Table of Contents

Overview
#

Single-stage amplifiers form the building blocks of analog circuit design. Understanding transconductance, the Miller effect, and common configurations is essential for designing high-performance analog systems.

Transconductance
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Definition
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Transconductance (\(g_m\)) describes how output current changes with input voltage:

$$ g_m = \frac{\partial I_D}{\partial V_{GS}} \bigg|_{V_{DS}=\text{const}} $$

Calculation Methods
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1. From Drain Current Equation (Saturation):

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

Taking the derivative:

$$ g_m = \mu_n C_{ox} \frac{W}{L}(V_{GS} - V_{th}) $$

2. In Terms of Drain Current:

$$ g_m = \sqrt{2\mu_n C_{ox} \frac{W}{L} I_D} $$

3. In Terms of Overdrive Voltage:

$$ g_m = \frac{2I_D}{V_{GS} - V_{th}} = \frac{2I_D}{V_{ov}} $$

4. Small-Signal Parameter:

$$ g_m = \frac{i_d}{v_{gs}} $$

MOSFET Operating Regions
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I-V Characteristics
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I_D
 │                    ___________  Saturation
 │                 __/
 │              __/
 │           __/
 │        __/
 │     __/  Triode (Linear)
 │  __/
 │_/
 └────────────────────────────────── V_DS
           V_DSAT

Region Boundaries
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RegionConditionI_D Expression
Cutoff\(V_{GS} < V_{th}\)\(\approx 0\)
Triode\(V_{DS} < V_{GS} - V_{th}\)\(\mu_n C_{ox} \frac{W}{L}[(V_{GS}-V_{th})V_{DS} - \frac{V_{DS}^2}{2}]\)
Saturation\(V_{DS} \geq V_{GS} - V_{th}\)\(\frac{1}{2}\mu_n C_{ox} \frac{W}{L}(V_{GS} - V_{th})^2(1+\lambda V_{DS})\)

Body Effect
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When the source-to-body voltage is non-zero:

$$ V_{th} = V_{th0} + \gamma(\sqrt{2\phi_F + V_{SB}} - \sqrt{2\phi_F}) $$

Where:

  • \(V_{th0}\): Zero-bias threshold voltage
  • \(\gamma\): Body effect coefficient
  • \(\phi_F\): Fermi potential
  • \(V_{SB}\): Source-body voltage

Impact:

  • Positive \(V_{SB}\) increases threshold voltage
  • Reduces drain current for fixed \(V_{GS}\)
  • Impedes channel formation

Miller Effect
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Concept
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The Miller effect describes how feedback capacitance appears larger due to voltage gain:

$$ C_{Miller} = C_{gd}(1 + |A_v|) $$

Where \(A_v\) is the voltage gain of the stage.

Impact on Frequency Response
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Gain (dB)
    │
    │   Low frequency            High frequency
    │   (improved by             (degraded by
    │    Miller effect)          increased capacitance)
    │_____
    │     \_
    │       \__
    │          \___
    │              \____
    └────────────────────────── Frequency (log)
                f_3dB

3dB Bandwidth:

$$ f_{3dB} = \frac{1}{2\pi R_{in}(C_{in} + C_{Miller})} $$

Common Amplifier Configurations
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Common-Source Amplifier
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        VDD
         │
        [RD]
         │
         ├───── Vout
         │
      ┌──┴──┐
Vin ──│ M1  │
      └──┬──┘
         │
        GND

Characteristics:

ParameterExpression
Voltage Gain\(A_v = -g_m R_D\)
Input Impedance\(R_{in} \approx \infty\)
Output Impedance\(R_{out} = R_D \parallel r_o\)

Operation:

  1. Input voltage rise increases \(V_{GS}\)
  2. Drain current increases proportionally (\(g_m\))
  3. Voltage drop across \(R_D\) increases
  4. Output voltage decreases (inverted)

Common-Source with Current Source
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        VDD
         │
      ┌──┴──┐
      │ M2  │ (current source)
      └──┬──┘
         │
         ├───── Vout
         │
      ┌──┴──┐
Vin ──│ M1  │
      └──┬──┘
         │
        GND

Benefits:

  • Higher output impedance: \(R_{out} = r_{o1} \parallel r_{o2}\)
  • Higher gain: \(A_v = -g_{m1}(r_{o1} \parallel r_{o2})\)
  • Better power supply rejection

Source Follower (Common-Drain)
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        VDD
         │
      ┌──┴──┐
Vin ──│ M1  │
      └──┬──┘
         │
         ├───── Vout
         │
        [RS]
         │
        GND

Characteristics:

ParameterExpression
Voltage Gain\(A_v \approx \frac{g_m R_S}{1 + g_m R_S} \approx 1\)
Input Impedance\(R_{in} \approx \infty\)
Output Impedance\(R_{out} \approx \frac{1}{g_m}\)

Operation:

  • Output follows input with unity gain
  • Low output impedance (buffer function)
  • No phase inversion

Common-Gate
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        VDD
         │
        [RD]
         │
         ├───── Vout
         │
      ┌──┴──┐
      │ M1  ├───── Vbias
      └──┬──┘
         │
Vin ─────┤
         │
        [RS]
         │
        GND

Characteristics:

ParameterExpression
Voltage Gain\(A_v = g_m R_D\)
Input Impedance\(R_{in} \approx \frac{1}{g_m}\)
Output Impedance\(R_{out} = R_D\)

Applications:

  • High-frequency circuits (no Miller effect on input)
  • Current sensing
  • Cascode stage

Cascode Configuration
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Combines common-source and common-gate for high gain:

        VDD
         │
        [RD]
         │
         ├───── Vout
         │
      ┌──┴──┐
Vbias─│ M2  │ (CG)
      └──┬──┘
         │
      ┌──┴──┐
Vin ──│ M1  │ (CS)
      └──┬──┘
         │
        GND

Gain:

$$ A_v = -g_{m1}(g_{m2}r_{o2}r_{o1} \parallel R_D) $$

Benefits:

  • Very high output impedance
  • Reduced Miller effect
  • Higher gain

Summary
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Key concepts in single-stage amplifiers:

  1. Transconductance: Links input voltage to output current
  2. Miller effect: Capacitance multiplication impacts bandwidth
  3. Common-source: Inverting, high gain
  4. Source follower: Unity gain, low output impedance
  5. Common-gate: Non-inverting, low input impedance
  6. Cascode: Combines CS and CG for optimal performance