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### Transcript of BJT High Frequency Distortion - Distortion Due to Di usion Capacitor C B = C ث‡ C je Q B = ث‌...

• Berkeley

BJT High Frequency Distortion

• High Frequency Distortion in BJTs

Assume current drive

Large Circuit Equivalent Circuit

neglect rb, rπ, ro

• Distortion Due to Diffusion Capacitor

CB = Cπ − Cje QB = τF IC

CB = dQB dVBE

= τF dIC dVBE

= τF qIC kT

= τF IS VT

eVBE/VT

CB is a non-linear diffusion capacitor

Cje is emitter-base depletion capacitor, assume constant

Cje ∝ 1

(φ+ VBE ) 1 n

If fT of a BJT is constant with IC , then we have no high frequency distortion in the device for current drive

⇒ Linear Differential Equation, fT constant → τF constant Niknejad Advanced IC’s for Comm

• Kirk Effect

iC = IC−IQ = IQ ( ev1/VT − 1

) ii = ix + ib + iµ

= Cje dv1 dt

+ dQB dt

+ dQµ dt

dQµ dt

= dQµ

d (v1 − vo) d (v1 − vo)

dt = Cµ

dv1 dt − Cµ

dvo dt

Cµ = k

(ϕ+ VCB) 1 n

= k

(ϕ+ VCBQ + VCB) 1 n

• Governing Differential Eq.

dQB dt

= τF dIc dt

= τF dic dt

ii = Cje dv1 dt

+ τF dic dt − Cµ

dvo dt

v1 = VT ln

( ic + IQ IQ

)

dv1 dt

= dv1 dic

dic dt

= VT IQ

IQ ic + IQ

dic dt

= VT

ic + IQ

dic dt

ii = Cje VT

ic + IQ

dic dt

+ τF dic dt − Cµ

dvo dt

• BJT Series Expansion

vo = −icRL

ii = −{ 1

RL

( τF +

CjeVT IQ + ic

) + Cµ}

dvo dt

= − 1 RL

[τF + CjeVT IQ + ic

+ CµRL] dvo dt

1

IQ

( 1 + icIQ

) ⇒ 1 1 + x

= 1− x + x2 + ...

Cµ = Cµo(

1 + voVQ

) 1 n

= Cµo

( 1− 1

n

vo VQ

+ 1

2n

( 1

n + 1

)( vo VQ

)2 + ...

)

• BJT Series Expansion (cont)

ii ∼= − 1

RL [τF +

CjeVT IQ

− CjeVT IQ

ic IQ

+ CjeVT IQ

( ic IQ

)2 +

CµoRL − CµoRL 1

n

vo VQ

+ CµoRL 1

2n

( 1 +

1

n

)( vo VQ

)2 ]× dvo

dt

Substitute vo = −icRL

ii ∼= ( a1 + a2vo + a3vo

2 ) dvo

dt (∗)

∼= a1 dvo dt

+ 1

2 a2

dvo 2

dt +

1

3 a3

dvo 3

dt (∗∗)

• Memoryless Terms

Non-linear differential equation for vo vs. ii where

a1 = − 1

RL

( τF +

CjeVT IQ

+ CµoRL

)

a2 = − 1

RL

( CjeVT

RLIQ 2 − CµoRL

1

n

1

VQ

)

a3 = − 1

RL

( CjeVT

RL 2IQ

3 + CµoRL

1

2n

( 1 +

1

n

) 1

VQ 2

) a2 term has a null but a3 terms always add.

• Memory Terms

In (**), put

vo = A1 (jωa) ◦ ii + A2 (jωa, jωb) ◦ ii 2 + ...

ii = a1 dvo dt

+ 1

2 a2

dvo 2

dt +

1

3 a3

dvo 3

dt

First Order:

ii = a1jωA1 (jω) ◦ ii

A1 (jω) = 1

jωa1

• Second Order Memory

Second Order:

0 = a1j (ωa + ωb)A2 (jωa, jωb)+ 1

2 a2 (jωa + jωb)A1 (jωa)A1 (jωb)

A2 (jωa, jωb) = − 1

2

a2 a1

A1 (jωa)A1 (jωb)

A2 (jωa, jωb) = − 1

2

a2 a1

1

jωaa1

1

jωba1

= 1

2

a2 a13

1

ωaωb

• Third Order Memory

Third Order:

0 = [a1A3 + 1

2 a22A1A2 +

1

3 a3A1A1A1] (jωa + jωb + jωc)

A3 = −12a22A1A2 +

1 3a3A1A1A1

a1

A3 = a3 3 −

a22

2a1

jωaωbωca14

No null because a3 is negative

• Second-Order Intermodulation (IM2)

ii = I1 cosω1t + I2 cosω2t

IM2 = |A2| I1I2

v̂o

2!

1!1!

1

2

v̂o = |A1 (jω1)| I1 = |A1 (jω2)| I2

IM2 = |A2 (jω1,±jω2)| |A1 (jω1)| |A1 (jω2)|

v̂o

IM2 = 1

2

a2 a13

1

ω1ω2 a1ω1a1ω2v̂o =

1

2

a2 a1

v̂o

IM2 independent of frequency at high frequency

• IM3 in BJT at High Frequency

IM3 = 3!

2!1!

1

22 I1I2

2 |A3| v̂o

Again |A1 (jω1)| I1 = |A1 (jω2)| I2 = v̂o

IM3 = 3

4

|A3 (jω2, jω2,−jω1)| |A1 (jω1)| |A1 (jω2)| |A1 (jω2)|

v̂2o = 3

4 v̂2o

a3 3 −

a32

2a1

a1 ≈ 3

4 v̂2o

a3 3a1

a1 = − 1

RL

( τF +

Cje gmQ

+ CµoRL

)

a3 = − 1

RL

( CjeVT

RL 2IQ

3 + CµoRL

1

2n

( 1

n + 1

) 1

VQ 2

• Example: 2 GHz LNA Front End

τF = 12ps Cµo = 37fF rb = 10Ω IC = 3mA Cje = 416fF RL = 50Ω η = 3 VQ = 2V

Supply ≈ VBE +VQ + 3mA×50Ω

≈ 750mV + 150mV = 2.9V

Find intercept point:

a3 = − 1

RL

( 1.6× 10−10 + 1× 10−13

) a1 = −

1

RL (12 + 3.6 + 1.9)× 10−12

IM3 = 3

4 v̂2o

a3 3a1

= 1

v̂o = 0.66V (+ 6 dBm output IP3 into 50Ω) Very close to SPICE and measurements

• LNA Example (cont)

Note:

rπ = β

gm =

150

gm ≈ 1.3kΩ

Cπ = Cje + CB

CB = τFgm = 1.38pF

Cje = 0.416pF

1

ωCπ = 44Ω

Cπ reactance is low compared to rπ

• LNA with Emitter Degen

548 IEEE JOURNAL OF SOLID-STATE CIRCUITS, VOL. 33, NO. 4, APRIL 1998

High-Frequency Nonlinearity Analysis of Common-Emitter and

Differential-Pair Transconductance Stages Keng Leong Fong, Member, IEEE, and Robert G. Meyer, Fellow, IEEE

Abstract—Equations describing the high-frequency nonlinear behavior of common-emitter and differential-pair transconduc- tance stages are derived. The equations show that transconduc- tance stages using inductive degeneration are more linear than those using capacitive or resistive degeneration, and that the common-emitter transconductance stages are more linear than the differential-pair transconductance stages with the same bias current and transconductance. The nonlinearity equations can also be used to explain the class AB behavior of the common- emitter transconductance stage with inductive degeneration.

Index Terms—Amplifier distortion, analog integrated circuits, circuit analysis, circuit optimization, nonlinear circuits, nonlinear distortion, nonlinear equations, Volterra series.

I. INTRODUCTION

THE rapid growth of portable wireless communicationsystems has led to a demand for low-power, high- performance, and highly integrated RF circuits. The bipolar- transistor common-emitter and differential-pair transcon- ductance stages shown in Figs. 1 and 2, respectively, are commonly used in many radio frequency (RF) building blocks, such as low-noise amplifiers (LNA) and mixers. To improve the linearity, the transconductance stages are usually degenerated by impedance , which can be implemented by using either resistors, capacitors, or inductors. It is easily shown that transconductance stages with reactive (inductive or capacitive) degeneration have lower noise figure (NF) than those with resistive degeneration since the degeneration reactance (apart from its loss resistance) does not introduce an additional noise source. Similarly, differential-pair transconductance stages have a higher noise figure than common-emitter transconductance stages since the former have more noise generators. However, there is little published information about linearity comparisons among resistive, capacitive, and inductive degeneration, and between common-emitter and differential-pair transconductance stages.

Manuscript received April 7, 1997; revised August 28, 1997. This material is based on work supported in part by the U.S. Army Research Office under Grant DAAH04-93-F-0200. K. L. Fong was with the Electronics Research Laboratory, Department

of Electrical Engineering and Computer Science, University of California, Berkeley, CA 94720-1772 USA. He is now with Philips Semiconductors, Sunnyvale, CA 94088-3409 USA. R. G. Meyer is with the Electronics Research Laboratory, Department

of Electrical Engineering and Computer Science, University of California, Berkeley, CA 94720-1772 USA. Publisher Item Identifier S 0018-9200(98)02326-9.

Fig.