Broadband Ampli ers - University of California, Berkeleyrfic.eecs.berkeley.edu/ee242/pdf/Module_3_4_AmpsBroad.pdf · Outline Broadband Ampli ers Shunt-Peaking Distributed Ampli ers

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


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


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


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)


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.


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


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


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


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


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


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


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


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).


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.


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?


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.


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


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


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.


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.


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


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


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 ...


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


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.


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.


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.


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.


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


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.


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


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.


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)


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


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)


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.


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.


Documents

Broadband Ampli ers - University of California, Berkeleyrfic.eecs.berkeley.edu/ee242/pdf/Module_3_4_AmpsBroad.pdf · Outline Broadband Ampli ers Shunt-Peaking Distributed Ampli ers