Mosfet equation: how to get it?

  • Thread starter Granger
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In summary, the conversation is about analyzing a circuit with a MOSFET and finding the dependence of the drain current over the threshold voltage. The equations for the gate current and drain current are provided, and the book takes the derivative of the expression to show the dependence. The correct application of the chain rule is discussed, and the conversation ends with the understanding of the mistake in assuming the current I_D was constant when taking the derivative.
  • #1
Granger
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Homework Statement


I'm studying the following circuit with a MOSFET

[![enter image description here][1]][1] [1]: https://i.stack.imgur.com/gEEdQ.png

Homework Equations


3. The Attempt at a Solution [/B]
Now for analyzing this circuit my book came out with various equations (which I totally understand and have no doubts about it).
Because the gate current is zero we have

$$V_G=V_{DD} \frac{R_2}{R_1+R_2}$$

Assuming the MOSFET is working on saturation we also have

$$I_D=k(V_{GS}-V_t)^2$$

And applying KVL:

$$V_{GS}=V_G-R_SI_D$$

Substituting equation (3) on equation (2) we have

$$I_D=k(V_G-R_SI_D -V_t)^2$$Now comes to the point I'm not understanding. To show us the dependence of the drain current over the threshold voltage, the book takes the derivative of the expression:

$$\frac{dI_D}{dV_t}=\frac{d}{dV_t}k(V_G-R_SI_D -V_t)^2$$

and then it writes

$$\frac{dI_D}{dV_t}=\frac{-2\sqrt{kI_D}}{1+2R_S\sqrt{kI_D}}$$

How on Earth did they go from equation (5) to equation (6) by taking the derivative. What kind of substitution are they making?
 
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  • #2
Did you try differentiating as indicated for the right hand side of your second-last equation? What do you get?
 
  • #3
NascentOxygen said:
Did you try differentiating as indicated for the right hand side of your second-last equation? What do you get?

I get
$$-2k(V_G - R_SI_D-V_t)$$
 
  • #4
Reminder: the chain rule says dy/dx =dy/dz * dz/dx

You have written the equivalent of dy/dz, and now need to multiply by the equivalent of dz/dx
 
  • #5
NascentOxygen said:
Reminder: the chain rule says dy/dx =dy/dz * dz/dx

You have written the equivalent of dy/dz, and now need to multiply by the equivalent of dz/dx

What do you mean? Why using the chain rule here? I'm supposed to take the derivative of I_D in relation to V_t and that's exactly what I'm taking.
 
  • #6
If you aren't confident in use of the chain rule, you can instead expand the squared term, i.e., get rid of the brackets on the right hand side, before attempting the differentiation. (This is a good way to check that you are going about it correctly, because obviously both methods should give identical answers.)
 
  • #7
NascentOxygen said:
If you aren't confident in use of the chain rule, you can instead expand the squared term, i.e., get rid of the brackets on the right hand side, before attempting the differentiation. (This is a good way to check that you are going about it correctly, because obviously both methods should give identical answers.)

Oh I got what I was doing wrong!
I was assuming the current I_D was constant when taking the derivative which is obviously wrong. Thanks!
 

Related to Mosfet equation: how to get it?

1. What is a Mosfet equation?

A Mosfet (Metal-Oxide-Semiconductor Field-Effect Transistor) equation is a mathematical formula that describes the behavior of a Mosfet, an electronic device used for amplifying or switching electronic signals. It relates the input voltage and current to the output voltage and current, and is essential for understanding and designing Mosfet circuits.

2. How do you derive the Mosfet equation?

The Mosfet equation can be derived using basic principles of semiconductor physics and circuit analysis. It involves analyzing the behavior of the Mosfet's three regions: the cutoff region, the linear region, and the saturation region. The equation is then simplified using various assumptions and approximations.

3. What parameters are included in the Mosfet equation?

The Mosfet equation includes parameters such as the threshold voltage, channel length, channel width, and various other physical and electrical characteristics of the Mosfet. These parameters vary depending on the specific Mosfet model and can be found in the manufacturer's datasheet.

4. How accurate is the Mosfet equation?

The accuracy of the Mosfet equation depends on the assumptions and approximations made during its derivation. In general, it provides a good approximation of the Mosfet's behavior in the linear and saturation regions. However, it may not accurately predict the behavior in extreme conditions or when the Mosfet's parameters vary significantly.

5. How can the Mosfet equation be used in circuit design?

The Mosfet equation is a useful tool for designing and analyzing Mosfet circuits. It can be used to determine the biasing conditions, calculate the gain and output characteristics, and analyze the stability of the circuit. It can also be used to optimize the circuit design and select the appropriate Mosfet for a specific application.

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