2.2 Long-Channel IV Characteristics

Regions of Operation

Region	Condition	Description
Cutoff (OFF)	$V_{gs} < V_t$	No channel is formed
Linear (Triode)	$V_{gs} > V_t$ and $V_{ds} < V_{dsat}$	Channel is formed, acts like a voltage controlled resistor
Saturation	$V_{gs}>V_t$ and $V_{ds} ≥ V_{dsat}$	Channel is pinched off near the drain, acts like voltage-controlled current source

Saturation voltage: boundary between linear and saturation regions: $V_{dsat} = V_{GT} = V_{gs} - V_t$

Equations

Gate Capacitance & Charge

Oxide capacitance per unit area

$$ C_{ox} = \frac{\kappa_{ox} \epsilon_0}{t_{ox}} $$
- $\kappa_{ox}$ dielectric constant of silicon oxide (~3.9)
- $t_{ox}$ gate oxide thickness
- $\epsilon_0$ permittivity of free space
Channel charge

$$ Q_{channel} = C_g \cdot (V_{gs}- V_t) = C_{ox} WL (V_{gs}-V_t) $$
Transconductance Parameter

$$ \beta = \mu C_{ox} \frac{W}{L} $$

Current Equations $I_{ds}$ by Region

$$ I_{ds} = \begin{cases} 0 & V_{gs} < V_t \quad \text{(Cutoff)} \\ \beta \left( V_{GT} - \frac{V_{ds}}{2} \right) V_{ds} & V_{ds} < V_{dsat} \quad \text{(Linear)} \\ \frac{\beta}{2} V_{GT}^{2} & V_{ds} \geq V_{dsat} \quad \text{(Saturation)}\end{cases} $$

Linear region current: $I_{ds} \approx \beta V_{GT} V_{ds}$
Saturation region current: $I_{ds} = \frac{\beta}{2} (V_{gs} - V_t)^2$
ON Current: $I_{on} = \frac{\beta}{2}(V_{DD} - V_t)^2$

pMOS

Same equations as above apply except for the following:

Threshold voltage $V_t$ is negative
Voltages $V_{gs}$ and $V_{ds}$ are typically negative
Current flows from source to drain