# How to calculate Vout from the MOSFET circuit

This circuit diagram is given in my book :

simulate this circuit – Schematic created using CircuitLab

and said that from the above figure , they calculate $\frac{V_{out}}{V_p}$ while considering $r_{o2}$ and write down the answer is:

$$\frac{V_{out}}{V_p} = \frac{1+g_{m2}r_{o2}}{1+\frac{r_{o2}}{r_{o4}}} \approx\frac{g_{m2}r_{o2}}{1+\frac{r_{o2}}{r_{o4}}}$$

No detail calculation was given and I tried my self but could not reach the solution. How can I do this kind of calculation?

can someone show me step by step solution for this so that I can do this kind of problems by my self in future. thanks in advance.

• Is ro2 in the answer given the R_o2 from circuit, or M2 output resistance? – Vicente Cunha Feb 3 '16 at 22:37
• I have attempted to solve your circuit, but ended up with the answer divided by a factor of 2. Is the definition of gm used by your book author 2(Id/Veff)? – Vicente Cunha Feb 3 '16 at 23:08
• sorry , my mistake. ro2 is from the circuit (R_o2) . I edited the mistake. – Anklon Feb 4 '16 at 3:59
• @vicenteCunha , yeah . definition of gm is 2(Id/Veff) . – Anklon Feb 4 '16 at 4:00

The output voltage Vout consists of two portions:

1.) Vout1 caused by the drain current Id of the transistor (voltage drop across r_o4),

2.) Vout2 caused (direct way) by the voltage divider between r_o4 and r_o2 .

Calculation:

1.) Transistor is in common base configuration with Vout1=gm*(r_o2||r_o4)

2.) Vout2=Vp*[r_o4/(r_o4+r_o2)

3.) The sum of both parts Vout=Vout1+Vout2 gives the total signal output voltage. Divided by the input Vp we get the given gain expression (after some mathematical manipulations).

• why two portion ? can you please explain the reason or provide me some link from where I can understand this. – Anklon Feb 5 '16 at 15:20
• The voltage Vp drives a current directly through r_o2 and causes a voltage drop across r_o4 (first portion). The other portion is a voltage caused by the transistors output current Id (at the parallel combination of r_04 and r_o2). Remember that the transistor acts as a current source. – LvW Feb 6 '16 at 10:06