The drain conductance of n-channel MOSFET in linear region is:
Drain Conductance in N-Channel MOSFET Linear Region
The drain conductance of an n-channel MOSFET in the linear region is a crucial parameter that describes how the drain current changes with respect to the drain-source voltage.
Understanding the Linear Region of a MOSFET
An n-channel MOSFET operates in the linear region (also known as the ohmic or triode region) when the drain-source voltage (\(V_{DS}\)) is less than the overdrive voltage (\(V_{GS} - V_{th}\)). In this region, the channel acts like a voltage-controlled resistor, and the drain current (\(I_D\)) is approximately proportional to \(V_{DS}\).
The drain current (\(I_D\)) for an n-channel MOSFET in the linear region is given by the formula:
\[I_D = \mu_n C_{ox} \frac{W}{L} \left( (V_{GS} - V_{th}) V_{DS} - \frac{1}{2} V_{DS}^2 \right)\]
Where:
- \(\mu_n\) is the electron mobility in the channel.
- \(C_{ox}\) is the gate oxide capacitance per unit area.
- \(W\) is the width of the channel.
- \(L\) is the length of the channel.
- \(V_{GS}\) is the gate-source voltage.
- \(V_{th}\) is the threshold voltage.
- \(V_{DS}\) is the drain-source voltage.
Derivation of Drain Conductance
The drain conductance (\(g_d\) or \(g_{ds}\)) is defined as the change in drain current with respect to the change in drain-source voltage, while keeping the gate-source voltage constant. Mathematically, it is the partial derivative of \(I_D\) with respect to \(V_{DS}\):
\[g_d = \frac{\partial I_D}{\partial V_{DS}}\]
Let's differentiate the drain current equation for the linear region:
\[g_d = \frac{\partial}{\partial V_{DS}} \left[ \mu_n C_{ox} \frac{W}{L} \left( (V_{GS} - V_{th}) V_{DS} - \frac{1}{2} V_{DS}^2 \right) \right]\]
Since \(\mu_n C_{ox} \frac{W}{L}\) is a constant (often denoted as \(K_n\) or \(k_n\)), we can take it out of the differentiation:
\[g_d = \mu_n C_{ox} \frac{W}{L} \frac{\partial}{\partial V_{DS}} \left[ (V_{GS} - V_{th}) V_{DS} - \frac{1}{2} V_{DS}^2 \right]\]
Now, differentiate the terms inside the bracket with respect to \(V_{DS}\):
- The derivative of \((V_{GS} - V_{th}) V_{DS}\) with respect to \(V_{DS}\) is \((V_{GS} - V_{th})\).
- The derivative of \(-\frac{1}{2} V_{DS}^2\) with respect to \(V_{DS}\) is \(-V_{DS}\).
Therefore, the general expression for the drain conductance in the linear region is:
\[g_d = \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th} - V_{DS})\]
Analyzing the Given Options
Let's compare this derived general formula with the provided options:
| Option | Expression |
|---|---|
| 1 | \(\rm \frac{\mu_n C_{ox}W}{L}(V_{gs}-V_{th})\) |
| 2 | \(\rm \frac{\mu_n C_{ox}W}{L}V_{ds}\) |
| 3 | \(\rm \frac{\mu_n C_{ox}W}{L}V_{gs}\) |
| 4 | \(\rm \frac{\mu_n C_{ox}W}{L}(V_{gs}-V_{th})^2\) |
The general drain conductance formula is \(g_d = \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th} - V_{DS})\). We need to find how this relates to Option 2, which is \(\rm \frac{\mu_n C_{ox}W}{L}V_{ds}\).
For the expression \(\mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th} - V_{DS})\) to be equal to \(\mu_n C_{ox} \frac{W}{L} V_{DS}\), the following equality must hold:
\[(V_{GS} - V_{th} - V_{DS}) = V_{DS}\]
Solving for the condition:
\[V_{GS} - V_{th} = 2V_{DS}\]
This shows that Option 2 represents the drain conductance of the n-channel MOSFET in the linear region under a very specific operating condition, where the overdrive voltage (\(V_{GS} - V_{th}\)) is exactly twice the drain-source voltage (\(V_{DS}\)). This specific point falls within the linear region, as \(V_{DS} < V_{GS} - V_{th}\) implies \(V_{DS} < 2V_{DS}\), which is true for any positive \(V_{DS}\).
Thus, while the general formula for drain conductance in the linear region is \(g_d = \mu_n C_{ox} \frac{W}{L} (V_{GS} - V_{th} - V_{DS})\), Option 2 provides the drain conductance for a particular operating point within that region.
Important Questions from MOSFET - Teaching
In enhancement type of MOSFET, at zero gate bias:
A. Channel conductance is very low
B. Channel conductance is very high
C. Channel resistance is very low
D. Channel resistance is very high
Choose the correct answer from the options given below:
The maximum operating frequency of a MOSFET is
A. \(\rm \frac{\omega_m}{2\pi}\)
B. \(\rm \frac{\mu_nV_D}{2\pi L^2}\)
C. \(\rm \frac{\mu_nL^2}{2\pi}\)
D. \(\rm \frac{\mu_nV_D}{2\pi L}\)
Choose the correct answer from the options given below:
In a MOSFET, the transconductance in linear region ean be expressed as:
For a thermally grown silicon dioxide layer, in a MOSFET, the leakage current between the gate and channel is:
If N is the doping level of substrate and V is the effective voltage across the junction in a MOSFET, then the depletion region width for the junction is given by:
