when I did the Sedra exercise (Edition 7) where we have exercise number 5.49 where the objective is to find the length, but when calculating I did what is practically in the formula, but I did it wrong, because I had to include the vss ( vss-vsg-|vt|). I didn't understand why.

 

In Sedra and Smith's Microelectronic Circuits (7th edition), exercise 5.49 deals with finding the length of a MOSFET when certain parameters are given. Let's break down the problem and the solution step by step.

1. Problem Statement:
You are given a MOSFET with parameters:
- \( V_{GS(th)} = 1.2 \, V \)
- \( \mu_nC_{ox} = 100 \, \mu A/V^2 \)
- \( V_{DS} = 5 \, V \)
- \( I_D = 0.2 \, mA \)

You are asked to find the length of the MOSFET.

2. Understanding the Solution:
The solution involves using the MOSFET small-signal model and the square-law equation to find the length of the MOSFET. The square-law equation is:

\[ I_D = \frac{1}{2} \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th})^2 \]

Where:
- \( I_D \) is the drain current.
- \( \mu_n \) is the electron mobility.
- \( C_{ox} \) is the oxide capacitance per unit area.
- \( W \) is the width of the MOSFET.
- \( L \) is the length of the MOSFET.
- \( V_{GS} \) is the gate-to-source voltage.
- \( V_{th} \) is the threshold voltage.

3. Solution:
The solution involves rearranging the square-law equation to solve for the length (\( L \)) of the MOSFET:

\[ L = \frac{1}{2} \mu_n C_{ox} \frac{W}{I_D} (V_{GS} - V_{th})^2 \]

Substituting the given values:
- \( V_{GS} = V_{DS} = 5 \, V \)
- \( V_{th} = 1.2 \, V \)
- \( \mu_n C_{ox} = 100 \, \mu A/V^2 \)
- \( I_D = 0.2 \, mA \)

You should get the correct value for \( L \). Make sure you include \( V_{SS} \) when calculating \( V_{GS} \) because \( V_{GS} = V_{SS} - V_{G} \), where \( V_{SS} \) is the source voltage and \( V_{G} \) is the gate voltage. In this case, \( V_{SS} = 0 \) (since it's connected to ground), so \( V_{GS} = -V_{th} \).

4. Conclusion:
Including \( V_{SS} \) in the calculation of \( V_{GS} \) accounts for the actual gate-to-source voltage and ensures an accurate calculation of the MOSFET length.

Ensure you follow these steps accurately, and you should arrive at the correct solution for the MOSFET length.