6. Cascode Amplifiers and Cascode Current Mirrors

6. Cascode Amplifiers and Cascode Current Mirrors Sedra & Smith Sec. 7 (MOS portion) (S&S 5th Ed: Sec. 6 MOS portion & ignore frequency response)

ECE 102, Fall 2012, F. Najmabadi

Cascode amplifier is a popular building block of ICs Cascode Configuration

Signal circuit: Current source becomes an open circuit

CG stage

signal

CS stage

Cascode amplifier is a two-stage, CS-CG configuration F. Najmabadi, ECE102, Fall 2012 (2/17)

bias

I D1 = I D 2

Small Signal Model of a Cascode Amplifier CG stage

Small Signal Model

CS stage  Lengthy analysis to find Av (and a complicated equation). Simpler to compute open-loop gain (Avo) and Ro.  Text book introduces Gm method to find Avo (See S&S Sec. 1)  Here will find Avo directly from the small signal model.  However, the solution of and insight into Cascode amplifiers are best obtained using fundamental MOS configurations!

Note that Avo and Ro calculated here are meant to find Av and guide the choice of the active load. Avo and Ro should be re-calculated for a practical circuit (see slides 14 & 15) F. Najmabadi, ECE102, Fall 2012 (3/17)

Open-Loop gain of a Cascode amplifier (using small signal model) Open Loop Gain (R’L → ∞, io = 0):

By KCL around Q2 Node Voltage Method: Node vo: Node v1:

vo − v1 − g m 2 v1 = 0 ro 2 v1 + g m1vi + 0 = 0 ro1 Avo =

F. Najmabadi, ECE102, Fall 2012 (4/17)

⇒ vo = (1 + g m 2 ro 2 ) v1

⇒ v1 = − g m1ro1 vi

vo = − g m1ro1 × (1 + g m 2 ro 2 ) ≈ − g m1ro1 g m 2 ro 2 vi

Output Resistance of a Cascode amplifier (using small signal model) Set vi = 0, attach a voltage source vx , compute ix, Ro = vx / ix

vi = vgs1 = 0 → gm1 vgs1 current source becomes open circuit

By KCL around Q2

KVL : vgs 2 = −ix ro1 KCL : i2 = ix − g m 2 vgs 2 = ix + ix g m 2 ro1 = ix (1 + g m 2 ro1 ) KVL : vx = i2 ro 2 + ix ro1 = ix (1 + g m 2 ro1 )ro 2 + ix ro1 vx = ix [(1 + g m 2 ro1 )ro 2 + ro1 ]

Ro =

vx = ro1 + ro 2 + g m 2 ro1 ro 2 ix

Note: Av = Avo × F. Najmabadi, ECE102, Fall 2012 (5/17)

RL′ + Ro RL′

Gain of a Cascode Amplifier (using MOS Fundamental Configurations) Cascode (signal circuit)

CG stages “reduces” the load seen by the CS stage by gm2ro2

CG stage

CS stage

Av 2 = vo / v1 ≈ g m 2 (ro 2 || RL′ )

Av1 = v1 / vi = − g m1 (ro1 || Ri 2 )

RL1 = Ri 2 =

ro 2 + RL′ 1 + g m 2 ro 2

Av = vo / vi = Av1 Av 2 = − g m1 g m 2 (ro1 || Ri 2 ) (ro 2 || RL′ )

ro 2 + RL′ = ∞ → Avo = − g m1ro1 g m 2 ro 2 Note: Open Loop Gain: (R’L → ∞) RL1 = Ri 2 = 1 + g m 2 ro 2

F. Najmabadi, ECE102, Fall 2012 (6/17-)

Output Resistance of a Cascode amplifier (from Elementary R forms) = ro (1 + g m R ) + R R

Ro = ro 2 (1 + g m 2 ro1 ) + ro1 Ro = ro1 + ro 2 + g m 2 ro1 ro 2 Ro = ro 2 (1 + g m 2 ro1 ) + ro1 Ro ≈ g m 2 ro 2 ro1 + ro1 = ro1 (1 + g m 2 ro 2 ) Ro ≈ g m 2 ro1 ro 2 F. Najmabadi, ECE102, Fall 2012 (7/17)

Cascode Amplifier needs a large load Av 2 = g m 2 (ro 2 || RL′ )

RL1 = Ri 2 =

ro 2 + RL′ 1 + g m 2 ro 2

Av1 = − g m1 (ro1 || Ri 2 )

Ro ≈ g m 2 ro1 ro 2

For simplicity assume ro1 = ro2 = ro and gm1 = gm2 = gm R’L

Av2(CG)

Ri2 = RL1

Av1 (CS)

Av= Av1 Av2

∞

gmro

∞

− gmro

− (gmro)2

Max. Gain

(gmro) ro = Ro

gmro

ro

− 0.5gmro

− 0.5(gmro)2

Practical Gain

ro

0.5 gmro

2/gm

−2

− gmro

Same gain as a single CS Amp.

 For comparison, a two-stage CS-amplifier (CS-CS) has a gain of 0.5 (gmro)2 for R’L = ro and a gain of (gmro)2 for R’L = gmro2. o Cascode amplifier needs a large load (R’L = gmro2 ).


Cascode amplifier needs a large load to get a high gain Current Mirror

Cascode

R’L = ro3 Av ≈ − gmro  Gain did not increase compared to a CS amplifier.  This is still a useful circuit because of its high gain-bandwidth (we see this later).  To get a high gain, Av = − 0.5(gmro)2 , we need to increase the small-signal resistance of the current mirror to ≈ (gmro) ro o Cascode current mirror


Cascode Current mirror  Identical MOS: Same µCox and Vt , & o vGS1 = vGS2 & vGS3 = vGS4

(W / L) 4 (W / L) 2 = (W / L) 3 (W / L)1

 Usually: (W/L)1 = (W/L)3 and (W/L)2 = (W/L)4 o vGS1 = vGS2 = vGS3 = vGS4 = vGS  Q1 and Q3 are always in saturation  Q2 and Q4 both have to be in saturation for current mirror to work o VDS2 > VGS – Vt o VDS4 > VGS – Vt  Straight forward to show

Io =

(W / L) 2 I ref (W / L)1

Exercise: Show that a single current mirror (no cascoding) works only if VD2 > VOV −VSS and a cascode current mirror requires VD4 > 2VOV −VSS F. Najmabadi, ECE102, Fall 2012 (10/17)

Small signal resistance of a cascode current mirror is quite large R = ro 4 (1 + g m 4 ro 2 ) + ro 2 R

Transistor numbering is different in different circuits Be careful in applying formulas! It is best to use elementary R forms to find R instead of formula above. F. Najmabadi, ECE102, Fall 2012 (11/17)

PMOS cascode current mirror is similar to NMOS version NMOS Cascode current mirror


PMOS Cascode current mirror

Cascode amplifier with a cascode current mirror/active load Bias

Cascode current mirror Signal

Cascode amplifier

I D1 = I D 2 = I D 3 = I D 4


Exercise: Draw the circuit for a PMOS cascode with a cascode current mirror (cascode current mirror would be made of NMOS).

Gain of a Cascode amplifier with a cascode current mirror/active load Value for the same gm and ro

Q2 (CG Amp)

Av 2 = vo / v1 ≈ g m 2 (ro 2 || RL′ ) ≈ g m 2 ro 2

Cascode current mirror

RL1 = Ri 2 =

ro 3 (1 + g m 3ro 4 ) + ro 4

Cascode amplifier

≈

Av 2 ≈ g m ro

ro 2 + ro 3 (1 + g m 3 ro 4 ) + ro 4 1 + g m 2 ro 2 g m 3 ro 3 ro 4 g m 2 ro 2

RL1 = Ri 2 ≈ ro

Q1 (CS Amp)

v1

Ri 2

Av1 = v1 / vi = − g m1 (ro1 || Avo = Av1 Av 2 = −


g m 3 ro 3 ro 4 ) g m 2 ro 2

g m1 g m 2 g m 3 ro1ro 2 ro 3 ro 4 g m 2 ro1ro 2 + g m 3 ro 3 ro 4

Av1 = −0.5 g m ro Avo = −0.5 ( g m ro ) 2

Output Resistance of a Cascode amplifier with a cascode current mirror/active load Value for the same gm and ro

R1

R1 = ro 3 (1 + g m 3 ro 4 ) + ro 4

R 1 ≈ g m ro2

R 2 = ro 2 (1 + g m 2 ro1 ) + ro1

R 2 ≈ g m ro2

R2 Ro = R 1 || R 2


Ro = 0.5 g m ro2

Why Cascode Amplifiers are popular? Cascode Amp. Avo = −0.5 ( g m ro ) 2 Ro = 0.5 g m ro2

2-stage CS Amp.

Drawbacks:  While Avo are similar, Cascode has a very Ro (MΩ level). o should be followed with a CS or CD stage (infinite load for cascode) o BJT cascodes are not useful.

Avo = +0.25( g m ro ) 2 Ro = r o

 Low voltage headroom (VDD across 4 MOS) o Folded cascodes solve this. Benefits:  Much better high-frequency response (high gain-bandwidth).*  Simpler biasing.


* We will see this later in our discussion of frequency response.

Folded Cascode increases voltage overhead* Bias

Signal CG stage

I2

I1 – I2

I2

I1 – I2

CS stage

NMOS CS stage Biased with I1 – I2 PMOS CG stage Biased with I2


* Folded cascode only helps the voltage overhead issue for difference amplifiers (see S&S pages 999-1000)