A standard CMOS inverter is designed with equal rise and fall times (βₙ = βₚ). If the width of the P MOS transistor in the inverter is increased, what would be the effect on the LOW noise margin (NM${}_L$) and the HIGH noise margin (NM${}_H$) ?

A CMOS inverter is a fundamental building block in digital circuits, consisting of a p-type Metal-Oxide-Semiconductor (PMOS) transistor and an n-type Metal-Oxide-Semiconductor (NMOS) transistor connected in series. Its primary function is to invert the input signal, meaning a high input results in a low output and vice versa.

CMOS Inverter Characteristics Explained

To understand the effect of changes in transistor sizing, it's crucial to grasp the key parameters and characteristics of a CMOS inverter:

- **Transistor Sizing and $\beta$ Parameter:** The current-driving capability of a MOSFET is proportional to its transconductance parameter, commonly denoted as $\beta$. For an NMOS transistor, ${\beta }_n={\mu }_n{C}_{ox}(W_n/L_n)$, and for a PMOS transistor, ${\beta }_p={\mu }_p{C}_{ox}(W_p/L_p)$, where $\mu$ is the carrier mobility, $C_{ox}$ is the gate oxide capacitance per unit area, $W$ is the transistor width, and $L$ is the transistor length. Increasing the width ($W$) of a transistor increases its $\beta$ value, making it "stronger" and able to conduct more current for a given voltage.
- Voltage Transfer Characteristic (VTC): This graph plots the output voltage ($V_{OUT}$) against the input voltage ($V_{IN}$). It shows how the inverter switches from a high output to a low output. Key points on the VTC include $V_{OH}$ (Output High Voltage), $V_{OL}$ (Output Low Voltage), $V_{IL}$ (Input Low Voltage, maximum input considered logic LOW), and $V_{IH}$ (Input High Voltage, minimum input considered logic HIGH).
- Noise Margins: These parameters quantify the noise immunity of the inverter. They indicate how much noise can be present on the input without causing an incorrect output state. - Low Noise Margin (NML): Defined as $N{M}_L=V_{IL}-V_{OL}$. A larger $N{M}_L$ means better noise immunity for a logic LOW input. - High Noise Margin (NMH): Defined as $N{M}_H=V_{OH}-V_{IH}$. A larger $N{M}_H$ means better noise immunity for a logic HIGH input.

Inverter Design: Equal Rise and Fall Times

The question states that the CMOS inverter is initially designed with equal rise and fall times, where ${\beta }_n={\beta }_p$. In a typical CMOS inverter, to achieve equal rise and fall times, the PMOS transistor is usually made wider than the NMOS transistor because hole mobility (${\mu }_p$) is lower than electron mobility (${\mu }_n$). The condition ${\beta }_n={\beta }_p$ implies a balanced initial state in terms of current driving capabilities under the given problem context.

PMOS Transistor Width: Impact on VTC

When the width of the PMOS transistor ($W_p$) is increased, its transconductance parameter ${\beta }_p$ increases. This signifies that the PMOS transistor becomes "stronger" in terms of its current-driving capability compared to the NMOS transistor.

Let's analyze how this change affects the Voltage Transfer Characteristic (VTC) and subsequently the noise margins:

- **Shift in Switching Threshold ($V_M$):** The switching threshold voltage ($V_M$) is the input voltage at which the output voltage equals the input voltage ($V_{OUT}=V_{IN}$). At this point, the drain currents of the NMOS and PMOS transistors are equal ($I_{Dn}=|I_{Dp}|$). If the PMOS transistor becomes stronger (due to increased $W_p$), it will pull the output voltage towards $V_{DD}$ more effectively. To balance the currents and reach the switching point, the input voltage ($V_{IN}$) needs to be higher to make the NMOS transistor stronger and pull the output down against the stronger PMOS. Therefore, the switching threshold voltage $V_M$ will shift towards $V_{DD}$ (i.e., it will increase).
- **Effect on $V_{IL}$ and $V_{IH}$:** As the entire VTC shifts to the right (towards higher input voltages) due to the stronger PMOS, both $V_{IL}$ and $V_{IH}$ will also shift to higher values. - $V_{IL}$ (Input Low Voltage) increases. - $V_{IH}$ (Input High Voltage) increases. (Note: $V_{OH}$ remains approximately $V_{DD}$ and $V_{OL}$ remains approximately $0\text{ V}$ in an ideal CMOS inverter).

Noise Margins: NML and NMH Changes

Based on the changes in $V_{IL}$ and $V_{IH}$:

- Low Noise Margin (NML): $$N{M}_L=V_{IL}-V_{OL}$$ Since $V_{IL}$ increases and $V_{OL}$ remains approximately $0\text{ V}$, the **low noise margin ($N{M}_L$) will increase. This indicates that the inverter can tolerate more noise when its input is a logic LOW.
- High Noise Margin (NMH):** $$N{M}_H=V_{OH}-V_{IH}$$ Since $V_{OH}$ remains approximately $V_{DD}$ and $V_{IH}$ increases, the difference $V_{OH}-V_{IH}$ will decrease. Therefore, the **high noise margin ($N{M}_H$) will decrease. This implies the inverter will be less tolerant to noise when its input is a logic HIGH.

Parameter	Effect
PMOS strength (${\beta }_p$)	Increases
Switching Threshold ($V_M$)	Increases (shifts right)
$V_{IL}$	Increases
$V_{IH}$	Increases
$V_{OH}$	No significant change ($\approx V_{DD}$)
$V_{OL}$	No significant change ($\approx 0\text{ V}$)
**Low Noise Margin ($N{M}_L$)**	Increases ($V_{IL}\uparrow -V_{OL}$)
**High Noise Margin ($N{M}_H$)**	Decreases ($V_{OH}-V_{IH}\uparrow$)

Thus, increasing the width of the PMOS transistor in a CMOS inverter leads to an increase in the low noise margin (NML) and a decrease in the high noise margin (NMH)**.