In a MOSFET, the transconductance in linear region ean be expressed as:
Calculating MOSFET Transconductance in the Linear Region
The question asks for the expression for the transconductance of a MOSFET when it is operating in the linear region, also known as the triode region.
Transconductance, denoted as \(g_m\), is a fundamental parameter of a transistor. It measures how effectively the input voltage (gate-source voltage, \(V_{GS}\) or \(V_G\)) controls the output current (drain current, \(I_D\)). Mathematically, transconductance is defined as the partial derivative of the drain current (\(I_D\)) with respect to the gate-source voltage (\(V_{GS}\)), keeping the drain-source voltage (\(V_{DS}\) or \(V_D\)) constant:
\[g_m = \frac{\partial I_D}{\partial V_{GS}} \Big|_{V_{DS}=\text{constant}}\]
In the linear region of a MOSFET, the drain current \(I_D\) is given by the following equation:
\[I_D = \mu_0 C_{ox} \frac{W}{L} \left( (V_{GS} - V_T)V_{DS} - \frac{1}{2}V_{DS}^2 \right)\]
Here:
- \(\mu_0\) is the carrier mobility (electron or hole mobility).
- \(C_{ox}\) is the gate oxide capacitance per unit area.
- \(W\) is the channel width.
- \(L\) is the channel length.
- \(V_{GS}\) is the gate-source voltage (\(V_G\) if the source is grounded).
- \(V_T\) is the threshold voltage.
- \(V_{DS}\) is the drain-source voltage (\(V_D\) if the source is grounded).
To find the transconductance \(g_m\) in the linear region, we need to differentiate this expression for \(I_D\) with respect to \(V_{GS}\), treating \(V_{DS}\) as a constant. Let's assume \(V_{GS} = V_G\) and \(V_{DS} = V_D\) for simplicity, as often done when the source is the common terminal.
\[g_m = \frac{\partial I_D}{\partial V_G} \Big|_{V_D=\text{constant}}\]
Differentiating the linear region \(I_D\) equation:
\[I_D = \mu_0 C_{ox} \frac{W}{L} \left( (V_G - V_T)V_D - \frac{1}{2}V_D^2 \right)\]
\[g_m = \frac{\partial}{\partial V_G} \left[ \mu_0 C_{ox} \frac{W}{L} \left( (V_G - V_T)V_D - \frac{1}{2}V_D^2 \right) \right]\]
Since \(\mu_0\), \(C_{ox}\), \(W\), \(L\), \(V_T\), and \(V_D\) are treated as constants during this differentiation:
\[g_m = \mu_0 C_{ox} \frac{W}{L} \frac{\partial}{\partial V_G} \left( (V_G - V_T)V_D - \frac{1}{2}V_D^2 \right)\]
Now, let's differentiate the terms inside the parenthesis with respect to \(V_G\):
- The term \((V_G - V_T)V_D\): The derivative with respect to \(V_G\) is \(1 \times V_D = V_D\). (\(V_T\) and \(V_D\) are constants).
- The term \(-\frac{1}{2}V_D^2\): This term is a constant with respect to \(V_G\), so its derivative is \(0\).
Substituting these results back into the expression for \(g_m\):
\[g_m = \mu_0 C_{ox} \frac{W}{L} (V_D - 0)\]
\[g_m = \mu_0 C_{ox} \frac{W}{L} V_D\]
Comparing this derived expression with the given options (assuming \(C_{0}x\) in the options represents \(C_{ox}\)):
- Option 1: \(\rm\frac{\mu_0 C_{0}xW}{L}\) (V G− V T)
- Option 2: \(\rm\frac{\mu_0 C_{0}xW}{2L}\) (V G− V T)V D
- Option 3: \(\rm\frac{\mu_0 C_{0}xW}{L}\) VD
- Option 4: \(\rm\frac{\mu_0 C_{0}xL}{W}\) (V G− V T)
Our derived expression \(g_m = \mu_0 C_{ox} \frac{W}{L} V_D\) matches Option 3.
Therefore, the transconductance in the linear region of a MOSFET is directly proportional to the mobility, the oxide capacitance per unit area, the ratio of channel width to length (\(W/L\)), and the drain-source voltage \(V_D\).
Revision Table: MOSFET Operating Regions and Parameters
| Parameter | Linear (Triode) Region | Saturation Region |
|---|---|---|
| Drain Current (\(I_D\)) | \(\mu_0 C_{ox} \frac{W}{L} \left( (V_{GS} - V_T)V_{DS} - \frac{1}{2}V_{DS}^2 \right)\) | \(\frac{1}{2} \mu_0 C_{ox} \frac{W}{L} (V_{GS} - V_T)^2\) |
| Transconductance (\(g_m = \frac{\partial I_D}{\partial V_{GS}}\)) | \(\mu_0 C_{ox} \frac{W}{L} V_{DS}\) | \(\mu_0 C_{ox} \frac{W}{L} (V_{GS} - V_T)\) or \(\sqrt{2 \mu_0 C_{ox} \frac{W}{L} I_D}\) |
| Output Conductance (\(g_d = \frac{\partial I_D}{\partial V_{DS}}\)) | \(\mu_0 C_{ox} \frac{W}{L} (V_{GS} - V_T - V_{DS})\) | Typically considered 0 (ideal), but non-zero due to channel length modulation. |
| Condition for Region | \(V_{GS} > V_T\) and \(V_{DS} < (V_{GS} - V_T)\) | \(V_{GS} > V_T\) and \(V_{DS} \ge (V_{GS} - V_T)\) |
Additional Information on MOSFET Transconductance and Operation
Understanding the MOSFET transconductance \(g_m\) is crucial for analyzing and designing amplifier circuits. It represents the gain of the transistor in terms of converting input voltage variations into output current variations.
- Linear Region \(g_m\): As calculated, \(g_m\) in the linear region is directly proportional to \(V_D\). This means \(g_m\) is not constant but varies with the drain voltage. This region is typically used when the MOSFET acts as a voltage-controlled resistor (for small \(V_D\)) or as a part of analog switches, not usually for linear amplification where a constant \(g_m\) is desired.
- Saturation Region \(g_m\): In contrast, the transconductance in the saturation region is \(g_m = \mu_0 C_{ox} \frac{W}{L} (V_{GS} - V_T)\). In this region, \(g_m\) is proportional to the effective gate voltage (\(V_{GS} - V_T\)), often called the overdrive voltage. This \(g_m\) is independent of \(V_D\) (ideally) and is the region where MOSFETs are commonly used as amplifiers.
- Parameters Affecting \(g_m\): Both mobility (\(\mu_0\)), oxide capacitance per unit area (\(C_{ox}\)), and the device geometry ratio (\(W/L\)) directly influence the transconductance in both regions. Increasing \(W/L\) ratio or using materials with higher mobility can increase the transconductance, leading to higher gain. \(C_{ox}\) is determined by the gate oxide thickness and permittivity.
- Cut-off Region: Below the threshold voltage (\(V_{GS} < V_T\)), the MOSFET is in the cut-off region. The channel is not formed, \(I_D\) is ideally zero, and thus \(g_m\) is also zero.
Important Questions from MOSFET - Teaching
Which of the following is true about a P-MOSFET?
I. Drain current consists of holes flowing from source to drain
II. To turn ON an enhanced P-MOSFET, gate voltage should be made more positive than source voltage at least by the magnitude of threshold voltage
III. The transconductance is directly proportional to oxide thickness
Which of the following statements about the MOS transistor is INCORRECT ?
MOSFET is used for _______ switching frequency application.
- The drain conductance of n-channel MOSFET in linear region 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:
