Simple Current Mirrors
I out
I in
v1
r out
Q2
Q1
A simple CMOS current-mirror.
Basic Current Mirrors and Gain Stages
Ken Martin
• Q 1 is a ‘diode-connected transistor; its small-signal model is simply a resistor of
Dept. of Elec. and Comp. Eng.
University of Toronto
Toronto, Canada M5S 1A4
• At low-frequencies, MOS transistors are unidirectional; voltage changes at drain
or source can not effect gate current or voltage. Thus, when calculating output
impedance one may assume V gs2 = 0 .
size 1 ⁄ g m .
Q1
martin@eecg.toronto.edu
(416) 978-6695
1
---------g m1
0V
Q2
ix
r ds2
v gs2 g v
m2 gs2
vx
(a)
ix
r ds2
vx
(b)
(a) A small-signal model for finding the output impedance of the simple currentmirror of and (b) a simplified small-signal model.
© D.A. Johns/K.W. Martin, 1995
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© D.A. Johns/K.W. Martin, 1995
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Source Follower
Common-Source Amplifier with Active Load
Active Load
Q3
Q2
V in
V out
I bias
I
r out
Q1
V in
Q
V out
bias
Q
1
Q2
3
Active Load
A common-source amplifier with a current-mirror active load.
A source-follower stage with a current-mirror used to supply the bias current.
KL i
• r out = r ds1 || r ds2 where r dsi = --------I Di
• This is one of the few circuits where the body-effect has a major effect on the
gain.
vd1
Rin
vout
vin = vg1
rds1
vgs1
v gs1+
-
vin
g m1 v gs1
gs1vs1
g m1v gs1
R 2 = r ds1 || r ds2
vs1
rds2
A small-signal equivalent circuit for the common-source amplifier.
•
A V = – g m1 r out = – g m1 ( r ds1 || r ds2 )
© D.A. Johns/K.W. Martin, 1995
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where g m1 =
2µ n C ox ( W ⁄ L ) 1 I D1
The low-frequency model of the source-follower amplifier.
© D.A. Johns/K.W. Martin, 1995
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vout = vs1
Common-Gate Amplifier
• The current source g s1 v s1 is equivalent to a resistor of size 1 ⁄ g s1 . This allows
us to simplify the model leading to
Active Load
Q3
vin
Q2
V out
r out
I bias
vgs1
g m1v gs1
Q1
r in
V bias
vout
R s1 = r ds1 ||r ds2 1|| ⁄ g s1
V in
A common-gate amplifier with a current-mirror active load.
An equivalent small-signal model for the source-follower.
• This stage is used extensively for high-speed design
g m1
A V = ---------------------------------------------------------g m1 + g s1 + g ds1 + g ds2
vout
(1)
vgs1
γ gm
where g s1 = -------------------------------------- is approximately g m1 ⁄ 5 to g m1 ⁄ 10
2 V SB + 2φ F
gm1vgs1
vs1
rds1
gs1vs1
RL
rin
Rs
vin
The small-signal model of the common-gate amplifier at low frequencies.
• Note that v s1 = – v gs1 and that therefore the two current sources can be
combined. This simplification can always be done at low frequencies when
considering signals at the drain or source irrespective of what is connected to the
gate. This often allows one to do analysis ignoring the body-effect and then
taking it into account at end of analysis.
© D.A. Johns/K.W. Martin, 1995
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© D.A. Johns/K.W. Martin, 1995
6
vout
vgs1
rds1
(gm1+gs1)vs1
vs1
Source-Degenerated Current Sources
RL
rin
I out
I in
Rs
vin
A simplified small-signal model of the common-gate amplifier.
• First analyzing for admittance looking into the source, we have
g m1 + g s1 + g ds1
g m1 ′
g in = ----------------------------------------- ≅ -------------------------------1 + g ds1 ⁄ G B
1 + g ds1 ⁄ G B
V1
r out
Q1
Q2
Rs
Rs
A current-mirror with source degeneration.
(2)
or equivalently
RL 
1 
r in ≅ -----------  1 + ---------
g m1 ′ 
r ds1
1
---------g m1
where R L = 1 ⁄ G L is the output impedance of the active load.
• Note that when R L is large, as is the case for ‘better’ active loads, r in can be
considerably greater than 1 ⁄ g m1
• Using the admittance divider rule, we now have
v s1
GS
------- = ---------------------v in
G S + g in
Gs
g m1 ′
≅ ---------------------------------------------------- -------------------------G L + g ds1
g m1 ′
G s + -------------------------------------1 + ( g ds1 ⁄ G L )
© D.A. Johns/K.W. Martin, 1995
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v gs
g m2 ′v gs
r ds2
vx
vs
Rs
ix
Rs
The small-signal model for the source-degenerated current source.
(4)
• Solving for r out , we get
and
g m1 + g s1 + g ds1
v s1 v out
Gs
A V = ------- --------- = ---------------------------------------------------------- -------------------------------------------G L + g ds1
v in v s1
g m1 + g s1 + g ds1
G s + -------------------------------------------1 + ( g ds1 ⁄ G L )
ix
0V
(3)
vx
r out = ----- = r ds2 [ 1 + R s ( g m2 ′ + g ds2 ) ] ≅ r ds2 ( 1 + R s g m2 ′ )
ix
(6)
• This formula is often very useful in estimating output impedances quickly.
(5)
• For a bipolar transistor this relationship is modified to be
r out = r ds2 [ 1 + g m2 ( R s || r π ) ]
r out ≤ r ds2 [ 1 + g m2 r π ] ≅ r ds2 [ 1 + β ]
© D.A. Johns/K.W. Martin, 1995
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(7)
(8)
Cascode Current Mirrors
Wilson Current Mirror
V out
I in
I out
r out
2 ( V eff + V tn )
Q3
( V eff + V tn )
Q4
I in
r in
Q2
Q3
Q4
Q1
Q2
( V eff + V tn )
Q1
A cascode current-mirror.
The Wilson current-mirror.
• Using analysis of source-degenerated mirror, we have
r out = r ds4 [ 1 + r ds2 ( g m4 ′ + g ds4 ) ]
≅ r ds4 ( 1 + r ds2 g m4 ′ )
(9)
≅ r ds4 ( r ds2 g m4 ′ )
• Assuming all transistors have the same current densities and therefore effective
gate-source voltages, V eff , we have V GSi = V eff + V tn and therefore
V G1 = V eff + V tn
V G3 = V G4 = 2 ( V eff + V tn )
(10)
V S4 = V G4 – V GS4 = V eff + V tn
and since to keep Q 4 in active region, we need V DS4 > V eff , we need
V out > V tn + 2V eff ∼ 1.3V
I out
(11)
• This is an example of series-series feedback which increases output impedance
by 1 + A L where A L is the loop gain. For Wilson current-mirror, the loop-gain is
given by
g m1 ( r ds1 || r in )
(12)
A L ≅ -------------------------------------2
where r in is the output impedance of the input current source, I in . Therefore
g m1 r
g m1 ( r ds1 || r in )
ds1
r out ≅ r ds4 -------------------------------------- ∼ r ds4  ---------------------


2
2
• This current mirror also requires V tn + 2V eff across to keep the cascode
transistor out of the triode region.
• This is for n-channel mirror; p-channel mirror would also require about 1.3V
leaving 0.4V for signal swings assuming 3V supply voltages. This is not
acceptable.
© D.A. Johns/K.W. Martin, 1995
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(13)
© D.A. Johns/K.W. Martin, 1995
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Cascode Gain Stage
MOS Differential Pair
Ibias
Ibias1
Vout
Vbias
Q2
Vin
Q1
CL
I D1
Q2
Vin
Q1
V
Vbias
(a)
+
V
Q1
Vout
Ibias2
I D2
Q2
I bias
CL
(b)
-
A MOS differential pair.
(a) A telescopic cascode amplifier and (b) a folded-cascode amplifier.
• Currently very popular because a) gain of single stage is increased without
significantly decreasing speed, and b) minimizes short-channel effects of modern
technology.
i d1 = i s1
+
v
• Telescopic implementation is fastest (assuming n-channels used), folded-cascode
has compatible bias voltages at input and output.
r s1
• The low-frequency gain is on the same order of the gain of a cascode of two gain
stages having simple active loads (assuming the active load of the cascode stage
is a ‘good’ current source.
1 2 2
(14)
A V ∼ --- g m
r ds
2
• Note, that at low frequencies there is considerable gain to v s2 , on the order of
g m r ds ⁄ 2 . At higher frequencies this gain decreases to -1.
© D.A. Johns/K.W. Martin, 1995
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i d2 = i s2
v
i s1 i s2
+
-
r s2
-
v –v
--------------------r s1 + r s2
The small-signal model of a MOS differential pair.
v in
g m1
v in
i d1 = i s1 = – i d2 = --------------------- = ----------------------------------------- ≅ ---------- v in
1 ⁄ g m1 + 1 ⁄ g m2
2
r s1 + r s2
© D.A. Johns/K.W. Martin, 1995
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(15)
Bipolar Current Mirrors
MOS Differential-Pair with Active Load
Q3 Q4
i s1
r out
i d4
i s1
v in
Q1
i s1
v out
I in
I out
Q1
Q2
A simple bipolar current-mirror.
Q2
1
I out = ------------------------ I in ≈ ( 1 – 2 ⁄ β )I in
(1 + 2 ⁄ β)
I bias
I out1
I in
A differential-input single-ended-output MOS gain stage.
r out = r ds4 || r ds2
(16)
A V = g m1 r out = g m1 ( r ds4 || r ds2 )
(17)
(18)
I out2
Q3
Q1
Q4
Q2
A current-mirror with less inaccuracies caused by finite base currents.
• Note that all nodes are low impedance (i.e. impedances on the order of 1 ⁄ g m )
except the output node.
2
I out ≈ I in ( 1 – 2 ⁄ β )
I in
(19)
I out
Q3
Q1
Q2
Re
Re
A current-mirror with emitter degeneration.
© D.A. Johns/K.W. Martin, 1995
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© D.A. Johns/K.W. Martin, 1995
14
V R ≈ 0.2V
e
r out = r o2 ( 1 + g m2 ( R e || r π ) ) ≅ r o2 ( 1 + g m2 R e )
(20)
V Re V Re
R e = ----------- ≅ ----------I e2
I c2
(21)
Bipolar Differential Pair
I C1
Using the relationship g m2 = I c2 ⁄ V T , where V T = kT ⁄ q ≅ 26 mV at 300°K,
gives
V Re

r out ≅ r o2 ( 1 + g m2 R ) = r o2  1 + ----------- ≅ 11r o2
(22)
e
VT 

• More importantly, the emitter degeneration greatly minimizes the thermal noise
due to the base resistance which is often the dominant noise source in wide-band
bipolar circuits
V
Iout
Q1
Q2
V
-
+
-V
be2
I EE
A bipolar differential pair.
( V BE ⁄ V T )
IC = IS e
V id = V BE1 – V BE2
Iin
Iout
Q3
Q4
Q3
Q4
Q1
Q2
Q1
Q2
I C1
-------- = e
I C2
 V + – V -
 ------------------
 VT 
= e
(23)
(24)
 V id
 --------
 VT 
(25)
αI EE = I C1 + I C2
(26)
αI EE
I C1 = -----------------------------------( – V id ⁄ V T )
1+e
αI EE
I C2 = ---------------------------------( V id ⁄ V T )
1+e
(b)
(a)
+
V be1
High-Impedance Bipolar Current Mirrors
Iin
+
I C2
High output impedance current-mirrors (a) Cascode (b) Wilson.
• For both mirrors r out ≅ ( βr o ) ⁄ 2
(27)
(28)
I C1, I C2
• For cascode mirror, I out ≅ 1 – 4 ⁄ β ; for Wilson current mirror, I out ≅ 1 – 2 ⁄ β 2 .
Moral: always use Wilson current mirrors for bipolar design
αI EE
I C2
I C1
0.5αI EE
–4
–2
0
2
Collector currents for a bipolar differential-pair.
© D.A. Johns/K.W. Martin, 1995
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© D.A. Johns/K.W. Martin, 1995
16
4
V id ⁄ V T
Frequency Response
sC gs1 + g m1
A ( s ) = A ( 0 ) ----------------------------------s
s2
1 + ----------- + ------ω0 Q ω 2
0
Common-Source Stage
Rin
v1
Cgd1
where
vout
+ C
gs1 g v
v gs1
m1 gs1
-
vin
R2
ω0 =
C2
1
ω – 3dB ≅ ----------------------------------------------------------------------------------------------------------------------------R in [ C gs1 + C gd1 ( 1 + g m1 R 2 ) ] + R 2 ( C gd1 + C 2 )
(29)
1
ω – 3dB ≅ ----------------------------------------------------------------R in [ C gs1 + C gd1 ( 1 + A ) ]
(30)
Rin
C' in
(32)
(33)
and
π
– ----------------------2
4Q – 1
(34)
% overshoot = 100e
• Source follower is fast, but can have overshoot especially when R in is small and
C L is large.
Common-Gate Stage
Source Follower
iin
G in ( g m1 + G s1 )
-------------------------------------------------------------------C gs1 C s + C' in ( C gs1 + C s )
G in ( g m1 + G s1 ) [ C gs1 C s + C' in ( C gs1 + C s ) ]
Q = -----------------------------------------------------------------------------------------------------------------------G in C s + C' in ( g m1 + G s1 ) + C gs1 G s1
A small-signal model for high-frequency analysis of the common-source amplifier.
vg1
(31)
• No Miller effect so time constant at input node seldom dominates. This makes it
typically much faster than common source stage. Time constant at output (i.e.
( r ds ⁄ 2 ) ( C L + C db ) ) usually dominates.
Yg
Cgs1
vgs1
g m1v gs1
vout
• Time constant at input (at high frequencies) is on the order of ( 1 ⁄ g m )C gs . This
can be important when determining approximate second pole frequency.
Cascode Gain Stage
Rs1
C' in = C in + C gd1
Cs
R s1 = r ds1 ||r ds2 1|| ⁄ g s1
• No Miller effect so often faster than common-source stage.
2 )(C
• Time constant at output (i.e ( g m r ds
db2 + C L ) ) dominates.
A simplified equivalent small-signal model for the source-follower.
• Time constant at source of cascode transistor is approximately
L 22
τ s2 ∼ ------------------µ p V eff2
(35)
This fundamental relationship assumes a folded-cascode stage with Q 2 being pchannel
© D.A. Johns/K.W. Martin, 1995
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© D.A. Johns/K.W. Martin, 1995
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References
• P. R. Gray and R. G. Meyer, Analysis and Design of Analog Integrated Circuits,
3rd ed., John Wiley and Sons, New York, 1993.
• D. Johns and K. Martin, Analog Integrated Circuits, Wiley, 1997.
• A.S. Sedra and K.C. Smith, Microelectronic Circuits, 3rd ed., Holt, Rinehart and
Winston, New York, 1991.
© D.A. Johns/K.W. Martin, 1995
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