MOSFET Fundamental Equations

1. Large-Signal Characteristics

Regions of Operation

• Cutoff Region
  Condition: $V_{GS} < V_{th}$
  $I_{DS} \approx 0$

• Linear (Triode) Region
  Condition: $V_{GS} > V_{th}, ; V_{DS} < (V_{GS} - V_{th})$

$$
I_{DS} = \mu_n C_{ox} \frac{W}{L} \left[(V_{GS}-V_{th})V_{DS} - \frac{1}{2} V_{DS}^2 \right]
$$

• Saturation Region
  Condition: $V_{GS} > V_{th}, ; V_{DS} \ge (V_{GS}-V_{th})$

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

where $\lambda$ models channel-length modulation.

Definitions

$$
K = \mu_n C_{ox} \frac{W}{L}
$$

Overdrive voltage:

$$
V_{OV} = V_{GS} - V_{th}
$$

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2. Small-Signal Parameters

• Transconductance

$$
g_m = \frac{\partial I_{DS}}{\partial V_{GS}} = K(V_{GS}-V_{th})(1+\lambda V_{DS})
= \sqrt{2K I_{DS} (1+\lambda V_{DS})}
$$

• Output resistance

$$
r_o = \left(\frac{\partial I_{DS}}{\partial V_{DS}}\right)^{-1} = \frac{1}{\lambda I_{DS}}
$$

• Small-signal current relation

$$
i_{ds} = g_m v_{gs} + \frac{v_{ds}}{r_o}
$$

• Intrinsic gain

$$
A_0 = g_m r_o
$$

• Input and output impedance

$$
Z_{in} \approx \infty, \quad Z_{out} \approx r_o
$$

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3. Current Mirror

Principle

Reference transistor M1 sets $V_{GS}$ through input current $I_{in}$:

$$
V_{GS} = V_{th} + \sqrt{\frac{2 I_{in}}{K_1}}
$$

Output transistor M2 shares the same $V_{GS}$, so:

$$
I_{out} = \frac{(W/L)_2}{(W/L)1} \cdot I{in}
$$

Small-Signal Model

• Current transfer ratio (mirror gain):

$$
G_{mirror} = \frac{g_{m2}}{g_{m1}}
$$

• Input impedance:

$$
Z_{in} \approx \frac{1}{g_{m1}}
$$

• Output impedance:

$$
Z_{out} \approx r_{o2}
$$

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