# three gm equations, how to use them

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#### surreyian

##### Member level 3 Hello

There are 3 gm equations that can be used

1/ gm=2Id/Vgs-Vth

2/ gm= sqrt(2k(W/L)Id)

3/ gm= k(W/L)(Vgs-Vth)

when do u know which equations to use to determine the gm?

Equation 1, if u have Id constant, gm will decrease with increasing vgs-vth. I can connect a common source with ideal current, increasing vgs will cause gm to decrease. Is that right?

Equation 3. what kind of circuits will use equation 3 to determine the gm. can u give me an example.

thanks

#### Junus2012 Hello
or the saturation region (which is the condition of the MOS amplifier circuits) gm = K'(W/L).(Vgs-Vth).......(1)

since vgs-vth=Overdrive voltage(Von) (at the edge of the saturation region).... Von=(2Id/K'.W/L)power0.5..........(2)

put 3 in 2 you get gm = (2.Id.(W/L).K')Power0.5...........(3)

it is usually easy to work with 3 as we know the current always in our design

#### monsoon

##### Member level 3 Equation 1, if u have Id constant, gm will decrease with increasing vgs-vth. I can connect a common source with ideal current, increasing vgs will cause gm to decrease. Is that right?
Hi
Yes. For a constant Id, if you increase vgs-vth, then at the same time W/L will have to be reduced. Since both vgs-vth and W/L are varying, it will not be proper to use equation 3. Either eqn 1 or eqn 2 will lead to the same result.

• surreyian

### surreyian

points: 2

#### surreyian

##### Member level 3 Hi
Yes. For a constant Id, if you increase vgs-vth, then at the same time W/L will have to be reduced. Since both vgs-vth and W/L are varying, it will not be proper to use equation 3. Either eqn 1 or eqn 2 will lead to the same result.
thanks.
in what condition will equation 3 be used?

#### monsoon

##### Member level 3 Hi
All the three equations are equally valid in all conditions. What equation is chosen depends on what is more 'convenient'. For example, if your common source stage had a resistive load (instead of current source), and again you ask the question what happens to gm as vgs-vth varies for a fixed W/L. Though you may use equation 1 and 2 to eventually arrive at the answer, eqn 3 is the easiest way to see that gm will increase linearly with respect to vgs-vth. I hope you get the point.

#### Junus2012 the only condition is that the transistor should be in the saturation region and this is a factor is only used for small signal condition

#### watersky

##### Member level 5 all of three equations are right. the different is the choice : which parameter is fixed, which parameter is variable.

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