# MOSFET as a VCCS or an variable resistance

I am a bit confused. In saturation region, sometimes, an MOSFET is called voltage-controlled current source and sometimes it is called variable resistor where its resistance is controlled by the gate-source voltage. Are the two concepts equivalent? I meant MOSFET operates in saturation region not triode.

• Where did you see someone say that a MOSFET can be used as a variable resistor in the saturation region? – The Photon Apr 18 '16 at 15:56

A MOSFET acts like a variable resistor in the linear region and a VCCS in the saturation region. In saturation, the VCCS has a small-signal output resistance due to channel length modulation, but it's not the same behavior as the linear region.

In saturation, the drain current is given by the equation:

$$I_D = k(V_{GS} - V_{T})^2(1 + \lambda V_{DS})$$

We can calculate the drain-source conductance (1/resistance) as:

$$G_{sat} = \frac {dI_D}{dV_{DS}} = \lambda k (V_{GS} - V_T)^2$$

In the linear region, the drain current is given by:

$$I_D = k[2(V_{GS} - V_T)V_{DS} - V_{DS}^2]$$

If $V_{DS} << V_{GS} - V_T$, this reduces to:

$$I_D = 2k(V_{GS} - V_T)V_{DS}$$

and the conductance is:

$$G_{lin} = \frac {dI_D}{dV_{DS}} = 2k(V_{GS} - V_T)$$

By themselves, $G_{sat}$ and $G_{lin}$ don't look all that different. To my mind, the main differences are:

1. Saturation assumes that $V_{DS} > V_{GS} - V_T$, and you need a current source to model that.

2. The linear region is used for switching. A resistor model is useful for estimating power dissipation when the switch is on.

3. Lambda is typically very small (~0.01), so the saturation output resistance is a second-order effect. In the linear region, $V_{DS}$ is just as important as $V_{GS}$.

I'm pretty sure they are the same as it's a voltage controlled current source, but I've never seen that exact description. A current constraining source would be a resistor. Usually, as MOSFET as a resistor is an ohmic device, so I pushed through the math as if the device was exclusively a saturated MOSFET resistor (ie: mobility controlled device) and you get the following: $$R_{ON}=res\left(\frac{L}{t_{inv}W}\right)=\left(\frac{1}{n_{inv}q\mu_n}\right)\left(\frac{L}{t_{inv}W}\right)$$ This then becomes $$R_{ON}=\frac{1}{\mu_n C_{ox}\frac{W}{L}\left(V_{gs}-V_{th}\right)}$$

because $q n_{inv} \mu_n =Q_{inv}=C_{ox} \left(V_{gs}-V_{th}\right)$ There's your zeroth order analysis.