Derive why small signal model of Pmos and Nmos are same
“Diode-connected” device (Chapters 9 and 10)
The small-signal model of PMOS device is identical to that of NMOS transistor; therefore, Ry equals Ry and hence (1/8m) || ro
The Correct Answer and Explanation is :
To understand why the small-signal model of a PMOS transistor is identical to that of an NMOS transistor, we need to analyze the small-signal parameters and their relationships for both types of devices.
Small-Signal Model Overview:
In small-signal analysis, we model a transistor as a combination of its small-signal parameters, which include:
- Transconductance (gm): This defines the relationship between the small change in the output current (drain current, (i_d)) and the small change in the input voltage (gate-to-source voltage, (v_{gs})).
- Output resistance (ro): This represents the drain-to-source resistance in the linear region of operation.
NMOS Transistor Small-Signal Model:
For an NMOS transistor operating in the active region, the small-signal parameters are:
- Transconductance (g_m = \frac{dI_{D}}{dV_{GS}}), where (I_D) is the drain current and (V_{GS}) is the gate-source voltage.
- Output resistance (r_o = \frac{1}{\lambda I_D}), where (I_D) is the drain current and (\lambda) is the channel-length modulation parameter.
In the small-signal model, we replace the transistor with a current source (g_m v_{gs}) in parallel with the output resistance (r_o). The model looks like a voltage-controlled current source with a parallel resistor.
PMOS Transistor Small-Signal Model:
The small-signal model for a PMOS transistor is similar to that of NMOS. In the case of a PMOS, the current flows from the drain to the source. For small-signal analysis, the PMOS parameters are:
- Transconductance (g_m = \frac{dI_{D}}{dV_{SG}}), where (I_D) is the drain current and (V_{SG}) is the source-gate voltage.
- Output resistance (r_o = \frac{1}{\lambda I_D}), similar to the NMOS case.
In this model, the small-signal PMOS transistor is also represented by a voltage-controlled current source (g_m v_{sg}) in parallel with the output resistance (r_o).
Identical Small-Signal Models:
The reason the small-signal models of PMOS and NMOS are similar lies in their mathematical structure. The only difference between the two is the polarity of the voltages (gate-to-source for NMOS and source-to-gate for PMOS), but both devices operate on similar principles in terms of transconductance and output resistance.
Since the small-signal model is based on the relationship between (g_m) and (r_o), the models are structurally identical. The diode-connected device setup for both PMOS and NMOS transistors implies that the device operates in a region where the drain and gate are connected, leading to similar behavior in both devices.
Therefore, the relationship (R_y = R_y) holds for both devices, and the small-signal models are effectively identical, implying that their small-signal equivalent circuits have the same form: a current source in parallel with a resistance.
In conclusion, both the NMOS and PMOS small-signal models have the same configuration, with only the sign difference in the gate-source or source-gate voltage, which does not affect the overall form of the small-signal model. Hence, the small-signal models of PMOS and NMOS are essentially identical.