Calculating Drain voltage of a mosfet

With no external resistors, is it possible to calculate drain voltage of a MOSFET with known values for Vg, Vs, Vb, and Ids?

• What is "Vb" in this circuit? – Spehro Pefhany Nov 21 '19 at 4:16
• Experiment and learn ...tinyurl.com/sdkwete – Mitu Raj Nov 21 '19 at 11:46
• Vb is the bulk voltage. For NMOS it's typically internally connected to Vs and for PMOS it's typically internally connected to Vd. This is not always the case, though. – jwolters Nov 25 '19 at 14:57

As drawn, the voltage on the drain of the MOSFET is $$\V_d\$$.

Sure, this is possible: it is one of the reasons for which it is possible to design a circuit with a MOSFET (or a BJT, JFET, etc...). Precisely, there exist a well defined relation $$I_D=f(V_{GS},V_{DS}, V_{BS})\label{1}\tag{1}$$ where

• $$\V_{GS}=V_G-V_S\$$ is the voltage between the gate and source of the MOSFET,
• $$\V_{DS}=V_D-V_S\$$ is the voltage between the drain and the source of the MOSFET,
• $$\V_{BS}=V_B-V_S\$$ is the voltage between the bulk and the source of the MOSFET
• $$\V_B\$$, $$\V_D\$$, $$\V_S\$$ are respectively the bulk (body terminal, usually inaccessible and electrically connected to the source in most modern MOSFETs) drain and source absolute (i.e. referred to a reference ground) voltages,

and this relation can be ideally used to determine one of the four quantities involved when the other three are known. However, equation \eqref{1} is not usually given in analytic form but is given in the form of several parametric diagrams in the data sheet of the device: by using the diagrams in the data sheet (or the ones the technology test laboratory guy gives you) you can determine, for given $$\I_D\$$, $$\V_{GS}\$$ and $$\V_{BS}\$$, the associated $$\V_{DS}\$$ value.

A real world example (further EDIT).

Following jwolter's request, let's show how what I said above applies to the determination of $$\V_{DS}\$$ when \eqref{1} is the well known standard SPICE model of the MOSFET: \left. \begin{align} I_D &=\frac{\beta}{2}(V_{GS}-V_\mathrm{th})^2(1+\lambda V_{DS})\\ V_\mathrm{th}&= V_{th_0}+\gamma\left(\sqrt{\phi-V_{BS}}-\sqrt{2\phi}\right) \end{align}\label{1'}\tag{1'} \quad\right\} where the quantities not previously defined have the following meaning:

• $$\\beta=\mu_nC_\mathrm{ox}\dfrac{W}{L}\equiv\mathrm{KP}\dfrac{W}{L}\$$ is the transconductance parameter multiplied by the channel length/channel width ratio,
• $$\V_{th_0}\equiv\mathrm{VTO}\$$ is the zero bias gate threshold voltage,
• $$\\gamma\equiv\mathrm{GAMMA}\$$ is the bulk threshold/backgate effect parameter and
• $$\\lambda\equiv\mathrm{LAMBDA}\$$ is the channel-length modulation parameter.
• $$\\phi\equiv\mathrm{PHI}\$$ is the surface potential of the MOSFET.

If you know exactly $$\V_{GS}, V_{BS}, I_D\$$, you know all the quantities in formula(s) \eqref{1'} except $$\V_{DS}\$$ and thus obtaining its value is a matter of simple algebraic manipulation: $$V_{DS}=\frac{1}{\lambda}\left[\frac{I_D}{\frac{\beta}{2}(V_{GS}-V_\mathrm{th})^2}-1\right]$$

Finally it is worth to note that the standard SPICE model could not descrive precisely the behavior of your MOSFET, and you may want to use more accurate models. And obviously, the more accurate the model, the more precise the value of $$\V_{DS}\$$ you will obtain: however, the model complexity could rise so much that an analytic solution simply cannot be found and you can use only numerical techniques in order to evaluate the sought for value.

• Thanks for the answer, and for any actual MOSFET with an available data sheet, I will keep this in mind for the future. In this specific case, though, I didn't have a data sheet. I did, however, have spice parameters. Do you know what the relationship would be with common spice paramters? (lambda, gamma, etc.) The drain was connected to an active loaded differential pair and I was trying to find the value of the drain voltage purely through hand analysis without simulating. The deadline for this task has come and gone (i got it from simulation), but I'm still curious. – jwolters Nov 25 '19 at 14:52
• @jwolters I'll update my answer as soon as possible, including an example where \eqref{1} is the equation of $I_D$ in a SPICE model. – Daniele Tampieri Nov 26 '19 at 6:42