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For the CMOS inverter circuit shown below, assume the following: Kn′ = Kp′, Ln = Lp = 1 μm, and |VTP| = VTN = VT. Let's define the threshold as the point (vI = VTH) in the vO versus vI transfer curve where vO = VDD/2. a) Sketch the inverter transfer curve and mark the VTH point. b) Find an expression for the inverter threshold voltage VTH in terms of VDD, VT, Wn and Wp. c) If Wn = Wp, what is VTH ? In what mode of operation are M1 and M2 when VI = VTH ?

For the CMOS inverter circuit shown below, assume the following: Kn′ = Kp′, Ln = Lp = 1 μm, and |VTP| = VTN = VT. Let's define the threshold as the point (vI = VTH) in the vO versus vI transfer curve where vO = VDD/2. a) Sketch the inverter transfer curve and mark the VTH point. b) Find an expression for the inverter threshold voltage VTH in terms of VDD, VT, Wn and Wp. c) If Wn = Wp, what is VTH ? In what mode of operation are M1 and M2 when VI = VTH ?

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  1. For the CMOS inverter circuit shown below, assume the following: K n ′ = K p ′ , L n = L p = 1 μ m , and / V T P / = V T N = V T . Let's define the threshold as the point ( v I = V T H ) in the v O versus v I transfer curve where v O = V D D / 2 . a) Sketch the inverter transfer curve and mark the V T H point. b) Find an expression for the inverter threshold voltage V T H in terms of V D D , V T , W n and W p . c) If W n = W p , what is V T H ? In what mode of operation are M1 and M2 when V I = V T H ?

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NOTE: For all the problems here, and from now on, we'll use the common terminology for K. Namely, we'll use Kn = Kn ' (W/L) for n−MOSFETs and Kp = Kp′(W/L) for p-MOSFETs. Kn′ = μnCox and Kp′ = μpCox. For the CMOS inverter circuit shown below, assume the following: Kn′ = Kp′, Ln = Lp = 1 μm, and /VTP/ = VTN = VT. Let's define the threshold as the point (vI = VTH) in the vO versus vI transfer curve where vO = VDD/2. a) Sketch the inverter transfer curve and mark the VTH point. b) Find an expression for the inverter threshold voltage VTH in terms of VDD, VT, Wn and Wp. c) If Wn = Wp, what is VTH ? In what mode of operation are M1 and M2 when VI = VTH ? What about when VI < VTH ?
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Consider the CMOS inverter circuit with a capacitor connected at its output as shown below. Here Kn′ = 2 mA/V2 and VTn = 1.5 V, Kp′ = 1 mA/V2 and VTp = −0.5 V, all MOSFETs have L = 1 μm, and remember that K = K′(W/L). For a MOSFET in triode, use the S-R model with RON = 1 /K(|VGS| − |VT| ) (works for both nFET and pFET). a) Design Wp and Wn such that the overall circuit threshold, VTH, occurs at VDD/2 as shown in the graph. Here you can assume that at the threshold point VIN = VOUT = VTH. Use the smallest integer values you can for Wp and Wn, assuming the minimum width allowed is 1 μm. b) When VIN = VDD, draw the equivalent circuit. What is the time constant? c) When VIN = 0, draw the equivalent circuit. What is the time constant? d) Based on your answers from (b) and (c), which transition do you expect to be fastest: when vOUT transitions from VDD to 0 V, or when vOUT transitions from 0 V to VDD ? Why?
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Transient Point (Threshold Voltage) In practice, in an inverter circuit, there is an unstable transient point at which both transistors are on and in saturation mode. The input voltage (connected to the gate) for this point is called the threshold voltage (Vin = VTH). At this point, VGS/VSG of the NMOS / PMOS is equal to Vin/VDD-Vin. Note that this threshold voltage is different from the intrinsic threshold voltage of transistors (VTN or VTP). Derive an expression for this threshold voltage [RP2]. Verify your work with the expression below: What happens to VTH when Kp > Kn, Kp = Kn, and Kp < Kn? What haapens when Kp approaches 0 or when Kn approaches 0? Use words to generalize how various Kn and Kp values change VTH and describe how the values of Kn and Kp can be tuned by parallel MOSFET connection [RP2]. Noise Margin Please read section 6.9.4 of the textbook or search on the internet about noise margin. Sketch a typical transfer function of an inverter (non-ideal one) with 1 PMOS and 1 NMOS and label VIL, VOH, VOL,VIH, NMH, and NML on your sketch [RP3]. Why is the metric of noise margin useful [RP3]? Make a reasoned guess about how the change in the threshold voltage VTH (or the change in Kn/Kp ) affects each of the noise margin parameters [RP4].

Transient Point (Threshold Voltage) In practice, in an inverter circuit, there is an unstable transient point at which both transistors are on and in saturation mode. The input voltage (connected to the gate) for this point is called the threshold voltage (Vin = VTH). At this point, VGS/VSG of the NMOS / PMOS is equal to Vin/VDD-Vin. Note that this threshold voltage is different from the intrinsic threshold voltage of transistors (VTN or VTP). Derive an expression for this threshold voltage [RP2]. Verify your work with the expression below: What happens to VTH when Kp > Kn, Kp = Kn, and Kp < Kn? What haapens when Kp approaches 0 or when Kn approaches 0? Use words to generalize how various Kn and Kp values change VTH and describe how the values of Kn and Kp can be tuned by parallel MOSFET connection [RP2]. Noise Margin Please read section 6.9.4 of the textbook or search on the internet about noise margin. Sketch a typical transfer function of an inverter (non-ideal one) with 1 PMOS and 1 NMOS and label VIL, VOH, VOL,VIH, NMH, and NML on your sketch [RP3]. Why is the metric of noise margin useful [RP3]? Make a reasoned guess about how the change in the threshold voltage VTH (or the change in Kn/Kp ) affects each of the noise margin parameters [RP4].
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