Derive an expression for the equivalent resistance offered by an N-MOSFET when a capacitorinitially charged to VDD is being discharged through it until VDD/2.
Question
Derive an expression for the equivalent resistance offered by an N-MOSFET when a capacitor initially charged to VDD is being discharged through it until VDD/2.
Solution
The equivalent resistance of an N-MOSFET during the discharge of a capacitor can be derived using the basic principles of electronics. Here are the steps:
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The N-MOSFET is in saturation mode during the discharge of the capacitor. In this mode, the current through the MOSFET is given by the equation:
Id = 1/2 * µn * Cox * (W/L) * (Vgs - Vth)^2
where µn is the mobility of the electrons, Cox is the oxide capacitance per unit area, W/L is the aspect ratio of the MOSFET, Vgs is the gate-source voltage, and Vth is the threshold voltage.
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The voltage across the capacitor during discharge is given by the equation:
Vc = VDD - Id * R
where VDD is the initial voltage across the capacitor, Id is the current through the MOSFET, and R is the equivalent resistance of the MOSFET.
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We are interested in the point where the voltage across the capacitor has dropped to VDD/2. At this point, the current through the MOSFET is given by:
Id = VDD / (2 * R)
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Substituting this into the equation for the current through the MOSFET in saturation mode, we get:
VDD / (2 * R) = 1/2 * µn * Cox * (W/L) * (Vgs - Vth)^2
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Solving this equation for R, we get the expression for the equivalent resistance of the N-MOSFET:
R = µn * Cox * (W/L) * (Vgs - Vth)^2 / VDD
This is the expression for the equivalent resistance of an N-MOSFET when a capacitor initially charged to VDD is being discharged through it until VDD/2.
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