To derive the expressions for the input low voltage, input high voltage, and inverter midpoint voltage, we need to consider the characteristics of an inverter circuit.

1. Input Low Voltage:
The input low voltage (VIL) is the maximum voltage level that is considered as a logic low input for the inverter. To derive the expression for VIL, we need to analyze the behavior of the inverter when the input is at a logic low level.

- In an inverter circuit, when the input is at a logic low level, the output is at a logic high level.
- The output of the inverter is determined by the pull-up resistor and the NMOS transistor.
- The NMOS transistor is in the cutoff region when the input is at a logic low level.
- In the cutoff region, the NMOS transistor acts as an open switch, and no current flows through it.
- Therefore, the output voltage is determined by the pull-up resistor, which is connected to the supply voltage (VDD).
- The output voltage (Vout) is equal to VDD.

Hence, the expression for the input low voltage (VIL) is VIL = VDD.

2. Input High Voltage:
The input high voltage (VIH) is the minimum voltage level that is considered as a logic high input for the inverter. To derive the expression for VIH, we need to analyze the behavior of the inverter when the input is at a logic high level.

- In an inverter circuit, when the input is at a logic high level, the output is at a logic low level.
- The output of the inverter is determined by the pull-down resistor and the PMOS transistor.
- The PMOS transistor is in the cutoff region when the input is at a logic high level.
- In the cutoff region, the PMOS transistor acts as an open switch, and no current flows through it.
- Therefore, the output voltage is determined by the pull-down resistor, which is connected to the ground (GND).
- The output voltage (Vout) is equal to GND.

Hence, the expression for the input high voltage (VIH) is VIH = GND.

3. Inverter Midpoint Voltage:
The inverter midpoint voltage (VM) is the voltage level at which the output of the inverter is at the midpoint between the logic high and logic low levels. To derive the expression for VM, we need to consider the threshold voltage of the NMOS and PMOS transistors.

- The threshold voltage (Vth) is the voltage level at which the NMOS or PMOS transistor starts to conduct.
- The threshold voltage is typically specified in the datasheet of the transistors used in the inverter circuit.
- The midpoint voltage (VM) is equal to the average of the input high voltage (VIH) and the input low voltage (VIL).
- Therefore, the expression for the inverter midpoint voltage (VM) is VM = (VIH + VIL) / 2.

By following these steps, we have derived the expressions for the input low voltage (VIL), input high voltage (VIH), and inverter midpoint voltage (VM).

Question

To derive the expressions for the input low voltage, input high voltage, and inverter midpoint voltage, we need to consider the characteristics of an inverter circuit.

1. Input Low Voltage:
The input low voltage (VIL) is the maximum voltage level that is considered as a logic low input for the inverter. To derive the expression for VIL, we need to analyze the behavior of the inverter when the input is at a logic low level.

- In an inverter circuit, when the input is at a logic low level, the output is at a logic high level.
- The output of the inverter is determined by the pull-up resistor and the NMOS transistor.
- The NMOS transistor is in the cutoff region when the input is at a logic low level.
- In the cutoff region, the NMOS transistor acts as an open switch, and no current flows through it.
- Therefore, the output voltage is determined by the pull-up resistor, which is connected to the supply voltage (VDD).
- The output voltage (Vout) is equal to VDD.

Hence, the expression for the input low voltage (VIL) is VIL = VDD.

2. Input High Voltage:
The input high voltage (VIH) is the minimum voltage level that is considered as a logic high input for the inverter. To derive the expression for VIH, we need to analyze the behavior of the inverter when the input is at a logic high level.

- In an inverter circuit, when the input is at a logic high level, the output is at a logic low level.
- The output of the inverter is determined by the pull-down resistor and the PMOS transistor.
- The PMOS transistor is in the cutoff region when the input is at a logic high level.
- In the cutoff region, the PMOS transistor acts as an open switch, and no current flows through it.
- Therefore, the output voltage is determined by the pull-down resistor, which is connected to the ground (GND).
- The output voltage (Vout) is equal to GND.

Hence, the expression for the input high voltage (VIH) is VIH = GND.

3. Inverter Midpoint Voltage:
The inverter midpoint voltage (VM) is the voltage level at which the output of the inverter is at the midpoint between the logic high and logic low levels. To derive the expression for VM, we need to consider the threshold voltage of the NMOS and PMOS transistors.

- The threshold voltage (Vth) is the voltage level at which the NMOS or PMOS transistor starts to conduct.
- The threshold voltage is typically specified in the datasheet of the transistors used in the inverter circuit.
- The midpoint voltage (VM) is equal to the average of the input high voltage (VIH) and the input low voltage (VIL).
- Therefore, the expression for the inverter midpoint voltage (VM) is VM = (VIH + VIL) / 2.

By following these steps, we have derived the expressions for the input low voltage (VIL), input high voltage (VIH), and inverter midpoint voltage (VM).

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