Chapter 3 Single-Stage

Zou Zhige 2006 HUST

Zou Zhige CMOS Chapter3 1 Table of contents

• Introduction • Common-Source Amplifiers • Source follower Amplifiers • Amplifiers • Amplifiers

Zou Zhige CMOS Chapter3 2 Overview

• Reading –Chapter 3 • Introduction In this lecture, we study the low-frequency behavior of single-stage CMOS amplifiers. Analyzing both the large-signal and the small-signal characteristics of each circuit, we develop intuitive techniques and models that prove useful in understanding more complex systems. Following a brief review of basic concepts, we describe in this chapter four types of amplifiers: ---Common-Source ---Common-Gate Amplifier ---Source Followers ---Cascode

Zou Zhige CMOS Chapter3 3 Questions

• What is amplifiers? • Why we need to learn amplifiers? • What will we learn about amplifiers? • How to learn?

Zou Zhige CMOS Chapter3 4 What is Amplifier

• The IO characteristic of an amplifier is a nonlinear function:

2 n y(t) » a0 + a1x(t) + a2 x (t) +K + an x (t) x1 £ x £ x2 • For a sufficiently narrow range of x:

y(t) » a0 + a1x(t)

• Where a0 is the operating (bias) point and a1 is the small-signal gain.

Zou Zhige CMOS Chapter3 5 Why we need to learn amplifiers?

• Microphone • Communications – Wireless – Optical-fiber • Disk Drive Electronics • Processing of Natural Signals – Sensors

• Amplifier is a key circuit

Zou Zhige CMOS Chapter3 6 What will we learn about amplifiers?

• Basic configurations of amp. • The main characteristics of amp. • How to analyze & design amp. – Signal-----Large Signal, Small Signal(*) – Model-----LSM, SSM(*), Two-Port Model – Parameter----- • Device parameter-----W,L • Circuit parameter------Rin, Rout, Av, Ai, Bias • Frequency characteristics-----Frequency response • Gain, speed, power dissipation, noise,…

Zou Zhige CMOS Chapter3 7 How to Learn?

• Lear the operation conditions of different amp. • Distinguish different characteristics of different amp. • DC and ac Analysis

Zou Zhige CMOS Chapter3 8 DC and ac Analysis

• DC Analysis (Large-Signal Analysis) --- Determine the exact . • AC Analysis (Small-Signal Analysis) --- Obtain the expression of the voltage gain, small-signal input and .

Zou Zhige CMOS Chapter3 9 Tradeoffs

Most of these parameters trade with each other. Such trade-offs lead to many challenges in the design of high performance amplifiers.

Zou Zhige CMOS Chapter3 10 Single-Stage Amplifiers

Zou Zhige CMOS Chapter3 11 Table of contents

• Introduction • Common-Source Amplifiers – CS with Resistive Load – CS with diode-connected MOS load – CS with Load – CS with Triode Region Load – CS with source degeneration • Source follower Amplifiers • Common Gate Amplifiers • Cascode Amplifiers

Zou Zhige CMOS Chapter3 12 3.2 Common Source

Zou Zhige CMOS Chapter3 13 3.2.1 Common Source (CS) with Resistive Load

• Resistive Load is often used in high-speed circuit because of the linearity of resistance, and also the output voltage swing

may reach up to VDD.

Fig. 3.1 (a) CS Stage; (b) input-output characteristic; (c) equivalent circuit in deep triode region

Zou Zhige CMOS Chapter3 14 Method of Analysis

• Large-Signal Analysis – Input range, output range, operation region – Plot of Vout V: Vin • Small-Signal Analysis – Small-signal equal circuit – Av of saturation region (easy to difficult: No gmb àgmb, No ro à ro) – Rin and Rou • Different load • Discussion

Zou Zhige CMOS Chapter3 15 CS with Resistive Load (cont.)

• DC Analysis (Large-Signal Analysis)

(1) When V in < VTH , M1 is in cut-off region, Id=0, Vout=VDD-IdRD=VDD

(2) When V in > VTH , and V in < V in1 , M1 is in saturation region. mnCox W V =V - (V -V )2 R out DD 2 L in TH D Here we have neglected channel length modulation.

(3) When Vin - VTH = Vout , M1 is at the boundary of saturation and triode regions. mnCox W V -V = V - (V -V )2 R in1 TH DD 2 L in1 TH D

From which Vin1-VTH and hence Vout can be calculated. 1 W 1 W 1+ 4aVDD -1 a = m C ( )R = k ( )R V = +V where n ox D N D in1 2a T H 2 L 2 L k = mCox ....Technology parameter

Zou Zhige CMOS Chapter3 16 CS with Resistive Load (cont.)

(4) For Vin > Vin1, M1 is in the triode region,

mnCox W V =V - R [2(V -V )V -V 2 ] out DD 2 L D in TH out out

As VGS has less control on Id when M1 works in triode region, we usually leave M1 in saturation for a large voltage gain.

If Vin is high enough to drive M1 into deep triode region. Vout<<2(Vin-VTH), and, R V V =V on = DD out DD R + R W on D 1+ m C R (V -V ) n ox L D in TH

In very deep triode range, Ron→0, Vout →0

Zou Zhige CMOS Chapter3 17 CS with Resistive Load (cont.)

• AC Analysis (Small-Signal Analysis) (1) Derivation at the operation point Assuming that the transistor is biased in strong inversion, active region

mnCox W V =V - (V -V )2 R out DD 2 L in TH D

Taking the derivative of Vout with respect to Vin, we get,

dVout W Av = = -RDmnCox (Vin -VTH) =-gmRD dVin L

Zou Zhige CMOS Chapter3 18 CS with Resistive Load (cont.) What is? How to draw? (2) Finding the gain from the small-signal equivalent circuit

Fig. 3.2 (a) CS Stage; (b) small-signal equivalent circuit

Zou Zhige CMOS Chapter3 19 CS with Resistive Load (cont.)

(3) Intuitive observation

Av = -g m RD

This result can be directly derived from the observation that

M1 converts an input voltage change ΔVin to a drain current change gm ΔVin , and hence an output voltage change - gmRD ΔVin .

Zou Zhige CMOS Chapter3 20 CS with Resistive Load (cont.)

• Taking the effect of channel length modulation in M1 into account, the small-signal equivalent circuit is modified as following,

Fig.3.3 small-signal equivalent circuit including the output resistance of M1

Av = -gm(RD ro )

How high can the gain reach ??

Zou Zhige CMOS Chapter3 21 CS with Resistive Load (cont.)

• Intrinsic Gain

If RD=∞, then Av = -gmro called the “intrinsic gain”of a transistor, this quantity represents the maximum voltage gain that can be achieved using a single device. For ideal long-channel

device, ro→ ∞, intrinsic gain → ∞; however, in today’s Fig 3.4 CMOS technology, intrinsic gain of short-channel device

is between roughly 10 and 30. Thus, we usually assume 1/gm << ro .

Question?: In Fig.3.4, Kirchhoff’s current law (KCL) requires that ID1 = I1. Then, how

can Vin change the current of M1 if I1 is constant?

Zou Zhige CMOS Chapter3 22 Small Signal Rin and Rout of CS

• What is small signal impedance and how to calculate? (Next page ) • of CS is infinite

• Output impedance of CS is ro

Zou Zhige CMOS Chapter3 23 Small Signal Input and Output Impedances

• Calculations of Small Signal Input and Output Impedances How to calculate input and output impedances (or admittances) of an amplifier? In the following sections, we assume that the amplifier is a voltage amplifier, whose input and output are both voltages. But we can easily extend the principles to any other types of amplifiers, such as current amplifiers (input and output are both currents), transimpedance amplifiers (input: current, output: voltage), and transconductance amplifiers (input: voltage, output: current).

Zou Zhige CMOS Chapter3 24 Small Signal Input and Output Impedances (cont.)

1) Input impedance

i) Apply vtst at the input (* see note below), draw the small signal diagram.

ii) Calculate itst = f (vtst) , or vtst = f (itst)

iii) The input impedance is given by zin = vtst / its

* Note: If the amplifier requires an output termination, we should terminate the output accordingly. The load condition may affect the input impedance.

Zou Zhige CMOS Chapter3 25 Small Signal Input and Output Impedances (cont.)

2) Output impedance

i) Set vin=0 , or if the input is a signal current, set iin= 0 (** see note below). ii) Apply vtst at the output, draw the small signal diagram. iii) Calculate itst = f (vtst) or vtst = f (itst) iv) The output impedance is given by zout = vtst / itst

** Note: If the amplifier requires some input termination, we should terminate the input accordingly. The input termination may affect the output impedance.

Zou Zhige CMOS Chapter3 26 Table of contents

• Introduction • Common-Source Amplifiers – CS with Resistive Load – CS with diode-connected MOS load – CS with Current Source Load – CS with Triode Region Load – CS with source degeneration • Source follower Amplifiers • Common Gate Amplifiers • Cascode Amplifiers

Zou Zhige CMOS Chapter3 27 3.2.2 CS stage with diode-connected MOS load

A MOST can operate as a small-signal resistor if its gate and drain are shorted, called “diode-connected”. • what is the impedance of the following circuit seen from the source side of transistor M1?

Zou Zhige CMOS Chapter3 28 CS stage with diode-connected MOS load (cont.)

is = -gmvgs - gmbvbs +vsgds

= gmvs + gmbvs + gdsvs

=(gm + gmb + gds)vs Seen from source terminal, the small signal conductance is given by

is yin = = (g m + g mb + gds ) vs or, the resistance is 1 1 1 rin = = » ym g m + g mb + g ds g m + g mb

R become smaller because of body effect ! How to reduce BE?

Zou Zhige CMOS Chapter3 29 CS stage with diode-connected MOS load (cont.)

1 gm1 1 Av = -gm1 = - gm2 + gmb2 gm2 1 + h

(W / L)1 1 Av = - (W / L)2 1 + h

Gain is independent of bias current!

Zou Zhige CMOS Chapter3 30 CS stage with diode-connected MOS load (cont.)

• Advantage of CS with diode-connected load: linearity

(W / L)1 1 Av = - (W / L)2 1 + h This equation imply: if the variation of ηwith the output voltage is neglected, the gain is independent of the bias currents and voltages (so long as the M1 stays in saturation. In other words, as the input and output signal levels vary, the gain remains relatively constant, indicating that the input-output characteristic is relatively linear.

•Homework: Please do large-signal analysis of this circuit to determine the biasing range.

Zou Zhige CMOS Chapter3 31 CS stage with diode-connected MOS load (cont.)

I D1 = I D2

æ W ö 2 æ W ö 2 un (VGS1 - VTH1) =up (VGS2 - VTH2) è ø è ø L 1 L 2

un (W / L) |VGS2 -VTH2 | - 1 = up (W / L)2 (VGS1 -VTH1)

un (W / L)1 Av = - up (W / L)2 This implies substantial voltage swing constraint.

Zou Zhige CMOS Chapter3 32 Example: CS with any type of load

Zou Zhige CMOS Chapter3 33 Example: CS with any type of load (cont.)

• Loads in series :

Zou Zhige CMOS Chapter3 34 Example: CS with any type of load (cont.)

• Loads in parallel :

Zou Zhige CMOS Chapter3 35 Example 3.3

What is the voltage gain of the amplifier ? I s = 0.75I1

Zou Zhige CMOS Chapter3 36 Short Channel Effect for the CS

• In today’s technology, channel-length modulation is quite significant. • The gain of CS should modified as:

æ 1 ö ç ÷ Av = -gm ç || ro1 || ro2 ÷ è gm2 ø

Zou Zhige CMOS Chapter3 37 Table of contents

• Introduction • Common-Source Amplifiers – CS with Resistive Load – CS with diode-connected MOS load – CS with Current Source Load – CS with Triode Region Load – CS with source degeneration • Source follower Amplifiers • Common Gate Amplifiers • Cascode Amplifiers

Zou Zhige CMOS Chapter3 38 3.2.3 CS with Current Source Load

Av = -gm ro1 || ro2

Assuming ro2 large,

æ ö ç W ÷ 1 A = -g r = - 2m C I ç ÷ v m o1 n ox D ç ÷ è L ø 1 lI D

Zou Zhige CMOS Chapter3 39 CS with Current Source Load (cont.)

• Advantage of CS with current source Load : the output

impedance and the minimum required VDS of M2 are less strongly coupled than the value and voltage drop of a resistor.

• Home work 2: Please do large-signal analysis of this circuit to determine the biasing range. How to increase the gain?

Zou Zhige CMOS Chapter3 40 CS with Current Source Load (cont.)

• Discussions: (1) Gain increasing:

a) increase ro1 and ro2 ,

1 1 L r = l µ r µ o1 lI L o D ID so increase L is an effective method.

b) increase gm1, W g=-m C()VV m1 noxL GSTH

Increasing W while keeping the Vov and L constant, gm1 increases. Note: increasing V is no use, as 2 ov IDµ-()VVGSTH

Zou Zhige CMOS Chapter3 41 CS with Current Source Load (cont.)

• (2) Design consideration: a) When requiring a large voltage gain, we usually design the

device with a large L2 of load M2 and a large W1 of M1, large L1 is not must.

b) If L1 is scaled by a factor n (>1), then W1 may need to be scaled proportionally as well. This is because, if W1 is not scaled, the overdrive voltage increases, limiting the output voltage swing.

Zou Zhige CMOS Chapter3 42 Table of contents

• Introduction • Common-Source Amplifiers – CS with Resistive Load – CS with diode-connected MOS load – CS with Current Source Load – CS with Triode Region Load – CS with source degeneration • Source follower Amplifiers • Common Gate Amplifiers • Cascode Amplifiers

Zou Zhige CMOS Chapter3 43 3.2.4 CS with Triode Region Load

Av = -gmRON2

1 R = ON 2 æ ö ç W ÷ m pCoxç ÷ (VDD -Vb - |VTHP |) è L ø 2

Zou Zhige CMOS Chapter3 44 Table of contents

• Introduction • Common-Source Amplifiers – CS with Resistive Load – CS with diode-connected MOS load – CS with Current Source Load – CS with Triode Region Load – CS with source degeneration • Source follower Amplifiers • Common Gate Amplifiers • Cascode Amplifiers

Zou Zhige CMOS Chapter3 45 3.2.5 CS with source degeneration

• In some applications, the square-law dependence of the drain current upon the overdrive voltage introductions excessive nonlinearity, making it desirable to “soften”the device characteristic. • In 3.2.2, we noted the linear behavior of a CS stage using a diode-connected load, but this topology has substantial voltage swing constraint. • Some other method?

Zou Zhige CMOS Chapter3 46 CS with source degeneration (cont.)

•CS with source degeneration

Define: Gm----the transconductance of the circuit when the output is shorted to ground.

¶¶IIDD¶¶VVGSGS Ggmm==×= ¶¶VVinGS¶¶VVinin

VGS=-VinIRDS

¶VGS ¶ID =11-RS=-GRmS ¶¶VVinin g G = m Gm=-gm(1 GRmS) m 1+ gRmS

gRmD AV=-GRmD=- 1+ gRmS

Zou Zhige CMOS Chapter3 47 CS with source degeneration (cont.)

If we ignore λ and γ :

Zou Zhige CMOS Chapter3 48 CS with source degeneration (cont.)

• With source degeneration, the transconductance is more linear !!

As Rs increases, Gm becomes a weaker function of gm and hence the drain current.

Zou Zhige CMOS Chapter3 49 CS with source degeneration (cont.)

• Intuitive analysis:

- gm RD RD Av = = - 1+ gm RS 1/ gm + RS

Drain current and transconductance of a CS device (a) without and (b) with source degenaration.

Zou Zhige CMOS Chapter3 50 Example 3.5

• Assuming λ =γ = 0 for both M1 and M2. Calculate the small signal gain.

Zou Zhige CMOS Chapter3 51 Example 3.5 (cont.)

• Recall the result from common source stage with source degeneration :

Zou Zhige CMOS Chapter3 52 Gm of CS with source degeneration

• What is the transconductance of the circuit (Gm) if we do not ignore λand γ ?

Zou Zhige CMOS Chapter3 53 Gm of CS stage with source degeneration (cont.)

gm ro or, Gm = Rs +[1+ (gm + gmb )Rs ]ro

Zou Zhige CMOS Chapter3 54 Output Resistance of the CS stage with source degeneration

•Another importance consequence of source degeneration is the increase of the output impedance of the stage.

rout = [1+ (gm + gmb )ro ]Rs + ro Much More Bigger !

Zou Zhige CMOS Chapter3 55 Gain (Av) of the CS stage with source degeneration

vout gmro RD Av = = - vin RD + Rs + ro + (gm + gmb )Rsro

Av = -Gm (RD || rout )

Lemma: In a linear circuit, the voltage gain is equal to -GmRout

Zou Zhige CMOS Chapter3 56 Example 3.6

Av = -Gm RD || rout

gmro Gm = RS +[1+ (gm + gmb )RS ]ro

RD || rout = ¥ ||{[1+ (gm + gmb )ro ]Rs + ro}

Av = -gmro !

Why? Please explain this result.

Zou Zhige CMOS Chapter3 57