Miller capacitance and
Gain-Bandwidth Product
Saeeid Tahmasbi Oskuii (saeta621@student.liu.se)
Behzad Mesgarzadeh (behme822@student.liu.se)
1
Miller Effect
• Miller Theorem
• Miller Capacitance
•
•
•
•
•
Miller Effect in digital circuits
Miller multiplier
Impact of Miller Capacitance on propagation delay
Impact on Power consumption
Applications of Miller Effect
• Gain Bandwidth Product
• Conclusion
2
1
Miller Theorem
An important phenomenon that occurs in many analog and digital
circuits is related to ”Miller Effect”, as described by Miller in a
theorem.
Miller theorem describes the way to convert a floating load into
two grounded loads, in such way that the voltages and currents
are remained unchanged.
X
Z Y
X
I
I
I
Z1
Z2
Y
3
Miller Theorem (Cont’d)
X
Z Y
X
I
I
I
Z1
Z2
Y
V −V
V
Y ,A= Y
I= X
Z
V
X
V
X
V −V
Y ŸZ =
=Z I =Z X
1
1
1
Z
Z
Ÿ
V
1− Y
V
X
V −V
X
Y ŸZ =
V =Z I =Z
Y
2
2
2
Z
Z
V
1− X
V
Y
Ÿ
Z
Z =
1 1− A
Z =
2
Z
1−
1
A
4
2
Miller Theorem (Cont’d)
It is important to know when this theorem is valid. Miller’s theorem
does not stipulate the conditions under which this coversion is
valid. If the impedance Z forms the only signal path between X and
Y, then the conversion is often invalid.
An example of improper use of Miller effect::
X
R1
Y
R2
X
Y
R1+R2
R2
-R2
5
Miller Capacitance
1
1
­
° Z1 = (1 + A)Cs = C s Ÿ C1 = C (1 + A)
1
1
Assume Z = Ÿ °®
1
1
1
=
Ÿ C2 = C (1 + )
Cs
°Z 2 =
1
A
(1 + )Cs C2 s
°
A
¯
C
-A
-A
C(1+A)
1
C(1+ A
-)
6
3
Miller Capacitance (Cont’d)
Miller capacitance for large gain:
C
-A
-A
AC
C
­°1 + A ≈ A
A >> 1 Ÿ ® 1
1+ ≈ 1
°̄ A
7
Miller Effect in digital circuits
Vout
Vin
C M = C gdp + C gdn
∆V
Vin
CM
Vout
∆V
Vin
Vout
2C M
2C M
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4
Miller Effect in digital circuits
Vout
Vin
Vout
CM
−
Instantanous Input Capacitance:
C (Vin ) =
iM
∂Q
M =C
∂V
in
M
∂Vout
∂Vin
Instantanous
Input Capacitance
Vin
Vin
[1 − A(Vin )]
Vin
9
Miller Multiplier
C (Vin ) =
iM
∂Q
M = C [1 − A(V )]
in
M
∂V
in
Miller Multiplier =
CiM
CM
As the input voltage varies,the Miller Capacitance
follows the gain curve.The Multiplier is 1 across nearly
the whole input range and rises to about 30 in a narrow
range about V inv .
10
5
Impact on propagation delay
C
M
2C
M
2C
M
• Although the average
Miller Multiplier for a full
swing signal is 2, for slowly
varying signal in the high
gain region of the inverter
can be 20 or more.
11
Impact on power consumption
CM
2C
M
2C
M
4 Times Greater Capacitance Increases Power Consumption
12
6
Frequency Response
A Three pole Transfer Function:
H ( s) =
K
s
s
s
(1 + )(1 + )(1 + )
ω1
ω2
ω3
∠H ( jω ) =
{
0°
ω << ω 1
270° ω >> ω 3
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Stability Issue (Phase Margin)
20 log10 H ( jω )
φ M = 180 + ∠H ( jω c )
-20 dB/Dec
0 dB
When: H ( jω c ) = 1
-40 dB/Dec
-60 dB/Dec
ω 2 ω3
ω1
∠H ( jω )
ω
ω
For Stability Phase
Margin Must be
Greater Than 0.
-90
Phase Margin
-180
-270
14
7
Negative Phase Margin
20 log10 H ( jω )
-20 dB/Dec
0 dB
An Unstable System
ω
ω1
∠H ( jω )
2
ω
ω3
jω
ω
-90
-180
Phase Margin
-270
15
Compensation
jω
Assume this node gives dominant pole (
ω3
ω1
ω2)
ω2
Before Compensation
CM
σ
jω
1
-A1
2
C2
-A 2
3
ω1
ω3
ω2
After Compensation
σ
16
8
Compensation (cont’d)
Effects of Miller Capacitance in Frequency Compensation:
1. Closes Dominant pole toward Origin
2. Causes to Split First and Second poles
3. Increases Phase Margin and Make System Stable
But:
REDUCES BANDWIDTH !!
17
Miller Effect Between Wires
∆V
C
∆V
2∆V
2C
2C
It seems that there is a 2 times larger capacitance
between the wires.
18
9
Gain Bandwidth Product
At high frequencies the gain of the static
CMOS gate is much smaller than DC gain.
Gain
A( f )
ADC
1
f
fc
f1
19
GBP (Cont’d)
Above cutoff frequency,gain rolls off linearly
with frequency crossing unity at unity-gain
frequency.
In this high frequency region, the gainbandwith product is constant:
fA( f ) = f1
( f > fc )
20
10
GBP (Cont’d)
Calculating
f1:
For small-signal sine wave:
dvout i g m vin
= =
C
C
dt
At the point of unity gain:
ω1 =
gm
I
1
1
≈ DSS = Ÿ f1 =
C VDDC τ
2πτ
21
GBP (Cont’d)
Example:
τinv = 30ps Ÿ f1 = 5.3GHz
As we know: ADC ≈
Assuming:
4
(VGS −VT )(λn + λp )
λn + λ p = 0.2,VGS − VT = 1
Ÿ ADC = 20 Ÿ f c = 265MHz
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11
Conclusion
•
•
•
•
Miller capacitance in digital
applications usually slows down the
speed of the circuits and increases the
power consumption.
In analog circuits Miller effect can be
used to stabilize the system.
Miller capacitance between two wires
reduces noise margin and speed.
Gain-Bandwith product is a constant
value above the cutoff frequency.
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12