broadband ampli ers - university of california,...
Post on 21-Mar-2018
233 Views
Preview:
TRANSCRIPT
Berkeley
Broadband Amplifiers
Prof. Ali M. Niknejad and Dr. Ehsan Adabi
U.C. BerkeleyCopyright c© 2014 by Ali M. Niknejad
Niknejad Advanced IC’s for Comm
Outline
Broadband Amplifiers
Shunt-Peaking
Distributed Amplifiers
Multi-section Matching (Bode-Fano Limits)
Transformer Matching Networks
Niknejad Advanced IC’s for Comm
Cascade Amplifier Bandwidth Shrinkage
Consider an amplifier consisting of a cascade of identicalsingle-pole stages
G (s) =G0
1 + sτ
The bandwidth of n stages can be derived as follows
Gn(s) = G (s)n =Gn
0
(1 + sτ)n
|Gn(jω0)| =Gn
0
|1 + jω0τ |n=
Gn0
|1 + ω20τ
2|n =Gn
0√2
21/n − 1 = ω20τ
2
ω0τ =√
21/n − 1
Bandwidth shrinks rapidly compared to the single stage.Three stages =⇒ bandwidth drops by half
Niknejad Advanced IC’s for Comm
Common Source Amplifier Bandwidth
RL
CL
Cgs
Cgd
Rs
Classic amplifier has several poles. The poles can becalculated as
τgs = CgsRs
τgd = Cgd(Rs + RL + gmRsRL)
τL = CLRL||ro
Niknejad Advanced IC’s for Comm
Minimizing the effect of Cin
RL
CL
Rs
RL
CL
Rs
M1
M2
M3
The effect of Cgd can be minimize with a cascodeconfiguration.
The load can be isolated with a buffer M3 (τL can be reduced)
Niknejad Advanced IC’s for Comm
More Buffers
RL
CLRs
Similarly, the input capacitance can be isolated with a buffer(τgs reduced)
We see that we can trade speed for power consumption.
Niknejad Advanced IC’s for Comm
Feedback Amplifiers
f
G0
ω0
0 dB
G0ω0
For low order systems (with one dominant pole), product ofgain and bandwidth is constant
G =G0
1 + sτ
GCL =G
1 + Gf=
G01+sτ
1 + G01+sτ f
=G0
1 + sτ + G0f=
G0G0f +1
1 + sτ 1G0f +1
=G0
1+T
1 + s τ1+T
Gain×BW =G0
1 + T×1 + T
τ=
G0
τ
Niknejad Advanced IC’s for Comm
Shunt-Series Amplifier
RL
CL
Cgs
Cgd
Rs
R
RF
By using feedback, Gain ↓Ri and Ro ↓, matching acquiredBW ↑
By using feedback, we reduce the gain, reduce Ri and Ro
(desired for output matching), and increase the bandwidth
Av =−RL
RE
RF − RE
RF + RE
RE =gm
1 + gmR1
Rin =RE (RF + RL)
RE + RL
Rout =RE (RF + RS)
RE + RS
BW × Av =1
Cgs
gm+
RLCgd
2
Niknejad Advanced IC’s for Comm
Taking Advantage of a Zero
RL CL
Rs
vs
vout
Consider the step function of a low pass circuit. The outputtracks the input with a time constant of τ :
τ = CL(Rs ||RL)
vout
t
Niknejad Advanced IC’s for Comm
Feedforward with a Capacitor
RL CL
Rs
vs
vout
Cs
Insight: Add a feedthrough capacitance CS so that the edgeof our signal propagates to the output immediately :
at t = 0+ −→ vout =Cs
CL + Csvs
at t =∞ −→ vout =RL
RL + Rsvs
τ = Rs ||RL(CL + Cs)
vout
t
Cs
Cs + CL
RL
RL +Rs
Niknejad Advanced IC’s for Comm
Transfer Function with Zero
To see this, we can derive the full transfer function:
voutvs
=
RL1+RLCLs
RL1+RLCLs
RS1+RSCLs
=RL
RL + Rs
1 + RsCss
1 + (RL||Rs)(CL + Cs)s
zero z =−1
RsCs
pole p =−1
(RL||Rs)(CL + Cs)
If we equalize the pole / zero, the pole zero cancel and wehave a perfect step !
if p = z −→ RsCs = RLCL
vout
t
Niknejad Advanced IC’s for Comm
Application of a Zero
RL
CL
Rs Cs
vs
If RsCs = RLCL, then the gain is approximately constant overa fraction of the device fT
voutvs≈ Zout
Zin−→ constant
Niknejad Advanced IC’s for Comm
Shunt Peaking Amplifier
RL
CL
L
vin
vout RC network
RL network
|Z|
f
Use an inductor at the drain to produce a zero. The zero“peaking” location should occur at a high frequency tocompensate for the gain roll-off due to the pole(s)
Note that inductor does not need to be a high Q componentsince it’s in series with a large resistor. Can build it usingmultiple layers in series to make the inductor compact.
ZL = (R + Ls)|| 1
sC=
R[s LR + 1
]s2LC + sRC + 1
Niknejad Advanced IC’s for Comm
Shunt Peaking Design Equations
|ZL|R
=
√1 + (ωτ)2
(1− ω2τ2m2) + (ωτm)2
m Normalized BW Normalized Peak
Max BW√
2 1.85 1.19|Z | = R at ω = 1/RC 2 1.8 1.03
Maximally flat 1 +√
2 1.72 1Best group delay 3.1 1.6 1No shunt peaking ∞ 1 1
Can trade off between bandwidth (85% increase) versus groupdelay variation (60% increase in bandwidth).
Niknejad Advanced IC’s for Comm
More Shunt Peaking
Cck
Shunt and series peaking
Shunt and double seriespeaking. T-coil bandwidthenhancement.
Basically the order of thematching network is increasingand it’s resembling asynthesized transmission line.
Niknejad Advanced IC’s for Comm
Why does this help?
In the above structures, parasitic capacitors are charged anddischarged serially so that the current available to charge acapacitor is more and hence the rise time is shorter at theexpense of delay
Can we take this idea to the limit?
Niknejad Advanced IC’s for Comm
Distributed Amplifier
+vs−
+vo−
Zd
Zg
M1 M2 M3 M4
γd, Zd
γg, Zg
�g
�d
Zd
Zg
The goal is to convert the lumped amplifier into a distributedstructure. The idea is to take a fixed gm (transistor widthW ), and split it into parallel fingers that are embedded into atransmission line at the gate and drain.
Both transmission lines need to be properly terminated to seeflat impedance with frequency. The propagation constant onthe gate and rain line need to be matched so that the wavesadd constructively.
Niknejad Advanced IC’s for Comm
Distributed Amplifier Gain
+vs−
+vo−
Zd
Zg
Zd
Zg
rogmv1 Corogmv2 Co
rogmv3 Corogmv4 Co
Cπ
+v1−
Ri
Cπ
+v2−
Ri
Cπ
+v3−
Ri
Cπ
+v4−
Ri
vgs,i =vs2
e−j(i−1)βg `g
βg is the propagation constant on the gate line. The loadcurrent is also a summation of N currents each coming fromthe input transistors
Id =1
2
N∑i=1
id ,ie−(N−i)jβd `d
Niknejad Advanced IC’s for Comm
Gain (cont)
id ,i = −gmvgs,i resulting in
Id = −gm4
vs
N∑i=1
e−(i−1)jβg `g e−(N−i)jβd `d
= −gm4
vse−Njβd `d e jβg `gN∑i=1
e−ij(βg `g−βd `d )
The above equation applies for any arbitrary line, butobviously we’d like to synchronize the delay on the gate anddrain line
βg `g = βd`d = θ
Id = −gm4
vse−(N−1)jθ · N
G =Pout
Pin=
12 |Id |2Zd
18 |vs |2/Zg
=g 2mN2ZdZg
4
Niknejad Advanced IC’s for Comm
Artificial Distributed Amplifiers
+vs−
+vo−
Zd
Zg
M1 M2 M3 M4
�g
�d
Zd
Zg
Additive gain versus multiplicative gain obtained in cascade.Bandwidth is extremely high, up to fT/2. In practice the gainwill vary due to the properties of the artificial transmissionline, particularly the cut-off frequency.
Niknejad Advanced IC’s for Comm
The Right Terminationsm-derived Sections
4 Stage DA Block
1
2 3
4
m-derived Sections
To improve the performance of a DA, it’s important to takeinto account the frequency variation of the impedance of theline due to the fact that it’s actually an artificial lumped linerather than a truly distributed line.The m-derived sections are loads that terminate the line inorder to to provide a match over a frequency rangeapproaching the line cut-off. Otherwise spurious reflectionswould occur and cause the gain to roll-off faster.
Niknejad Advanced IC’s for Comm
A New Twist on Distributed Amplifiers
Core DAOutput DA
Input DA1
2
4
3
1
2
4
3
1
2
4
3
Filter
MM
MM M
M
M
LOAD
Zx
ZY
M-Derived Sections
Internal Feedback
Notice that a drain line wave can be fed back into the DA andit can travel back through the gate line in the oppositedirection, thereby generating a cascade gain from the sameDA!
Input and output DA’s used to provide broadband match.
A. Arbabian and A. M. Niknejad “A broadband distributed amplifier with internal feedback providing 660 GHzGBW in 90 nm CMOS,” Int. Solid-State Circuits Conf. Tech. Dig., pp.196 -197 2008
Niknejad Advanced IC’s for Comm
Tapering the Line TaperedTapered--Line Amplifier Line Amplifier
o o
o
Forward traveling currents add constructivelyReverse traveling currents cancel
D1 D2I I if and properly delayed:
D12I
ID1 ID2
� � D22 /3 I� � D21/3 I
� � D14 /3 I� � D11/3 I�
Two section (n=2) example
© 2009 IEEE International Solid-State Circuits Conference © 2009 IEEE
In a DA, half the power is wasted on the second draintermination. In a PA, that’s a lot of power to throw away.By tapering the line, we can eliminate reverse-wavepropagation and hence termination.In this two-section example, the reflection and transmissioncoefficient are given byρ1 =
(Z0/2) − Z0
(Z0/2) + Z0
=−1
3τ1 =
2Z0
(Z0/2) + Z0
=4
3
Note that if the currents are properly delayed, the reversecurrents can cancel
J. Roderick and H. Hashemi “A 0.13 µm CMOS power amplifier with ultra-wide instantaneous bandwidth forimaging applications”, IEEE Int. Solid-State Circuits Conf. Tech. Dig., vol. 1, pp.374 -376 2009
Niknejad Advanced IC’s for Comm
Broadband Matching Networks
+vs−
YS
YL
Y ∗in
Yin
Y ∗out
Yout
Input
Match
Output
Match
Consider that many core amplifiers are broadband but toobtain the optimal gain requires matching, and the LCmatching networks introduce bandwidth limitations.
Can we make broadband matching networks? Bode-Fanoprovides a clue ...
Niknejad Advanced IC’s for Comm
Bode-Fano Criterion
What’s the best we can do with a matching network in termsof the quality of the match Γ ∼ 0 and bandwidth?
Surprisingly, there is a theoretical answer to this question andthe answer depends on the load (reactance versus resistance)[Bode][Fano]. In other words, if we’re trying to match to adevice that has capacitance input impedance with some realpart, there is a fundamental limit to the bandwidthachievable. For an RC shunt load∫ ∞
0ln
1
|Γ(ω)|dω ≤π
RC
and for an RC series load∫ ∞0
ln1
|Γ(ω)|dω ≤ πRC
Niknejad Advanced IC’s for Comm
Bode-Fano Criterion
For example, imagine a Brickwall match shown over abandwidth B. This result implies that∫ ∞
0ln
1
|Γ(ω)|dω = B ln1
|Γ0|≤ πRC
So we cannot in particular go to zero reflection (perfectmatch) over an interval of frequencies, only at a finite numberof frequencies. Moreover, there’s a trade-off between theobtainable bandwidth and the quality of the match. They arenot independent quantities.
Niknejad Advanced IC’s for Comm
Active Load Filter Matching
active filter
+vs−
+vo−
L1C1
L2C2
Lg
Ls
Cp
LL
RL
Note that the input impedance of an inductively degeneratedamplifier looks like an LCR network. Make that thetermination of a ladder section (broadband) filter !
The core amplifier can be made broadband by using inductivepeaking.
Niknejad Advanced IC’s for Comm
Balanced Amplifiers
0o
90o0o
90o
Z0Z0 Z0
Zin
Z1
If a core amplifier is broadband but poorly matched, we canalso use a coupler to drive two amplifiers in parallel as shown.
Note that the reflected signal from the top amplifier is 180◦
out of phase with the reflected signal from the bottomamplifier ! The reflected signals cancel out.
The bandwidth limitation now comes from the design of abroadband quadrature coupler.
Niknejad Advanced IC’s for Comm
Stagger Tuning
0.25 0.5 0.75 1 1.25 1.5 1.75 2
0.2
0.4
0.6
0.8
1
0 0.25 0.5 0.75 1 1.25 1.5
-30
-25
-20
-15
-10
-5
0
A multi-stage amplifier can be made broadband by staggertuning the various stages
There’s a trade-off in bandwidth versus gain and gain flatness.
Niknejad Advanced IC’s for Comm
Transformer Matching
Writing the mesh equations for the transfer function of anideal transformer
voutvs
=RLMs
s2(L1L2 −M2) + s(RsL2 + RLL1) + RsRL
voutvs
=RLM
L1L2 −M2
s
(s − p1)(s − p2)
If p2 � p1, we can simplify the transfer function
p1 =−RsRL
RsL2 + RLL1
p1 + p2 ≈ p2 =−RsL2 + RLL1
L1L2 −M2
Mid band gainvoutvs
=RLM
RsL2 + RLL1
Niknejad Advanced IC’s for Comm
Low k Coupling Transformers
If we include the capacitance in the transformer, it’s actually afourth order circuit. Intuitively, there are two modes due toresonance and anti-resonance.
In resonance the mutual coupling adds to the effectiveinductance and the coupling capacitance is neutralized.
In anti-resonance, the mutual coupling subtracts from theeffective inductance and the coupling is excited. We can seethat the anti-resonance mode is at a higher frequency thanthe resonance mode.
If the coupling is strong, these modes are very far apart infrequency. If the coupling is weak, these modes can be movedclose together to give two peaks in the transfer function.Similar to stagger tuning, we can broadband the response bymoving peaks close together in an optimal fashion.
Niknejad Advanced IC’s for Comm
Coupled Resonator Matching
C1 L1R1 C2L2 R2
Cc
The transfer function is a 4th order circuit. We can build it asis or convert it into a transformer coupled circuit by usingDuality (and Y-∆ transformation):
C1R1 C2 R2
L1 −M L2 −M
M
Niknejad Advanced IC’s for Comm
Capacitively Coupled Resonator Stages
Vecchi et. al., “A Wideband Receiver for Multi-Gbit/s Communications in 65 nm CMOS”, JSSC 2011
The transfer function of two coupled resonators can beapproximated by the product of two second-order transferfunctions.
Depending on the strength of the coupling, the poles of thesystem can be staggered optimally to provide a flat passband.
Niknejad Advanced IC’s for Comm
Transfer Function Equations
30 40 50 60 70 80 90 100
10
20
30
40
5040f
30f20f
10f
GCR(s) = −gms3kQω0
√R1R2
(Q(1 + k)s2 + sω0 + Qω2)(Q(1− k)s2 + sω0 + Qω2)
ω0 = 1/√
L1(C1 + Cc) = 1/√
L2(C2 + Cc)
Q = R1/ω0L1 = R2/ω0L2
k = Cc/√
(C1 + Cc)(C2 + Cc)
Niknejad Advanced IC’s for Comm
Matching Network “Filter” Design
Matching network design is very similar to filter design.
Given a transfer function, you can trade-off flatness for groupdelay variation, or try to maximize the attenuation andparticular frequencies.
For example, for maximally flat, make as many derivativeszero near ω = 1
H(s) =s
s4 + a3s3 + a2s2 + a1s + a0
Niknejad Advanced IC’s for Comm
Optimal Fourth Order Transfer Function
C1
L1 −M L2 −M
M
R1
C2 RL
R2
is
vout
vo
io= −RL
sM
s3M2C1(1 + sRLC2) − RL(1 + sR1C1 + s2L1C1) − (R2 + sL2)(1 + sRLC2)(1 + sR1C1 + s2L1C1)
For M � 1, (low coupling factor k)
voutis
=sM
(1 + sR1C1 + s2L1C1)(1 + R2RL
+ s( L2RL
+ R2C2) + s2L2C2)
Niknejad Advanced IC’s for Comm
Maximizing Gain
voutis
=sM
(1 + sR1C1 + s2L1C1)(1 + R2RL
+ s( L2RL
+ R2C2) + s2L2C2)
L1C1 ≈ L2C2 ≈1
ω0
|Z (jω0)| =k√
L1
1Q1
(√L2
RL+ 1√
L2ω0Q2
)Increasing L1 and L2 increases the gain. Limited by qualityfactor and resonance.
Niknejad Advanced IC’s for Comm
References
1 H. W. Bode, Network Analysis and Feedback AmplifierDesign, Van Nostrand, N.Y., 1945.
2 R. M. Fano, “Theoretical Limitations on the Broad-BandMatching of Arbitrary Impedances,” Journal of the FranklinInstitute, vol. 249, Jan. 1950.
Niknejad Advanced IC’s for Comm
top related