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

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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

    Prof. Ali M. Niknejad

    U.C. Berkeley Copyright c© 2014 by Ali M. Niknejad

    Niknejad Advanced IC’s for Comm

  • High Frequency Distortion in BJTs

    Assume current drive

    Large Circuit Equivalent Circuit

    neglect rb, rπ, ro

    Niknejad Advanced IC’s for Comm

  • 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

    Niknejad Advanced IC’s for Comm

  • 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

    Niknejad Advanced IC’s for Comm

  • 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 + ...

    )

    Niknejad Advanced IC’s for Comm

  • 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 (∗∗)

    Niknejad Advanced IC’s for Comm

  • 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.

    Niknejad Advanced IC’s for Comm

  • 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

    Niknejad Advanced IC’s for Comm

  • 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

    Niknejad Advanced IC’s for Comm

  • 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

    Niknejad Advanced IC’s for Comm

  • 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

    Niknejad Advanced IC’s for Comm

  • 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

    ) Niknejad Advanced IC’s for Comm

  • 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

    Niknejad Advanced IC’s for Comm

  • 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

  • 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.