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									EECS 105 Fall 2003, Lecture 15




                                      Lecture 15:

                                 Small Signal Modeling




                                           Prof. Niknejad




Department of EECS                                          University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                       Prof. A. Niknejad



                                         Lecture Outline



                                    Review: Diffusion Revisited
                                    BJT Small-Signal Model
                                    Circuits!!!
                                    Small Signal Modeling
                                    Example: Simple MOS Amplifier




            Department of EECS                                       University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                                      Prof. A. Niknejad



                                      Notation Review
                                                                  Large signal
                                           iC  f (vBE , vCE )
                                 I C  iC  f (VBE  vBE ,VCE  vCE )
    Quiescent Point                                                                  small signal
        (bias)                                                                    DC (bias)
                                     I C  ic  f (VBE  vbe ,VCE  vce )
                                                                                  small signal
                                             f             f                    (less messy!)
            Q  (VBE ,VCE )            ic           vbe             vce
                                            vBE   Q
                                                           vCE   Q

                          transconductance                                  Output conductance

                   Since we’re introducing a new (confusing) subject, let’s adopt some
                    consistent notation
                   Please point out any mistakes (that I will surely make!)
                   Once you get a feel for small-signal analysis, we can drop the notation
                    and things will be clear by context (yeah right! … good excuse)
            Department of EECS                                                      University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                                Prof. A. Niknejad



                                 Diffusion Revisited
                  Why is minority current profile a linear function?
                  Recall that the path through the Si crystal is a zig-zag series
                   of acceleration and deceleration (due to collisions)
                  Note that diffusion current density is controlled by width of
                   region (base width for BJT):
                                 Density here fixed by potential (injection of carriers)
                                 Physical interpretation: How many electrons (holes) have
  Half go left,                  enough energy to cross barrier? Boltzmann distribution give
  half go right                  this number.

                                                           Density fixed by
                                                            metal contact


                                    Wp

                  Decreasing width increases current!
            Department of EECS                                                University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                                        Prof. A. Niknejad



                                 Diffusion Capacitance
                  The total minority carrier charge for a one-sided
                   junction is (area of triangle)
                                                                                 qV
                                      1          1                         D

                             Qn  qA  bh2  qA  (W  xdep , p )(n p 0e kT  n p 0 )
                                      2          2

                  For a one-sided junction, the current is dominated
                   by these minority carriers:
                                                                 qVD
                                              qADn
                                       ID                (n p 0e kT  n p 0 )
                                            Wp  xdep , p

                                        ID     Dn
                                                                                 Constant!
                                        Qn W  x
                                                 dep , p 
                                                           2
                                             p

            Department of EECS                                                        University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                                     Prof. A. Niknejad



                         Diffusion Capacitance (cont)
                  The proportionality constant has units of time
                                                                                Distance across
                                              Qn W p  xdep , p 
                                                                         2
                                                                                  P-type base
                                         T     
                                              ID       Dn

                                               q W p  xdep , p 
                                                                     2       Diffusion Coefficient

                                         T                                 Mobility
                                              kT       n
                           Temperature

                  The physical interpretation is that this is the transit
                   time for the minority carriers to cross the p-type
                   region. Since the capacitance is related to charge:
                                 Qn   T I D          Qn      I
                                                  Cd       T     g d T
                                                       V       V
            Department of EECS                                                     University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                             Prof. A. Niknejad



                             BJT Transconductance gm




                  The transconductance is analogous to diode
                   conductance
            Department of EECS                             University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                                Prof. A. Niknejad



                                 Transconductance (cont)

              Forward-active large-signal current:

                                    iC  I S e   vBE / Vth
                                                             (1  vCE VA )


         • Differentiating and evaluating at Q = (VBE, VCE )
                                   iC          q
                                                 I S e qVBE / kT (1  VCE VA )
                                  vBE   Q
                                               kT

                                                   iC               qI C
                                             gm                   
                                                  vBE         Q
                                                                     kT
             Department of EECS                                               University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                                                      Prof. A. Niknejad



                                        BJT Base Currents
             Unlike MOSFET, there is a DC current into the
             base terminal of a bipolar transistor:

                                 I B  IC F   I S F  eqVBE / kT (1  VCE VA )

             To find the change in base current due to change
             in base-emitter voltage:
                                        iB                     iB            iB       iC               1
                                 ib               vbe                                                         gm
                                        vBE   Q                vBE   Q
                                                                               iC       vBE   Q
                                                                                                         F
                                                                                     Q



                                                                gm
                                                         ib         vbe
                                                                F
            Department of EECS                                                                      University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                         Prof. A. Niknejad



                                 Small Signal Current Gain
                                                                         iC
                                                                  0         F
                                                                         iB

                                                                   iC   0 iB

                                                                     ic   0ib




                  Since currents are linearly related, the derivative is a
                   constant (small signal = large signal)
            Department of EECS                                         University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                                         Prof. A. Niknejad



                                 Input Resistance rπ
                                                   iB          1 iC             gm
                                   r 
                                           1
                                                                            
                                                  vBE   Q
                                                                F vBE   Q
                                                                                  F
                                                                F
                                                         r 
                                                                gm

                   In practice, the DC current gain F and the small-signal
                    current gain o are both highly variable (+/- 25%)
                   Typical bias point: DC collector current = 100 A

                                                          25 mV
                                            r  100             25 k
                                                          .1mA
                                                Ri                 MOSFET

            Department of EECS                                                         University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                         Prof. A. Niknejad



                                 Output Resistance ro
             Why does current increase slightly with increasing vCE?
                                 Collector (n)



                           WB    Base (p)




                                 Emitter (n+)


                Answer: Base width modulation (similar to CLM for MOS)
                Model: Math is a mess, so introduce the Early voltage

                                 iC  I S e vBE / Vth (1  vCE V A )
            Department of EECS                                         University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                      Prof. A. Niknejad



                         Graphical Interpretation of ro



                                 slope~1/ro




                                       slope




            Department of EECS                      University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                             Prof. A. Niknejad



                                 BJT Small-Signal Model




                                         ib  r vbe
                                                    1
                                      ic  g m vbe  vce
                                                    ro
            Department of EECS                             University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                               Prof. A. Niknejad



                                 BJT Capacitors
                  Emitter-base is a forward biased junction 
                   depletion capacitance:
                                  C j , BE  1.4C j , BE 0
                  Collector-base is a reverse biased junction 
                   depletion capacitance
                  Due to minority charge injection into base, we have
                   to account for the diffusion capacitance as well

                                        Cb   F g m




            Department of EECS                               University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                               Prof. A. Niknejad



                                  BJT Cross Section
                                 Core Transistor




                                                        External Parasitic
                   Core transistor is the vertical region under the
                    emitter contact
                   Everything else is “parasitic” or unwanted
                   Lateral BJT structure is also possible
            Department of EECS                               University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                          Prof. A. Niknejad



                                  Core BJT Model
                                              Reverse biased junction

                          Base                                      Collector


                                                    g m v

                                                                  Fictional Resistance
                                                                        (no noise)
                  Reverse biased junction &
                   Diffusion Capacitance      Emitter


                  Given an ideal BJT structure, we can model most of the
                   action with the above circuit
                  For low frequencies, we can forget the capacitors
                  Capacitors are non-linear! MOS gate & overlap caps are
                   linear
            Department of EECS                                          University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                               Prof. A. Niknejad



                    Complete Small-Signal Model
                                 “core” BJT           Reverse biased junctions




     Real Resistance
       (has noise)




                                          External Parasitics



            Department of EECS                                               University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                               Prof. A. Niknejad



                                    Circuits!




                 When the inventors of the bipolar transistor first
                  got a working device, the first thing they did was to
                  build an audio amplifier to prove that the transistor
                  was actually working!
            Department of EECS                               University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                                          Prof. A. Niknejad



                                                     Modern ICs

                                                            Source: Intel Corporation
                                                              Used without permission




                       Source: Texas Instruments
                           Used without permission




                  First IC (TI, Jack Kilby, 1958): A couple of transistors
                  Modern IC: Intel Pentium 4 (55 million transistors, 3 GHz)
            Department of EECS                                                          University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                          Prof. A. Niknejad



                 A Simple Circuit: An MOS Amplifier


                  Input signal                   VDD
                                       RD                Supply “Rail”

                                        vo

                                        I DS
                                  vs
vGS  VGS  vs                                 Output signal
                                 VGS




            Department of EECS                          University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                 Prof. A. Niknejad



                    Selecting the Output Bias Point
                  The bias voltage VGS is selected so that the output is
                   mid-rail (between VDD and ground)
                  For gain, the transistor is biased in saturation
                  Constraint on the DC drain current:
                                        VDD  Vo VDD  VDS
                                 IR            
                                          RD        RD
                  All the resistor current flows into transistor:
                                          I R  I DS , sat
                  Must ensure that this gives a self-consistent
                   solution (transistor is biased in saturation)
                                          VDS  VGS  VT
            Department of EECS                                 University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                                     Prof. A. Niknejad



                      Finding the Input Bias Voltage
                  Ignoring the output impedance
                                                       W     1
                                         I DS , sat    nCox (VGS  VTn )2
                                                       L     2
                  Typical numbers: W = 40 m, L = 2 m,
                   RD = 25k, nCox = 100 A/V2, VTn = 1 V,
                   VDD = 5 V
                                          VDD                W     1
                                 I RD          I DS , sat  nCox (VGS  VTn ) 2
                                          2 RD               L     2
                                     5V                   μA 1
                                          100μA  20 100 2  (VGS  1) 2
                                    50k                  V 2
                      .1  (VGS  1) 2                VGS  1.32   VGS  VT  .32  VDS  2.5            
            Department of EECS                                                     University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                            Prof. A. Niknejad




            Applying the Small-Signal Voltage

            Approach 1. Just use vGS in the equation for the total
            drain current iD and find vo
                                            vGS  VGS  vs

                                            vs  vs cos t
                                                 ˆ

                                                             W1
                         vO  VDD  RDiDS    VDD  RD nCox     (VGS  vs  VT )2
                                                             L 2

                   Note: Neglecting charge storage effects. Ignoring
                   device output impedance.


            Department of EECS                                            University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                              Prof. A. Niknejad



                   Solving for the Output Voltage vO
                                                          W1
                      vO  VDD  RDiDS    VDD  RD nCox     (VGS  vs  VT )2
                                                          L 2
                                                                                                   2
                                                  W 1            2       vs    
            vO  VDD  RDiDS      VDD  RD nCox     (VGS  VT ) 1           
                                                  L 2                 VGS  VT 

                                                        I DS
                                                                        2
                                                             vs    
                                   vO  VDD  RD I DS 1           
                                                          VGS  VT 

                                                 VDD
                                                  2


            Department of EECS                                              University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                                                         Prof. A. Niknejad



                                        Small-Signal Case
                  Linearize the output voltage for the s.s. case
                  Expand (1 + x)2 = 1 + 2x + x2 … last term can be
                   dropped when x << 1
                                                        2
                                      vs                            2v s                           vs             2
                       1 + ------------------------- = 1 + ------------------------- +  ------------------------- 
                                                    -                                -                             -
                           V G S – V Tn                    V GS – V Tn  V – V 
                                                                                               GS              Tn


                                                                                        Neglect

                                                                2vs 
                                         vO  VDD  RD I DS 1        
                                                             VGS  VT 


            Department of EECS                                                                         University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                                              Prof. A. Niknejad



                             Linearized Output Voltage
             For this case, the total output voltage is:
                                              VDD     2vs 
                                   vO  VDD      1       
                                               2  VGS  VT 
                                             VDD   vsVDD
                                        vO      
                                              2 VGS  VT
                                 “DC”
                                                                    Small-signal output

             The small-signal output voltage:
                                                  vsVDD
                                        vo               Av vs
                                                 VGS  VT

                                                     Voltage gain
            Department of EECS                                              University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                         Prof. A. Niknejad


                  Plot of Output Waveform (Gain!)
            Numbers: VDD / (VGS – VT) = 5/ 0.32 = 16      output



                                                                            input


    mV




            Department of EECS                         University of California, Berkeley
EECS 105 Fall 2003, Lecture 15                                               Prof. A. Niknejad



                                 There is a Better Way!
                     What’s missing: didn’t include device output
                      impedance or charge storage effects (must solve
                      non-linear differential equations…)
                     Approach 2. Do problem in two steps.
                     DC voltages and currents (ignore small signals
                      sources): set bias point of the MOSFET ... we had
                      to do this to pick VGS already
                     Substitute the small-signal model of the MOSFET
                      and the small-signal models of the other circuit
                      elements …
                     This constitutes small-signal analysis
            Department of EECS                               University of California, Berkeley

								
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