33
Example: Push-Pull Pav
VP  VCC VP 
Pav 


RL  
4 
If Vp = 4VCC/π → Pav=0
Impossible since Vp cannot go
above supply (VCC)
CH 13 Output Stages and Power Amplifiers
34
Heat Sink
 Heat sink, provides large surface area to dissipate heat from
the chip.
CH 13 Output Stages and Power Amplifiers
35
Thermal Runaway Mitigation
I C1 I C 2
I D1I D 2

I S , D1I S , D 2 I S ,Q1I S ,Q 2
 Using diode biasing prevents thermal runaway since the
currents in Q1 and Q2 will track those of D1 and D2 as long as
theie Is’s track with temperature.
CH 13 Output Stages and Power Amplifiers
36
Efficiency
Pout

Pout  Pckt
Emitter Follower
 EF 
 EF
VP2
VP2 2 RL
2 RL  I1  2VCC  VP 2 
VP

4VCC
I1=VP/RL
Push-Pull Stage
 PP 
VP2
VP2 2 RL
2 RL  2 I1 VCC /   VP 4 

 PP  VPVCC
4
I1=VP/RL
 Efficiency is defined as the average power delivered to the load
divided by the power drawn from the supply
CH 13 Output Stages and Power Amplifiers
37
Example: Efficiency
Emitter Follower
VP=VCC/2
1

15
CH 13 Output Stages and Power Amplifiers
Push-Pull
I1=(VP/RL)/β
VP
1

4 VCC   VP 
38
Power Amplifier Classes
Class A: High linearity, low efficiency
Class B: High efficiency, low linearity
Class AB: Compromise between
Class A and B
CH 13 Output Stages and Power Amplifiers
39
Chapter 14 Analog Filters





14.1
14.2
14.3
14.4
14.5
General Considerations
First-Order Filters
Second-Order Filters
Active Filters
Approximation of Filter Response
40
Outline of the Chapter
CH 14 Analog Filters
41
Why We Need Filters
 In order to eliminate the unwanted interference that
accompanies a signal, a filter is needed.
CH 14 Analog Filters
42
Filter Characteristics
 Ideally, a filter needs to have a flat pass band and a sharp rolloff in its transition band.
 Realistically, it has a rippling pass/stop band and a transition
band.
CH 14 Analog Filters
43
Example: Filter I
Given: Adjacent channel Interference is 25 dB above the signal
Design goal: Signal to Interference ratio of 15 dB
Solution: A filter with stop band of 40 dB
CH 14 Analog Filters
44
Example: Filter II
Given: Adjacent channel Interference is 40 dB above the signal
Design goal: Signal to Interference ratio of 20 dB
Solution: A filter with stop band of 60 dB at 60 Hz
CH 14 Analog Filters
45
Example: Filter III
 A bandpass filter around 1.5 GHz is needed to reject the
adjacent Cellular and PCS signals.
CH 14 Analog Filters
46
Classification of Filters I
CH 14 Analog Filters
47
Classification of Filters II
Continuous-time
CH 14 Analog Filters
Discrete-time
48
Classification of Filters III
Passive
CH 14 Analog Filters
Active
49
Summary of Filter Classifications
CH 14 Analog Filters
50
Filter Transfer Function
A
B
 Filter a) has a transfer function with -20dB/dec roll-off
 Filter b) has a transfer function with -40dB/dec roll-off, better
selectivity.
CH 14 Analog Filters
51
General Transfer Function
 s  Z1  s  Z 2   s  Z m 
H ( s)  
 s  P1  s  P2   s  Pm 
CH 14 Analog Filters
Zm=m’th zero
Pn =n’th pole
52
Pole-Zero Diagram
CH 14 Analog Filters
53
Position of the Poles
Poles on the RHP
Unstable
(no good)
CH 14 Analog Filters
Poles on the jω axis
Oscillatory
(no good)
Poles on the LHP
Decaying
(good)
54
Imaginary Zero
 Imaginary zero is used to create a null at certain frequency.
CH 14 Analog Filters
55
Sensitivity
dP
P
SC 
P
dC
C
P=Parameter
C=Component
 Sensitivity measures the variation of a filter parameter due to
variation of a filter component.
CH 14 Analog Filters
56
Example: Sensitivity
0  1 / R1C1 
d0
1
dR1

R12C1
d0
dR1

0
R1
S R10  1
CH 14 Analog Filters
57
First-Order Filters
s  z1
H ( s)  
s  p1
 First-order filters are represented by the transfer function
shown above.
 Low/high pass filters can be realized by changing the relative
positions of poles and zeros.
CH 14 Analog Filters
58
Example: First-Order Filter I
R2C 2 < R1C1
CH 14 Analog Filters
R2C2 > R1C1
59
Example: First-Order Filter II
CH 14 Analog Filters
R2C2 < R1C 1
R2C2 > R1C1
60
Second-Order Filters
 s2   s  
H (s) 
2 n
s 
s  n2
Q
p1,2  
n
2Q
 jn 1 
1
4Q 2
 Second-order filters are characterized by the “biquadratic”
equation with two complex poles shown above.
CH 14 Analog Filters
61
Second-Order Low-Pass Filter
H ( j ) 
2
2

n2
CH 14 Analog Filters

2

2
 n 
 
Q 
2
α=β=0
62
Example: Second-Order LPF
Q3
Q / 1  1/(4Q 2 )  3
n 1  1/(2Q )  n
2
CH 14 Analog Filters
63
Second-Order High-Pass Filter
H (s) 
s2 
CH 14 Analog Filters
 s2
n
Q
s  n2
β=γ=0
64
Second-Order Band-Pass Filter
H (s) 
s 
2
CH 14 Analog Filters
s
n
Q
2
s  n
α=γ=0
65
Example: -3-dB Bandwidth
R1L1s
Z2 
R1L1C1s 2  L1s  R1
CH 14 Analog Filters
66
LC Realization of Second-Order Filters
L1s
Z1 
L1C1s 2  1
 An LC tank realizes a second-order band-pass filter with two
imaginary poles at ±j/(L1C1)1/2 , which implies infinite
impedance at ω=1/(L1C1)1/2.
CH 14 Analog Filters
67
Example: Tank
 ω=0, the inductor acts as a short.
 ω=∞, the capacitor acts as a short.
CH 14 Analog Filters
68
RLC Realization of Second-Order Filters
R1L1s
Z2 
R1L1C1s 2  L1s  R1
p1,2  
L
1
1
j
1  21
2 R1C1
4 R1 C1
L1C1
 With a resistor, the poles are no longer pure imaginary which
implies there will be no infinite impedance at any ω.
CH 14 Analog Filters
69
Voltage Divider Using General Impedances
Vout
ZP
(s) 
Vin
ZS  ZP
Low-pass
CH 14 Analog Filters
High-pass
Band-pass
70
Low-pass Filter Implementation with Voltage Divider
Vout
R1
s 
2
Vin
R1C1L1s  L1s  R1
CH 14 Analog Filters
71
Example: Frequency Peaking
Vout
R1
s 
Vin
R1C1L1s 2  L1s  R1
1
Q
2
CH 14 Analog Filters
Peaking exists
Voltage gain larger than unity
72
Low Pass Circuit Comparison
Good
Bad
 The circuit on the left has a sharper roll-off at high frequency
than the circuit on the right.
CH 14 Analog Filters
73
High-pass Filter Implementation with Voltage Divider
Vout
L1C1R1s 2
s 
2
Vin
R1C1L1s  L1s  R1
CH 14 Analog Filters
74
Band-pass Filter Implementation with Voltage Divider
2
Vout
L1s
s 
2
Vin
R1C1L1s  L1s  R1
CH 14 Analog Filters
75
Sallen and Key (SK) Filter: Low-Pass
Vout
1
s 
Vin
R1R2C1C2 s 2   R1  R2  C2 s  1
1
Q
R1  R2
C
R1R2 1
C2
n 
1
R1R2C1C2
 Sallen and Key filters are examples of active filters. This
particular filter implements a low-pass, second-order transfer
function.
CH 14 Analog Filters
76
Sallen and Key (SK) Filter: Band-pass
Vout
s 
Vin
CH 14 Analog Filters
R3
1
R4

R3 
R1R2C1C2 s   R1C2  R2C2  R1 C1  s  1
R4 

2
77
Example: SK Filter Poles
C1=C2
R1=R2
CH 14 Analog Filters
78
Sensitivity in Band-Pass SK Filter
n
n
n
n
S R  S R  SC  SC
1
S RQ
1
2

S RQ
2
1
2
1

2
R2C2
1
  Q
2
R1C1
CH 14 Analog Filters
SCQ
1

 SCQ
2
 R2C2
R1C2
1
   Q 

2
R2C1
 R1C1
S KQ  QK
R1C1
R2C2
K=1+R3/R4
79



Example: SK Filter Sensitivity I
R1  R2  R
C1  C2  C
S RQ
1

 S RQ
2
SCQ   SCQ
1
S KQ
CH 14 Analog Filters
2
1
1
 
2 3 K
1
2
 
2 3 K
K

3 K
80
Example: SK Filter Sensitivity II
Q2
K 2
R2C2 3
  S RQ  1
1
R1C1 4
R1C1 1
5
Q
  SC 
1
R2C2 8
4
S KQ
CH 14 Analog Filters
8

1.5
81
Integrator-Based Biquads
Vout
 s2
s 
Vin
2 n
s 
s  n2
Q
Vout  s   Vin  s  
n 1
n2
. Vout  s   2 Vout  s 
Q s
s
 It is possible to use integrators to implement biquadratic
transfer functions.
 The block-diagram above illustrates how.
CH 14 Analog Filters
82
KHN Biquads
Vout  s   Vin  s  
R5  R6 

1  
R4  R5  R3 
CH 14 Analog Filters
n
n 1
n2
. Vout  s   2 Vout  s 
Q s
s
R4
1

.
Q R4  R5 R1C1
n2 
R6
1
.
R3 R1R2C1C2
83
Versatility of KHN Biquads
High-Pass
Vout
 s2
s 

Vin
s 2  n s  n2
Q
Band-Pass
VX
 s2
1
.
s 

Vin
s 2  n s  n2 R1C1s
Q
CH 14 Analog Filters
Low-Pass
VY
 s2
1
.
s 
2
Vin
2 n
2 R1R2C1C2 s
s 
s  n
Q
84
Sensitivity in KHN Biquads
S Rn, R ,C ,C , R , R , R , R  0.5
1
2
1
2
4
5
3
6
SRQ , R ,C ,C  0.5
1
S RQ , R 
3
6
Q R3  R6
2 1  R5
R4
CH 14 Analog Filters
R2C2
R 3 R6 R1C1
S RQ , R
4
5
2
1
2
R5

1
R4  R5
85
Tow-Thomas Biquad
Vout
RRR
C2 s
 2 3 4.
Vin
R1
R2 R3 R4C1C2 s 2  R2 R4C2 s  R3
Band-Pass
CH 14 Analog Filters
VY R3 R4
1

.
Vin
R1 R2 R3 R4C1C2 s 2  R2 R4C2 s  R3
Low-Pass
86
Example: Tow-Thomas Biquad
n 
1
R2 R4C1C2
Adjusted by R2 or R4
CH 14 Analog Filters
1
Q
R3
R2 R4C2
C1
Adjusted by R3
87
Differential Tow-Thomas Biquads
 By using differential integrators, the inverting stage is
eliminated.
CH 14 Analog Filters
88
Simulated Inductor (SI)
Z1Z 3
Z in 
Z5
Z2Z4
 It is possible to simulate the behavior of an inductor by using
active circuits in feedback with properly chosen passive
elements.
CH 14 Analog Filters
89
Example: Simulated Inductor I
Zin 
2
RX RY Cs
 By proper choices of Z1-Z4, Zin has become an impedance that
increases with frequency, simulating inductive effect.
CH 14 Analog Filters
90
Example: Simulated Inductor II
Z1  Z 2  Z3  RY
CH 14 Analog Filters
1
Z4 
Cs
Zin  RX RY2Cs
91
High-Pass Filter with SI
Vout
L1s 2
s 
Vin
R1C1L1s 2  L1s  R1
 With the inductor simulated at the output, the transfer function
resembles a second-order high-pass filter.
CH 14 Analog Filters
92
Example: High-Pass Filter with SI
Node 4 is also an output node
CH 14 Analog Filters

RY 
V4  Vout 1 

 RX 
93
Low-Pass Filter with Super Capacitor
1
Zin 
Cs  RX Cs  1
Vout
Zin
1


Vin Zin  R1 R1RX C 2 s 2  R1Cs  1
Low-Pass
CH 14 Analog Filters
94
Example: Poor Low Pass Filter
V4  Vout  2  RX Cs 
 Node 4 is no longer a scaled version of the Vout. Therefore the
output can only be sensed at node 1, suffering from a high
impedance.
CH 14 Analog Filters
95
Frequency Response Template
 With all the specifications on pass/stop band ripples and
transition band slope, one can create a filter template that will
lend itself to transfer function approximation.
CH 14 Analog Filters
96
Butterworth Response
H ( j ) 
1
 
1  
 0 
2n
ω-3dB=ω0, for all n
 The Butterworth response completely avoids ripples in the
pass/stop bands at the expense of the transition band slope.
CH 14 Analog Filters
97
Poles of the Butterworth Response
j
 2k  1 
pk  0 exp
exp  j
  , k  1, 2,
2
 2n

CH 14 Analog Filters
2nd-Order
,n
nth-Order
98
Example: Butterworth Order
 f2 
 
 f1 
2n
 64.2
f 2  2 f1
n=3
 The Butterworth order of three is needed to satisfy the filter
response on the left.
CH 14 Analog Filters
99
Example: Butterworth Response
2
2 

p1  2 *(1.45MHz ) *  cos
 j sin

3
3 

2
2 

p3  2 *(1.45MHz ) *  cos
 j sin

3
3 

CH 14 Analog Filters
RC section
2nd-Order SK
p2  2 *(1.45MHz )
100
Chebyshev Response
1
H  j  
1 
2
2
Cn 
 


 0
Chebyshev Polynomial
 The Chebyshev response provides an “equiripple” pass/stop
band response.
CH 14 Analog Filters
101
Chebyshev Polynomial
Chebyshev Polynomial for
n=1,2,3
Resulting Transfer function for
n=2,3
 

1  
Cn    cos  n cos
 ,   0


0
 0

CH 14 Analog Filters

1  
 cosh  n cosh
 ,   0

0

102
Example: Chebyshev Attenuation
H  j  
1
   3

2
1  0.329  4    3 
0 
  0 

ω0=2π X (2MHz)
 A third-order Chebyshev response provides an attenuation of 18.7 dB a 2MHz.
CH 14 Analog Filters
103
2
Example: Chebyshev Order




Passband Ripple: 1 dB
Bandwidth: 5 MHz
Attenuation at 10 MHz: 30 dB
What’s the order?
1
1  0.509 cosh
2
CH 14 Analog Filters
2
 n cosh 2
1
 0.0316
n>3.66
104
Example: Chebyshev Response
pk  0 sin
 2k 1  sinh  1 sinh 1 1   j
2n

n
 
0 cos
 2k 1  cosh  1 sinh 1 1 
2n

n
 
K=1,2,3,4
p1,4  0.1400  0.983 j0
SK1
CH 14 Analog Filters
p2,3  0.3370  0.407 j0
SK2
105
Chapter 15 Digital CMOS Circuits
 15.1 General Considerations
 15.2 CMOS Inverter
 15.3 CMOS NOR and NAND Gates
106
Chapter Outline
CH 15 Digital CMOS Circuits
107
Inverter Characteristic
_
XA
 An inverter outputs a logical “1” when the input is a logical “0”
and vice versa.
CH 15 Digital CMOS Circuits
108
NMOS Inverter
Ron1 
1
nCox
W
(VDD  VTH )
L
 The CS stage resembles a voltage divider between RD and Ron1
when M1 is in deep triode region. It produces VDD when M1 is
off.
CH 15 Digital CMOS Circuits
109
Transition Region Gain
Infinite Transition Region Gain
Finite Transition Region Gain
 Ideally, the VTC of an inverter has infinite transition region
gain. However, practically the gain is finite.
CH 15 Digital CMOS Circuits
110
Example: Transition Gain
 Transition Region: 50
mV
 Supply voltage: 1.8V
V0 – V2: Transition Region
CH 15 Digital CMOS Circuits
Av 
1.8
 36
0.05
111
Logical Level Degradation
 Since real power buses have losses, the power supply levels at
two different locations will be different. This will result in
logical level degradation.
CH 15 Digital CMOS Circuits
112
Example: Logic Level Degradation
Supply B=1.675V
Supply A=1.8V
V  5 A 25m  125mV
CH 15 Digital CMOS Circuits
113
The Effects of Level Degradation and Finite Gain
 In conjunction with finite transition gain, logical level
degradation in succeeding gates will reduce the output swings
of gates.
CH 15 Digital CMOS Circuits
114
Small-Signal Gain Variation of NMOS Inverter
 As it can be seen, the small-signal gain is the largest in the
transition region.
CH 15 Digital CMOS Circuits
115
Above Unity Small-Signal Gain
 The magnitude of the small-signal gain in the transition region
can be above 1.
CH 15 Digital CMOS Circuits
116
Noise Margin
 Noise margin is the amount of input logic level degradation that
a gate can handle before the small-signal gain becomes -1.
CH 15 Digital CMOS Circuits
117
Example: NMOS Inverter Noise Margin
1: NM L  VIL 
1
 VTH
W
nCox RD
L
1
W
2 
Vout  VDD  nCox RD  2 Vin  VTH Vout  Vout

2
L
Vout
V V

 in TH
W
2
2nCox RD
L
CH 15 Digital CMOS Circuits
1
Vin=VIH
2: NM H  VDD  VIH
118
Example: Minimum Vout
RD 
19
nCox
W
VDD  VTH 
L
 To guarantee an output low level that is below 0.05VDD, RD is
chosen above.
CH 15 Digital CMOS Circuits
119
Dynamic Behavior of NMOS Inverter Gates
 Since digital circuits operate with large signals and experience
nonlinearity, the concept of transfer function is no longer
meaningful. Therefore, we must resort to time-domain analysis
to evaluate the speed of a gate.
 It usually takes 3 time constants for the output to transition.
CH 15 Digital CMOS Circuits
120
Rise/Fall Time and Delay
CH 15 Digital CMOS Circuits
121
Example: Time Constant
19 L2
  RDC X 
n VDD  VTH 
 Assuming a 5% degradation in output low level, the time
constant at node X is shown above.
CH 15 Digital CMOS Circuits
122
Example: Interconnect Capacitance
Wire Capacitance per Mircon: 50x10-18 F/µm
Total Interconnect Capacitance: 15000X50x10-18 =750 fF
Equivalent to 640 MOS FETs with W=0.5µm, L=0.18µm, Cox =13.5fF/µm2
CH 15 Digital CMOS Circuits
123
Power-Delay Product
2
PDP  VDD
CX
 The power delay product of an NMOS Inverter can be loosely
thought of as the amount of energy the gate uses in each
switching event.
CH 15 Digital CMOS Circuits
124
Example: Power-Delay Product
TPLH  3RDC X
PDP   I DDVDD  3RDC X 
2
PDP  3VDD
WLCox
CH 15 Digital CMOS Circuits
125
Drawbacks of the NMOS Inverter
 Because of constant RD, NMOS inverter consumes static power
even when there is no switching.
 RD presents a tradeoff between speed and power dissipation.
CH 15 Digital CMOS Circuits
126
Improved Inverter Topology
 A better alternative would probably have been an “intelligent”
pullup device that turns on when M1 is off and vice versa.
CH 15 Digital CMOS Circuits
127
Improved Falltime
 This improved inverter topology decreases falltime since all of
the current from M1 is available to discharge the capacitor.
CH 15 Digital CMOS Circuits
128
CMOS Inverter
 A circuit realization of this improved inverter topology is the
CMOS inverter shown above.
 The NMOS/PMOS pair complement each other to produce the
desired effects.
CH 15 Digital CMOS Circuits
129
CMOS Inverter Small-Signal Model
vout
   g m1  g m 2  rO1 || rO 2 
vin
 When both M1 and M2 are in saturation, the small-signal gain is
shown above.
CH 15 Digital CMOS Circuits
130
Switching Threshold
 The switching threshold (VinT) or the “trip point” of the inverter
is when Vout equals Vin.
 If VinT =Vdd/2, then W2/W1=µn/µp
CH 15 Digital CMOS Circuits
131
CMOS Inverter VTC
CH 15 Digital CMOS Circuits
132
Example: VTC
W2
 As the PMOS device is made stronger, the VTC is shifted to the
right.
CH 15 Digital CMOS Circuits
133
Noise Margins
VIL 
NML =VIL
NMH =Vdd-VIH
2 a Vdd  VTH 1  VTH 2
 a  1
a3
  Vdd  aVTH1  VTH 2
a 1
VIL is the low-level input voltage
at which (δVout/ δVin)=-1
VIH 
2a Vdd  VTH 1  VTH 2
 a  1
1  3a
  Vdd  aVTH 1  VTH 2
a 1
VIH is the high-level input voltage
at which (δVout/ δVin)=-1
W 

 L 1
a
W 
p  
 L 2
n 
CH 15 Digital CMOS Circuits
134
VIL of a Symmetric VTC
VIL 
2 a VDD  2VTH 1   a  3 VDD   a  1VTH 1 
 a  1
a3
Symmetric VTC: a=1
3
1
VIL  VDD  VTH 1
8
4
CH 15 Digital CMOS Circuits
135
Noise Margins of an Ideal Symmetric VTC
NM H ,ideal  NM L ,ideal 
CH 15 Digital CMOS Circuits
VDD
2
136
Floating Output
VTH 1  VDD / 2
VTH 2  VDD / 2
 When Vin=VDD/2, M2 and M1 will both be off and the output floats.
CH 15 Digital CMOS Circuits
137
Charging Dynamics of CMOS Inverter
 As Vout is initially charged high, the charging is linear since M2
is in saturation. However, as M2 enters triode region the charge
rate becomes sublinear.
CH 15 Digital CMOS Circuits
138
Charging Current Variation with Time
 The current of M2 is initially constant as M2 is in saturation.
However as M2 enters triode, its current decreases.
CH 15 Digital CMOS Circuits
139
Size Variation Effect to Output Transition
 As the PMOS size is increased, the output exhibits a faster
transition.
CH 15 Digital CMOS Circuits
140
Discharging Dynamics of CMOS Inverter
 Similar to the charging dynamics, the discharge is linear when
M1 is in saturation and becomes sublinear as M1 enters triode
region.
CH 15 Digital CMOS Circuits
141
Rise/Fall Time Delay
Rise Time Delay
TPLH 
 2 VTH 2


V
 ln  3  4 TH 2  

V V
VDD  
W 

 pCox   VDD  VTH 2   DD TH 2
 L 2
CL
Fall Time Delay
TPHL 
CH 15 Digital CMOS Circuits
 2 VTH 1

V 
 ln  3  4 TH 1  

V V
VDD  
W 

nCox   VDD  VTH 1   DD TH 1
 L 1
CL
142
Example: Averaged Rise Time Delay
I AVG
TPLH 2 
1
W 
  pCox   VDD  VTH 2
4
 L 2
CL
W 
 VDD  VTH 2
 L 2
 pCox 
TPLH 2
CH 15 Digital CMOS Circuits

.

2
VDD / 2  VDD  VTH 2

VDD  VTH 2
4
 Ron 2CL
3
143
Low Threshold Improves Speed
1st Term
TPLH / HL 
 2 VTH 2 /1


V
 ln  3  4 TH 2 /1  

V V
VDD  

VDD  VTH 2 /1   DD TH 2 /1
CL
W 

 L 2 /1
 p / nCox 
2nd Term
 The sum of the 1st and 2nd terms of the bracket is the smallest
when VTH is the smallest, hence low VTH improves speed.
CH 15 Digital CMOS Circuits
144
Example: Increased Fall Time Due to Manufacturing
Error
'
Ron1 || Ron
1
1
  W   W  ' 

nCox        VDD  VTH 1  
  L 1  L 1 




 2 RON1
Since pull-down resistance is doubled, the fall time is also doubled.
CH 15 Digital CMOS Circuits
145
Power Dissipation of the CMOS Inverter
1
2
PDissipation _ PMOS  CLVDD
fin
2
2
Psupply  CLVDD
fin
1
2
PDissipation _ NMOS  CLVDD
fin
2
CH 15 Digital CMOS Circuits
146
Example: Energy Calculation
1
2
Estored  CLVDD
2
1
2
Edissipated  CLVDD
2
2
Edrawn  CLVDD
CH 15 Digital CMOS Circuits
147
Power Delay Product
2
 2 VTH 1

finCL2VDD
VTH 1  
PDP 
 ln  3  4


W
V

V
V
 

DD  
nCox   VDD  VTH 1   DD TH 1
 L 1
Ron1=Ron2
CH 15 Digital CMOS Circuits
148
Example: PDP
4
Ron 
3
PDP 
CH 15 Digital CMOS Circuits
1
W
nCox 
L

 VDD

2
7.25WL2Cox finVDD
n
149
Crowbar Current
 When Vin is between VTH1 and VDD-|VTH2|, both M1 and M2 are on
and there will be a current flowing from supply to ground.
CH 15 Digital CMOS Circuits
150
NMOS Section of NOR
 When either A or B is high or if both A and B are high, the
output will be low. Transistors operate as pull-down devices.
CH 15 Digital CMOS Circuits
151
Example: Poor NOR
 The above circuit fails to act as a NOR because when A is high
and B is low, both M4 and M1 are on and produces an ill-defined
low.
CH 15 Digital CMOS Circuits
152
PMOS Section of NOR
 When both A and B are low, the output is high. Transistors
operate as pull-up devices.
CH 15 Digital CMOS Circuits
153
CMOS NOR
 Combing the NMOS and PMOS NOR sections, we have the
CMOS NOR.
CH 15 Digital CMOS Circuits
154
Example: Three-Input NOR
Vout   A  B  C 
'
Equal Rise & Fall (µn≈2µp)
W1=W2=W3=W
W4=W5=W6=6W
CH 15 Digital CMOS Circuits
155
Drawback of CMOS NOR
 Due to low PMOS mobility, series combination of M3 and M4
suffers from a high resistance, producing a long delay.
 The widths of the PMOS transistors can be increased to
counter the high resistance, however this would load the
preceding stage and the overall delay of the system may not
improve.
CH 15 Digital CMOS Circuits
156
NMOS NAND Section
 When both A and B are high, the output is low.
CH 15 Digital CMOS Circuits
157
PMOS NOR Section
 When either A or B is low or if both A and B are low, the output
is high.
CH 15 Digital CMOS Circuits
158
CMOS NAND
 Just like the CMOS NOR, the CMOS NAND can be implemented
by combining its respective NMOS and PMOS sections,
however it has better performance because its PMOS
transistors are not in series.
CH 15 Digital CMOS Circuits
159
Example: Three-Input NAND
Vout   ABC 
'
Equal Rise & Fall (µn≈2µp)
W1=W2=W3=3W
W4=W5=W6=2W
CH 15 Digital CMOS Circuits
160
NMOS and PMOS Duality
C is in “series” with the
“parallel” combination of A and B
C is in “parallel” with the
“series” combination of A and B
 In the CMOS philosophy, the PMOS section can be obtained
from the NMOS section by converting series combinations to
the parallel combinations and vice versa.
CH 15 Digital CMOS Circuits
161