Table of Contents

Overview
#

In analog circuit design, achieving high gain often comes at the expense of linearity. This analysis explores the fundamental reasons for this trade-off and techniques to optimize both parameters.

The Fundamental Trade-off
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Linearity Definition
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A perfectly linear amplifier satisfies:

$$ V_{out} = A \cdot V_{in} $$

Real amplifiers have nonlinear transfer characteristics:

$$ V_{out} = a_1 V_{in} + a_2 V_{in}^2 + a_3 V_{in}^3 + ... $$

Where \(a_1\) is the desired gain and \(a_2, a_3, …\) represent distortion.

Sources of Nonlinearity
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  1. Transistor \(g_m\) variation: \(g_m\) depends on \(V_{GS}\)
  2. Output resistance variation: \(r_o\) changes with \(V_{DS}\)
  3. Saturation limits: Clipping at supply rails
  4. Body effect: Threshold varies with signal

Transistor-Level Analysis
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MOSFET Transfer Characteristic
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In saturation:

$$ I_D = \frac{1}{2}\mu C_{ox}\frac{W}{L}(V_{GS} - V_{th})^2(1 + \lambda V_{DS}) $$

The square-law relationship is inherently nonlinear.

Small-Signal Linearity
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For small signals around operating point \(Q\):

$$ i_d \approx g_m v_{gs} + \frac{1}{2}g_m' v_{gs}^2 + ... $$

Where:

$$ g_m' = \frac{\partial g_m}{\partial V_{GS}} = \mu C_{ox}\frac{W}{L} $$

High Gain Increases Nonlinearity
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Higher gain requires:

  • Larger \(g_m\) → steeper transfer curve
  • Larger voltage swings → more nonlinear region traversed
Vout
  │
  │      ╱───── Saturation (linear region)
  │    ╱
  │   ╱  ← Operating point
  │  ╱
  │ ╱
  │╱
  └──────────────────── Vin

Large swings → traverse nonlinear regions

Quantifying Nonlinearity
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Total Harmonic Distortion (THD)
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$$ THD = \frac{\sqrt{V_2^2 + V_3^2 + V_4^2 + ...}}{V_1} \times 100\% $$

Where \(V_n\) is the amplitude of the \(n\)th harmonic.

Third-Order Intercept Point (IP3)
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$$ IIP3 = P_{in} + \frac{\Delta P}{2} $$

Where \(\Delta P\) is the difference between fundamental and third-order product power levels.

1-dB Compression Point
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Input level where gain drops 1 dB from linear:

$$ P_{1dB} = \text{IIP3} - 9.6 \text{ dB} $$

Linearity Enhancement Techniques
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1. Source Degeneration
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        VDD
         │
        [RD]
         │
         ├───── Vout
         │
      ┌──┴──┐
Vin ──│ M1  │
      └──┬──┘
         │
        [RS]  ← Degeneration resistor
         │
        GND

Effect on Gain:

$$ A_v = \frac{-g_m R_D}{1 + g_m R_S} $$

Effect on Linearity:

The effective transconductance becomes:

$$ G_m = \frac{g_m}{1 + g_m R_S} $$
ParameterWithout RSWith RS
Gain\(g_m R_D\)\(\frac{g_m R_D}{1 + g_m R_S}\)
LinearityBaselineImproved
BandwidthBaselineImproved

2. Feedback
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Negative feedback reduces distortion:

$$ THD_{CL} = \frac{THD_{OL}}{1 + A\beta} $$

Trade-off: Gain reduced by the same factor.

3. Differential Topology
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        VDD
         │
    ┌────┴────┐
   [RD]      [RD]
    │         │
    ├─Vout+   ├─Vout-
    │         │
 ┌──┴──┐   ┌──┴──┐
─│ M1  │   │ M2  │─
 └──┬──┘   └──┬──┘
    │         │
    └────┬────┘
         │
       [ISS]
         │
        GND

Benefits:

  • Cancels even-order harmonics
  • Better PSRR
  • Higher output swing

4. Operating Point Optimization
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Choose bias point for best linearity:

Bias RegionGainLinearity
Weak inversionLowBest
Moderate inversionMediumGood
Strong inversionHighWorst

Mathematical Analysis
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Power Series Expansion
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For input \(v_{in} = V_m \cos(\omega t)\):

$$ v_{out} = a_1 V_m \cos(\omega t) + \frac{a_2 V_m^2}{2}[1 + \cos(2\omega t)] + ... $$

DC offset: \(\frac{a_2 V_m^2}{2}\)

Second harmonic: \(\frac{a_2 V_m^2}{2}\cos(2\omega t)\)

Third harmonic: \(\frac{a_3 V_m^3}{4}\cos(3\omega t)\)

HD3 (Third Harmonic Distortion)
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$$ HD3 = \frac{a_3 V_m^2}{4a_1} $$

Increases with signal amplitude squared.

Design Guidelines
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For High Gain Priority
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  1. Accept higher THD
  2. Use minimum degeneration
  3. Limit input signal amplitude
  4. Apply post-amplifier filtering

For High Linearity Priority
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  1. Accept lower gain
  2. Use source degeneration
  3. Apply negative feedback
  4. Use differential topology
  5. Bias in weak inversion

Balanced Approach
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TechniqueGain ImpactLinearity Improvement
10% degeneration-0.8 dB~10×
DifferentialSame2× (even harmonics)
10× feedback-20 dB10×

Practical Applications
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RF Amplifiers
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  • High IP3 required for blocking signals
  • Moderate gain acceptable
  • Use multiple stages

Audio Amplifiers
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  • Low THD critical (<0.01%)
  • Feedback extensively used
  • Class AB for efficiency

Sensor Interfaces
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  • High gain for weak signals
  • Linearity important for accuracy
  • Chopper techniques for DC

Summary
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Key insights on gain-linearity trade-off:

  1. Inherent conflict: Higher gain → larger swings → more nonlinearity
  2. Source degeneration: Trades gain for linearity
  3. Feedback: Reduces both gain and distortion
  4. Differential: Cancels even harmonics
  5. Bias point: Weak inversion most linear
  6. Application-specific: Balance based on requirements