MOS Circuits

Q1: What states can a MOS be in?
A1: A NMOS may be in:

1. Cutoff
• $v_{OV}\leq 0$
• $i_D=0$
2. Saturation
• $v_{OV}\geq 0$ and $v_{DS}\geq v_{OV}$
• $i_D=0.5\mu_nC_{ox}\frac{W}{L}v_{OV}^2(1+\lambda v_{DS})$
3. Triode
• $v_{OV}\geq 0$ and $v_{DS}\leq v_{OV}$
• $i_D=0.5\mu_nC_{ox}\frac{W}{L}(2v_{OV}v_{DS}- v_{DS}^2)$

A couple of notes:

• $v_{OV}=v_{GS}-v_{tn}$ where $v_{OV}$ is the overdrive voltage and $v_{tn}$ is the threshold voltage associated with the NMOS
• The expressions for $i_D$ in saturation and triode both contain the pre-factor $0.5\mu_nC_{ox}\frac{W}{L}$ 
• $\mu_n$ is the negative charge carrier effective mobility
• $C_{ox}$ is gate-oxide capacitance per unit area
• $W$ is the gate width
• $L$ is the gate length
• The expression for $i_D$ in saturation contains the channel-length modulation parameter $\lambda$. 
• If channel-length modulation can be neglected, then $\lambda=0$
• The channel-length modulation parameter is related to early voltage by: $\lambda=\frac{1}{v_A}$
• The phenomenon of channel-length modulation in MOSFETs can be compared to Early effect in BJTs

For a PMOS:

1. $v_{OV}=v_{GS}-v_{tn}\to v_{OV}=v_{SG}-|v_{tp}|$
2. $\mu_n \to \mu_p$
3. $v_{DS}\to v_{SD}$
4. $v_{GS}\to v_{SG}$

Q2: How do I solve MOS circuits?

A2: For an NMOS circuit:

1. Write the KVLs for $v_{GS}$ and $v_{DS}$
2. Assume cutoff by setting $i_D$=0. Check that you are in cutoff.
3. Assume saturation. Check that $v_{DS}\geq v_{OV}$
4. You're in triode mode. 

Note that the equations for $i_D$ in saturation and triode mode are quadratic. Although you may find 2 roots, only keep the 1 that's valid for that particular state.