Download - BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

Transcript
Page 1: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

Berkeley

BJT High Frequency Distortion

Prof. Ali M. Niknejad

U.C. BerkeleyCopyright c© 2014 by Ali M. Niknejad

Niknejad Advanced IC’s for Comm

Page 2: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

High Frequency Distortion in BJTs

Assume current drive

Large Circuit Equivalent Circuit

neglect rb, rπ, ro

Niknejad Advanced IC’s for Comm

Page 3: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

Distortion Due to Diffusion Capacitor

CB = Cπ − Cje

QB = τF IC

CB =dQB

dVBE= τF

dICdVBE

= τFqICkT

= τFISVT

eVBE/VT

CB is a non-linear diffusion capacitor

Cje is emitter-base depletion capacitor, assume constant

Cje ∝1

(φ+ VBE )1n

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

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

Page 4: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

Kirk Effect

iC = IC−IQ = IQ

(ev1/VT − 1

)ii = ix + ib + iµ

= Cjedv1

dt+

dQB

dt+

dQµdt

dQµdt

=dQµ

d (v1 − vo)

d (v1 − vo)

dt= Cµ

dv1

dt− Cµ

dvodt

Cµ =k

(ϕ+ VCB)1n

=k

(ϕ+ VCBQ + VCB)1n

Niknejad Advanced IC’s for Comm

Page 5: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

Governing Differential Eq.

dQB

dt= τF

dIcdt

= τFdicdt

ii = Cjedv1

dt+ τF

dicdt− Cµ

dvodt

v1 = VT ln

(ic + IQIQ

)

dv1

dt=

dv1

dic

dicdt

=VT

IQ

IQic + IQ

dicdt

=VT

ic + IQ

dicdt

ii = CjeVT

ic + IQ

dicdt

+ τFdicdt− Cµ

dvodt

Niknejad Advanced IC’s for Comm

Page 6: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

BJT Series Expansion

vo = −icRL

ii = − 1

RL

(τF +

CjeVT

IQ + ic

)+ Cµ

dvodt

= − 1

RL[τF +

CjeVT

IQ + ic+ CµRL]

dvodt

1

IQ

(1 + ic

IQ

) ⇒ 1

1 + x= 1− x + x2 + ...

Cµ =Cµo(

1 + voVQ

) 1n

= Cµo

(1− 1

n

voVQ

+1

2n

(1

n+ 1

)(voVQ

)2

+ ...

)

Niknejad Advanced IC’s for Comm

Page 7: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

BJT Series Expansion (cont)

ii ∼= −1

RL[τF +

CjeVT

IQ−

CjeVT

IQ

icIQ

+CjeVT

IQ

(icIQ

)2

+

CµoRL − CµoRL1

n

voVQ

+ CµoRL1

2n

(1 +

1

n

)(voVQ

)2

]× dvodt

Substitute vo = −icRL

ii ∼=(a1 + a2vo + a3vo

2) dvo

dt(∗)

∼= a1dvodt

+1

2a2

dvo2

dt+

1

3a3

dvo3

dt(∗∗)

Niknejad Advanced IC’s for Comm

Page 8: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

Memoryless Terms

Non-linear differential equation for vo vs. ii where

a1 = − 1

RL

(τF +

CjeVT

IQ+ CµoRL

)

a2 = − 1

RL

(CjeVT

RLIQ2− CµoRL

1

n

1

VQ

)

a3 = − 1

RL

(CjeVT

RL2IQ

3+ CµoRL

1

2n

(1 +

1

n

)1

VQ2

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

Niknejad Advanced IC’s for Comm

Page 9: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

Memory Terms

In (**), put

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

ii = a1dvodt

+1

2a2

dvo2

dt+

1

3a3

dvo3

dt

First Order:

ii = a1jωA1 (jω) ii

A1 (jω) =1

jωa1

Niknejad Advanced IC’s for Comm

Page 10: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

Second Order Memory

Second Order:

0 = a1j (ωa + ωb)A2 (jωa, jωb)+1

2a2 (jωa + jωb)A1 (jωa)A1 (jωb)

A2 (jωa, jωb) = −1

2

a2

a1A1 (jωa)A1 (jωb)

A2 (jωa, jωb) = −1

2

a2

a1

1

jωaa1

1

jωba1

=1

2

a2

a13

1

ωaωb

Niknejad Advanced IC’s for Comm

Page 11: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

Third Order Memory

Third Order:

0 = [a1A3 +1

2a22A1A2 +

1

3a3A1A1A1] (jωa + jωb + jωc)

A3 =−1

2a22A1A2 + 13a3A1A1A1

a1

A3 =a33 −

a22

2a1

jωaωbωca14

No null because a3 is negative

Niknejad Advanced IC’s for Comm

Page 12: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

Second-Order Intermodulation (IM2)

ii = I1 cosω1t + I2 cosω2t

IM2 =|A2| I1I2

vo

2!

1!1!

1

2

vo = |A1 (jω1)| I1 = |A1 (jω2)| I2

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

vo

IM2 =1

2

a2

a13

1

ω1ω2a1ω1a1ω2vo =

1

2

a2

a1vo

IM2 independent of frequency at high frequency

Niknejad Advanced IC’s for Comm

Page 13: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

IM3 in BJT at High Frequency

IM3 =3!

2!1!

1

22

I1I22 |A3|vo

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

IM3 =3

4

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

v2o =

3

4v2o

a33 −

a32

2a1

a1≈ 3

4v2o

a3

3a1

a1 = − 1

RL

(τF +

Cje

gmQ+ CµoRL

)

a3 = − 1

RL

(CjeVT

RL2IQ

3+ CµoRL

1

2n

(1

n+ 1

)1

VQ2

)Niknejad Advanced IC’s for Comm

Page 14: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

Example: 2 GHz LNA Front End

τF = 12psCµo = 37fFrb = 10ΩIC = 3mACje = 416fFRL = 50Ωη = 3VQ = 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

4v2o

a3

3a1= 1

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

Niknejad Advanced IC’s for Comm

Page 15: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

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

Niknejad Advanced IC’s for Comm

Page 16: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

LNA with Emitter Degen

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

High-Frequency NonlinearityAnalysis of Common-Emitter and

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

Abstract—Equations describing the high-frequency nonlinearbehavior of common-emitter and differential-pair transconduc-tance stages are derived. The equations show that transconduc-tance stages using inductive degeneration are more linear thanthose using capacitive or resistive degeneration, and that thecommon-emitter transconductance stages are more linear thanthe differential-pair transconductance stages with the same biascurrent and transconductance. The nonlinearity equations canalso 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, nonlineardistortion, 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, arecommonly used in many radio frequency (RF) buildingblocks, such as low-noise amplifiers (LNA) and mixers. Toimprove the linearity, the transconductance stages are usuallydegenerated by impedance , which can be implementedby using either resistors, capacitors, or inductors. It iseasily shown that transconductance stages with reactive(inductive or capacitive) degeneration have lower noisefigure (NF) than those with resistive degeneration sincethe degeneration reactance (apart from its loss resistance)does not introduce an additional noise source. Similarly,differential-pair transconductance stages have a higher noisefigure than common-emitter transconductance stages since theformer have more noise generators. However, there is littlepublished information about linearity comparisons amongresistive, capacitive, and inductive degeneration, and betweencommon-emitter and differential-pair transconductance stages.

Manuscript received April 7, 1997; revised August 28, 1997. This materialis based on work supported in part by the U.S. Army Research Office underGrant 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. 1. Common-emitter transconductance stage.

Fig. 2. Differential-pair transconductance stage.

Fig. 3. Third-order intermodulation product corrupts desired channel.

In this paper, high-frequency nonlinearity equations in Volterraseries [1], [2] for both common-emitter and differential-pair transconductance stages are derived. The emphasisis on interpretation and application of the equations. Thenonlinearity equations are also used to explain the class ABbehavior described in [3] and [4].

II. THIRD-ORDER INTERMODULATION DISTORTIONDue to the third-order nonlinearity of a transconductance

stage, two undesired signals in adjacent channels generatethird-order intermodulation ( ) products at the output ofthe transconductance stage. As illustrated in Fig. 3, theproduct can corrupt the desired signal if it falls within thedesired channel. If the two adjacent channel frequencies are

and , respectively, two products are generated atfrequencies ( ) and ( ), respectively.

0018–9200/98$10.00 1998 IEEE

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 16, 2009 at 17:31 from IEEE Xplore. Restrictions apply.

FONG AND MEYER: ANALYSIS OF COMMON-EMITTER AND DIFFERENTIAL-PAIR TRANSCONDUCTANCE STAGES 555

[5] C. D. Hull, “Analysis and optimization of monolithic RF downcon-version receivers,” University of California at Berkeley, Berkeley, CA,Ph.D. dissertation, 1992.

[6] D. O. Pederson and K. Mayaram, Analog Integrated Circuits for Com-munication. Norwell, MA: Kluwer, 1991.

Keng Leong Fong (S’93–M’97) was born in KualaLumpur, Malaysia, on March 6, 1970. He receivedthe B.A.Sc. degree in engineering science (com-puter engineering option) and the M.A.Sc. degree inelectrical engineering, both from the University ofToronto, Ontario, in 1992 and 1993, respectively. Hereceived the Ph.D. degree in electrical engineeringfrom the University of California at Berkeley in1997.During the summer of 1995, he worked at Rock-

well International Corporation, Newport Beach, CA,where he was involved in designing class AB mixer and evaluating CAD toolsfor nonlinear noise analysis. During the summer of 1996, he worked in thesame company, designing class AB power amplifiers. Since August 1997, hehas been employed by Philips Semiconductors, Sunnyvale, CA. His currentresearch interest is in the area of analog integrated circuit design for RFapplications.

Robert G. Meyer (S’64–M’68–SM’74–F’81) wasborn in Melbourne, Australia, on July 21, 1942. Hereceived the B.E., M.Eng.Sci., and Ph.D. degreesin electrical engineering from the University ofMelbourne in 1963, 1965, and 1968, respectively.In 1968, he was employed as an Assistant Lec-

turer in electrical engineering at the University ofMelbourne. Since September 1968, he has been em-ployed in the Department of Electrical Engineeringand Computer Sciences, University of California,Berkeley, where he is now a Professor. His current

research interests are high-frequency analog integrated-circuit design anddevice fabrication. He has acted as a consultant on electronic circuit design fornumerous companies in the electronics industry. He is coauthor of the bookAnalysis and Design of Analog Integrated Circuits (Wiley, 1993) and Editorof the book Integrated Circuit Operational Amplifiers (IEEE Press, 1978).Dr. Meyer was President of the IEEE Solid-State Circuits Council and was

an Associate Editor of the IEEE JOURNAL OF SOLID-STATE CIRCUITS and of theIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS.

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 16, 2009 at 17:31 from IEEE Xplore. Restrictions apply.

FONG AND MEYER: ANALYSIS OF COMMON-EMITTER AND DIFFERENTIAL-PAIR TRANSCONDUCTANCE STAGES 555

[5] C. D. Hull, “Analysis and optimization of monolithic RF downcon-version receivers,” University of California at Berkeley, Berkeley, CA,Ph.D. dissertation, 1992.

[6] D. O. Pederson and K. Mayaram, Analog Integrated Circuits for Com-munication. Norwell, MA: Kluwer, 1991.

Keng Leong Fong (S’93–M’97) was born in KualaLumpur, Malaysia, on March 6, 1970. He receivedthe B.A.Sc. degree in engineering science (com-puter engineering option) and the M.A.Sc. degree inelectrical engineering, both from the University ofToronto, Ontario, in 1992 and 1993, respectively. Hereceived the Ph.D. degree in electrical engineeringfrom the University of California at Berkeley in1997.During the summer of 1995, he worked at Rock-

well International Corporation, Newport Beach, CA,where he was involved in designing class AB mixer and evaluating CAD toolsfor nonlinear noise analysis. During the summer of 1996, he worked in thesame company, designing class AB power amplifiers. Since August 1997, hehas been employed by Philips Semiconductors, Sunnyvale, CA. His currentresearch interest is in the area of analog integrated circuit design for RFapplications.

Robert G. Meyer (S’64–M’68–SM’74–F’81) wasborn in Melbourne, Australia, on July 21, 1942. Hereceived the B.E., M.Eng.Sci., and Ph.D. degreesin electrical engineering from the University ofMelbourne in 1963, 1965, and 1968, respectively.In 1968, he was employed as an Assistant Lec-

turer in electrical engineering at the University ofMelbourne. Since September 1968, he has been em-ployed in the Department of Electrical Engineeringand Computer Sciences, University of California,Berkeley, where he is now a Professor. His current

research interests are high-frequency analog integrated-circuit design anddevice fabrication. He has acted as a consultant on electronic circuit design fornumerous companies in the electronics industry. He is coauthor of the bookAnalysis and Design of Analog Integrated Circuits (Wiley, 1993) and Editorof the book Integrated Circuit Operational Amplifiers (IEEE Press, 1978).Dr. Meyer was President of the IEEE Solid-State Circuits Council and was

an Associate Editor of the IEEE JOURNAL OF SOLID-STATE CIRCUITS and of theIEEE TRANSACTIONS ON CIRCUITS AND SYSTEMS.

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 16, 2009 at 17:31 from IEEE Xplore. Restrictions apply.

Reference: K. L. Fong and R. G. Meyer, “High-Frequency Nonlinearity Analysis of Common-Emitter and

Differential-Pair Transconductance Stages, ” IEEE J. of Solid-State Circuits, vol. 33, no. 4, April 1998.

Analysis of BJT transconductor as a non-linear Gm cell

Include emitter degeneration.

Niknejad Advanced IC’s for Comm

Page 17: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

KCL EquationsFONG AND MEYER: ANALYSIS OF COMMON-EMITTER AND DIFFERENTIAL-PAIR TRANSCONDUCTANCE STAGES 549

Fig. 4. Model of common-emitter transconductance stage.

Fig. 4 shows the large-signal model used to derive the non-linearity equations for the common-emitter transconductancestage shown in Fig. 1. This model ignores the effect of base-collector junction capacitance of . Inclusion ofgreatly complicates the analysis without adding significantaccuracy to the results in typical situations. is the voltagesignal source. is the impedance at the base of whichincludes source resistance ( ), base resistance ( ) of ,shunt impedance of bias circuit, and impedance of impedance-matching network. is the impedance at the emitter ofwhich includes the parasitic emitter resistance ( ) of andthe impedance of the degeneration elements (resistor, capac-itor, and/or inductor). is the base-charging capacitance of

which is linearly proportional to the collector current ( )and the forward transit time ( ) of . is the base-emitterjunction capacitance which is assumed to be constant in thismodel. is the base current which is equal to whereis the small-signal low-frequency current gain of . cannotbe ignored since the high-frequency nonlinearity also dependson the low-frequency characteristics of (this will becomeobvious later). Using the model in Fig. 4, a Kirchhoff’s voltagelaw equation can be written as

(1)

where is the collector signal current (collector current minusbias current) where the collector is assumed to be ac grounded,

is the base-emitter signal voltage drop across and ,and is the Laplace variable. Solving this equation(derivation is shown in Appendix A) results in the followingVolterra series expression:

(2)where is the th power of the voltage source signal and

is the Volterra series coefficient which is a linear functionof number of frequencies. The operator “ ” indicates mul-tiplying each frequency component in by the magnitudeof and shifting each frequency component in by the

phase of . The first three Volterra series coefficients are

(3)

(4)

(5)

where and are as shown in the equations atthe bottom of the page, is the bias current of , and isthe thermal voltage. The coefficient is the small-signaltransconductance of the transconductance stage.The product at frequency ( ) can be calculated

by using the Volterra series coefficient , andletting and . Similarly, theproduct at frequency ( ) can calculated by letting

and . Typically, the frequencydifference between and is so small thatcan be assumed. Using two input signals of the same amplitude, the magnitude ( ) of the input-referred products

(the two products have about the same magnitude) of thecommon-emitter transconductance stage is given by

(6)

where .The depends on the magnitude of

(7)

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 16, 2009 at 17:31 from IEEE Xplore. Restrictions apply.

Vs = (sCjeVπ + sτF Ic + Ic/β0)(Zb + Ze) + IcZc + Vπ

Ic = IQ exp

(VπVT

)= IQ

[(VπVT

)+

1

2

(VπVT

)2

+1

6

(VπVT

)3

+ · · ·

]Vπ = C1(s1) Vs + C2(s1, s2) V 2

s + C3(s1, s2, s3) V 3s + · · ·

Niknejad Advanced IC’s for Comm

Page 18: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

IM3 Calculation

C1(s) =1

sCjeZ (s) + sτFgmZ (s) + gmZ (s)/β0 + 1 + gmZe(s)

C2(s) = −C1(s1 + s2)C1(s1)C1(s2) IQ2V 2

T×[

(s1 + s2)τFZ (s1 + s2) + Z(s1+s2)β0

+ Ze(s1 + s2)]

C3(s1, s2, s3) = −A1(s1+s2+s3)IQ6V 3

T×[

A1(s1)A1(s2)A1(s3) + 6VTA1A2

]×[

(s1 + s2 + s3)τFZ (s1 + s2 + s3) + Z(s1+s2+s3)β0

+ Ze(s1 + s2 + s3)]

Ic = A1(s) Vs + A2(s1, s2) V 2s + A3(s1, s2, s3) V 3

s + · · ·

A1(s) = gmC1(s)

A2(s1, s2) = gmC2(s1, s2) +IQ

2V 2T

C1(s1)C1(s2)

A3(s1, s2, s3) = gmC3(s1, s2, s3) +IQ

6V 3T

C1(s1)C1(s2)C1(s3) +IQV 2T

C1C2

Niknejad Advanced IC’s for Comm

Page 19: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

IM3 Calculation

IM3 =

∣∣∣∣34 A3(sa, sa,−sb)

A1(2sa − sb)

∣∣∣∣ |Vs |2

≈∣∣∣A1(s)

IQ

∣∣∣3 ∣∣∣VT4 (1 + sCjeZ (s))×

−1 + A1(∆s)gm

(1 + ∆sCjeZ (∆s)) + A1(2s)2gm

(1 + 2sCjeZ (2s))∣∣∣ |Vs |2

With Z (s) = Zb(s) + Ze(s), note that these terms canresonate:

(1 + sCjeZb(s) + sCjeZe(s))

For small offset frequencies (1 + ∆sCjeZ (∆s)) ≈ 1

IM3 ≈∣∣∣∣A1(s)

IQ

∣∣∣∣3 ∣∣∣∣VT

4(1 + sCjeZ(s))×

−1 +

A1(∆s)

gm+

A1(2s)

2gm(1 + 2sCjeZ(2s))

∣∣∣∣ |Vs |2

Niknejad Advanced IC’s for Comm

Page 20: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

Typical Example

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

With inductive degeneration, the term is a nega-tive real number which cancels the “1” term in (7) partially.There is no such cancellation with resistive degeneration sincethe term is a positive imaginary number whichadds to the imaginary part of the term in (7). Forthe same reason, capacitive degeneration would increase the

because the term is a positive real numberwhich adds to the “1” term in (7).The also depends on the magnitude of

(8)

where the “ 1” and

terms come from the third-order nonlinearity ( ) andthe second-order interaction ( ), respectively. Using thefollowing approximation (for practical design values),

equation (6) can be simplified to

(9)

where the value of the termis typically small, compared to the value of theterm. However, it is not so small that it can be ignored.As shown in (9), the is independent of if the small-

signal transconductance is kept constant. On the otherhand, it increases with for the resistive and capacitivedegeneration cases since the term is a positiveimaginary number and a positive real number, respectively, butdecreases with for the inductive degeneration case becausethe term is a negative real number. Furthermore,the is proportional to the cube of the ratio of small-signal transconductance to bias current ( ).The can be lowered by increasing the

term. This increases the second-order interaction to cancelthe third-order nonlinearity. Since the degeneration inductorhas low impedance at low frequency, the term inthe inductive degeneration case is much larger than those inthe resistive and capacitive degeneration cases. Similarly, thedegeneration capacitor has high impedance at low frequency,and hence the term in the capacitive degeneration caseis much smaller than that in the resistive degeneration case.This is the second reason why the inductively degeneratedtransconductance stage is more linear than the resistivelydegenerated transconductance stage, which in turn is morelinear than the capacitive degenerated transconductance stagewith the same transconductance and bias current.

Fig. 5. Inductively degenerated common-emitter transconductances stage.

Fig. 6. Input IP3 versus bias current.

Fig. 5 shows the basic topology of a typical common-emitter transconductance stage with inductive degeneration.The degeneration inductor is typically implemented bybond wires. serves as a dc blocking capacitor. It is alsoused to tune out the bond wire inductance . is a biasresistor used to isolate the bias circuit from the RF port. Atlow frequency , the impedance of isnegligible. The impedance of is dominated by thebias resistor since the blocking capacitor has highimpedance at low frequency. In this case, theterm in (9) can be simplified to

(10)

where is the small-signal base-emitter resistance of .Therefore, in order to increase the linearity, should be keptsmall (relative to ) to increase the second-order interaction.However, has to be large enough avoid significant loadingon the RF port, which would cause impedance mismatch andnoise figure degradation.Fig. 6 shows the simulated (using HSPICE) and analytical

[using (6)] input of the common-emitter transconductancestage shown in Fig. 5 as a function of bias current ( ). Thesimulation results include the nonlinear effects of andof . Neglecting these effects does not seem to introducesignificant errors. The two RF sinusoidal signals used are at900 and 910 MHz, respectively. The component values usedare: ps pF nH

nH pF and .Similarly, the model shown in Fig. 7 is used to derive the

nonlinearity equations for the differential-pair transconduc-

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 16, 2009 at 17:31 from IEEE Xplore. Restrictions apply.

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

With inductive degeneration, the term is a nega-tive real number which cancels the “1” term in (7) partially.There is no such cancellation with resistive degeneration sincethe term is a positive imaginary number whichadds to the imaginary part of the term in (7). Forthe same reason, capacitive degeneration would increase the

because the term is a positive real numberwhich adds to the “1” term in (7).The also depends on the magnitude of

(8)

where the “ 1” and

terms come from the third-order nonlinearity ( ) andthe second-order interaction ( ), respectively. Using thefollowing approximation (for practical design values),

equation (6) can be simplified to

(9)

where the value of the termis typically small, compared to the value of theterm. However, it is not so small that it can be ignored.As shown in (9), the is independent of if the small-

signal transconductance is kept constant. On the otherhand, it increases with for the resistive and capacitivedegeneration cases since the term is a positiveimaginary number and a positive real number, respectively, butdecreases with for the inductive degeneration case becausethe term is a negative real number. Furthermore,the is proportional to the cube of the ratio of small-signal transconductance to bias current ( ).The can be lowered by increasing the

term. This increases the second-order interaction to cancelthe third-order nonlinearity. Since the degeneration inductorhas low impedance at low frequency, the term inthe inductive degeneration case is much larger than those inthe resistive and capacitive degeneration cases. Similarly, thedegeneration capacitor has high impedance at low frequency,and hence the term in the capacitive degeneration caseis much smaller than that in the resistive degeneration case.This is the second reason why the inductively degeneratedtransconductance stage is more linear than the resistivelydegenerated transconductance stage, which in turn is morelinear than the capacitive degenerated transconductance stagewith the same transconductance and bias current.

Fig. 5. Inductively degenerated common-emitter transconductances stage.

Fig. 6. Input IP3 versus bias current.

Fig. 5 shows the basic topology of a typical common-emitter transconductance stage with inductive degeneration.The degeneration inductor is typically implemented bybond wires. serves as a dc blocking capacitor. It is alsoused to tune out the bond wire inductance . is a biasresistor used to isolate the bias circuit from the RF port. Atlow frequency , the impedance of isnegligible. The impedance of is dominated by thebias resistor since the blocking capacitor has highimpedance at low frequency. In this case, theterm in (9) can be simplified to

(10)

where is the small-signal base-emitter resistance of .Therefore, in order to increase the linearity, should be keptsmall (relative to ) to increase the second-order interaction.However, has to be large enough avoid significant loadingon the RF port, which would cause impedance mismatch andnoise figure degradation.Fig. 6 shows the simulated (using HSPICE) and analytical

[using (6)] input of the common-emitter transconductancestage shown in Fig. 5 as a function of bias current ( ). Thesimulation results include the nonlinear effects of andof . Neglecting these effects does not seem to introducesignificant errors. The two RF sinusoidal signals used are at900 and 910 MHz, respectively. The component values usedare: ps pF nH

nH pF and .Similarly, the model shown in Fig. 7 is used to derive the

nonlinearity equations for the differential-pair transconduc-

Authorized licensed use limited to: Univ of Calif Berkeley. Downloaded on March 16, 2009 at 17:31 from IEEE Xplore. Restrictions apply.

The two RF sinusoidal signals used are at 900 and 910 MHz,respectively. The component values used are: τF = 10.5ps,Cje = 1.17pF, β = 73, and Le = 2.4nH, L1 = 3.5nH,C1 = 20pF, and R1 = 150Ω.

The simulation results include the nonlinear effects of Cµ andCje , and neglecting these effects does not seem to introducesignificant errors.

Niknejad Advanced IC’s for Comm

Page 21: BJT High Frequency Distortion - RFICrfic.eecs.berkeley.edu/ee242/pdf/Module_4_3_bjtdisto.pdf · Distortion Due to Di usion Capacitor C B = C ˇ C je Q B = ˝ FI C C B = dQ B dV BE

References

UCB EECS 242 Class Notes, Robert G. Meyer, Spring 1995

K. L. Fong and R. G. Meyer, “High-Frequency NonlinearityAnalysis of Common-Emitter and Differential PairTransconductance Stages,” JSSC, pp. 548-555, vol. 33, no.4, April 1998.

Niknejad Advanced IC’s for Comm