Lecture 15: Small Signal Modeling

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Lecture 15 Small Signal Modeling

• Prof. Niknejad

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Lecture Outline

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

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Notation Review
small signal
Quiescent Point (bias)
DC (bias)
small signal (less messy!)
transconductance
Output conductance

• Since were introducing a new (confusing)
  subject, lets 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)

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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)
• Decreasing width increases current!

Density here fixed by potential (injection of
carriers) Physical interpretation How many
electrons (holes) have enough energy to cross
barrier? Boltzmann distribution give this number.
Density fixed by metal contact
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Diffusion Capacitance

• The total minority carrier charge for a one-sided
  junction is (area of triangle)
• For a one-sided junction, the current is
  dominated by these minority carriers

Constant!
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Diffusion Capacitance (cont)

• The proportionality constant has units of time
• 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

Distance across P-type base
Diffusion Coefficient
Mobility
Temperature
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BJT Transconductance gm

• The transconductance is analogous to diode
  conductance

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Transconductance (cont)

• Forward-active large-signal current

• Differentiating and evaluating at Q (VBE, VCE )

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BJT Base Currents
Unlike MOSFET, there is a DC current into
the base terminal of a bipolar transistor
To find the change in base current due to change
in base-emitter voltage
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Small Signal Current Gain

• Since currents are linearly related, the
  derivative is a constant (small signal large
  signal)

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Input Resistance rp

• 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

MOSFET
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Output Resistance ro
Why does current increase slightly with
increasing vCE?
Collector (n)
Base (p)
Emitter (n)
Answer Base width modulation (similar to CLM
for MOS) Model Math is a mess, so introduce the
Early voltage
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Graphical Interpretation of ro
slope1/ro
slope
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BJT Small-Signal Model
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BJT Capacitors

• Emitter-base is a forward biased junction ?
  depletion capacitance
• 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

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BJT Cross Section

• Core transistor is the vertical region under the
  emitter contact
• Everything else is parasitic or unwanted
• Lateral BJT structure is also possible

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Core BJT Model
Reverse biased junction
Fictional Resistance (no noise)
Reverse biased junction Diffusion Capacitance

• 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

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Complete Small-Signal Model
core BJT
Reverse biased junctions
Real Resistance (has noise)
External Parasitics
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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!

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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)

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A Simple Circuit An MOS Amplifier
Input signal
Output signal
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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
• All the resistor current flows into transistor
• Must ensure that this gives a self-consistent
  solution (transistor is biased in saturation)

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Finding the Input Bias Voltage

• Ignoring the output impedance
• Typical numbers W 40 ?m, L 2 ?m,
  RD 25k??, ?nCox 100 ?A/V2, VTn 1 V,
  VDD 5 V

?
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Applying the Small-Signal Voltage
Approach 1. Just use vGS in the equation for the
total drain current iD and find vo
   
Note Neglecting charge storage effects.
Ignoring device output impedance.
     
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Solving for the Output Voltage vO
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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 ltlt 1

Neglect
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Linearized Output Voltage

• For this case, the total output voltage is
• The small-signal output voltage

     
Voltage gain
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Plot of Output Waveform (Gain!)
Numbers VDD / (VGS VT) 5/ 0.32 16
output

input
mV
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There is a Better Way!

• Whats missing didnt 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