# Negative Resistance equation

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#### GGAPBE96

##### Member level 2
resistance equation

Many references say negative resistance of colpitts oscillator can describe the following equation.

-R = -gm/ ω^2/C1/C2 (gm :: trans conductance of FET)

However,negative resistance is also proportional to the ratio "C1/C2".

Does anyone tell me how to develop the expression for the exact negative resistance?

Thanks,

colpitts oscillator equation

That is the expression for the negative resistance. Except the equation is:

R= -gm/(ω^2(C1C2))

Wikipedia for Colpitts oscillator shows the derivation of the negative resistance

### GGAPBE96

Points: 2
equation for resistance

snafflekid said:
That is the expression for the negative resistance. Except the equation is:

R= -gm/(ω^2(C1C2))

Wikipedia for Colpitts oscillator shows the derivation of the negative resistance

Thank you for the comment.

I know the equation,but I believe this expression is not exact.
In fact,negative resistance is the function of C1,C2,and C1/C2,
but the equation means the negative resistance is only inversely proportional to　C1,C2.
It does not depend on the ratio C1/C2.

negative resistance oscillators:

GGAPBE96 said:
I know the equation,but I believe this expression is not exact.
In fact,negative resistance is the function of C1,C2,and C1/C2,
but the equation means the negative resistance is only inversely proportional to　C1,C2.
It does not depend on the ratio C1/C2.

The equation must contain the PRODUCT (and not the ratio of both capacitors).
One simple reason: Otherwise the units of the right side of the equation give not "ohms".

### GGAPBE96

Points: 2
math equation negative 35

Yes,I agree with you.The equation itself is correct.
But I think it is not perfect.

I believe there is more accurate expression to describe the negative resistance.
For example,

R = -gm/(ω^2C1C2) + f(C1,C2,C1/C2)

Of cource,the dimension of the additional term is ohm.

negative equation oscillator circuit

the complex impedance that the inductor sees is

Zin= 1/sC1 + 1/sC2 - gm/(ω^2C1C2) Is that what you mean?

The real component of this impedance is R = - gm/(ω^2C1C2)
the imaginary component is X = 1/sC1 + 1/sC2 which you could rewrite as
(C1+C2)/sC1C2

And also this is a small signal analysis, where gm does not vary. Maybe C1/C2 affects the bias point of the transistor? I am not familiar with that.

### GGAPBE96

Points: 2
colpitts negative resistance

>Zin= 1/sC1 + 1/sC2 - gm/(ω^2C1C2) Is that what you mean?
>Maybe C1/C2 affects the bias point of the transistor?

No,I'm focusing on the real part.
As you said,C1/C2 may change the transistor condition..

I might mistake the premise.

Maybe I ought to consider the open loop gain.
The equation "R = -gm/(ω^2C1C2)" is just derived from the circuit impedance.
The feedback is not considered..

what does negative resistance mean

GGAPBE96 said:
>Zin= 1/sC1 + 1/sC2 - gm/(ω^2C1C2) Is that what you mean?
>Maybe C1/C2 affects the bias point of the transistor?

Maybe I ought to consider the open loop gain.
The equation "R = -gm/(ω^2C1C2)" is just derived from the circuit impedance.
The feedback is not considered..

May I jump into the discussion again ?
Here is a general remark: Each oscillator circuit can be viewed as a negative resistance type or as a 4-pole type (gain stage with feedback). It depends which sight you prefer resp. which approach is more suitable.

Therefore: If you decide to handle the colpitt oscillator from the negative resistance viewpoint (which sounds reasonable) you must not consider any feedback again (that means: the second time). The circuitry with the negative input resistance is shunted with a lossy inductance. That´s all !
And from the CLAPP oscillator which I have analyzed already (and which is very similar to COLPITT) I know that it has the same neg. input resistance as mentioned above, that is: Zin= 1/sC1 + 1/sC2 - gm/(ω^2C1C2)

### GGAPBE96

Points: 2
explanation of a resistance equation

Thank you for the detailed comment!

I didn't know an oscillator can be defined as 2 different viewpoints.

I simulated the negative resistance of some oscillators,
and the results showed the negative resistance was proportional to C2/C1.
So,I doubted the equation was not accurate.

I'll study the basics of oscillators again.

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