The pull up to pull down ratio for an n mos inverter driven by another n mos inverter can be evaluated from: (A) Vi nv = V t − \(\rm\frac{V_{t d}}{\sqrt{Zpu/Zpd}}\) (B) V inv = \(\rm\frac{−V_{t d}}{\sqrt{Zpu/Zpd}}\) (C) \(\rm\frac{W_{pd}}{L_{pd}}\) (V inv − V t) 2= \(\rm\frac{W_{p u}}{L_{p u}}\) (−V td )2 (D) \(\rm\frac{W_{p d}}{L_{p d}}\) (V inv − V t ) = \(\rm\frac{W_{p u}}{L_{pu}}\) (−V td )2 Choose the correct answer from the options given below:
(A), (C) only
Understanding NMOS Inverter Pull-Up to Pull-Down Ratio
The pull-up to pull-down ratio (often related to the ratio of the widths-to-lengths, W/L, of the transistors) is a critical parameter in determining the performance and characteristics of a logic gate, such as an NMOS inverter. This ratio influences the voltage transfer characteristic (VTC), switching threshold (VINV), noise margins, and propagation delay.
An NMOS inverter typically consists of an enhancement-mode NMOS transistor as the pull-down network and a pull-up network, which can be a resistor, another enhancement-mode NMOS (acting as a load), or more commonly, a depletion-mode NMOS transistor (acting as a load). The question refers to an NMOS inverter, implying one of these configurations. We will consider the common depletion-load NMOS inverter setup as the equations provided in the options strongly suggest this context, especially the mention of Vtd which usually relates to the depletion threshold voltage.
The question asks how to evaluate the pull up to pull down ratio for this inverter when it is driven by another NMOS inverter. This setup is standard in NMOS logic circuits. The ratio can be evaluated by analyzing the circuit at specific operating points on its Voltage Transfer Characteristic (VTC), such as the inverter threshold voltage VINV, where the input voltage Vin equals the output voltage Vout (Vin = Vout = VINV).
At the VINV point, the current through the pull-down transistor (Ipd) must equal the current through the pull-up transistor (Ipu), assuming no current is drawn by the next stage (which is the case when driven by another NMOS inverter due to the high input impedance of a MOSFET gate).
Let's analyze the provided options:
Analyzing Option (C)
Option (C) states: \(\rm\frac{W_{pd}}{L_{pd}}\) (V inv − V t) 2= \(\rm\frac{W_{p u}}{L_{p u}}\) (−V td )2
This equation resembles the current equation for an NMOS transistor operating in the saturation region:
\(I_{DS} = \frac{1}{2} k'_n \frac{W}{L} (V_{GS} - V_t)^2\)
where \(k'_n\) is the transconductance parameter, \(W/L\) is the width-to-length ratio, \(V_{GS}\) is the gate-to-source voltage, and \(V_t\) is the threshold voltage.
- For the pull-down transistor (enhancement NMOS):
- Let \((W/L)_{pd}\) be its dimensions ratio.
- Its threshold voltage is \(V_t\).
- At Vin = Vout = VINV, \(V_{GS,pd} = V_{in} = V_{INV}\).
- Assuming the pull-down is in saturation (VDS > VGS - Vt, i.e., VINV > VINV - Vt, which means Vt > 0, true for enhancement NMOS), its current is \(I_{pd} = \frac{1}{2} k'_n (\frac{W}{L})_{pd} (V_{INV} - V_t)^2\).
- For the pull-up transistor (depletion NMOS load):
- Let \((W/L)_{pu}\) be its dimensions ratio.
- Its threshold voltage is \(V_{t,dep} = V_{td}\) (usually a negative value for depletion NMOS).
- In the common depletion-load configuration, the gate is tied to the source (or drain), resulting in \(V_{GS,pu} = 0\) (if gate tied to source) or \(V_{GS,pu} = V_{DS,pu}\) (if gate tied to drain). The form of the equation in (C) suggests \((V_{GS,pu} - V_{t,pu})^2\) term is \((-V_{td})^2\). This happens if \(V_{GS,pu} = 0\) and \(V_{t,pu} = V_{td}\). So, let's assume the pull-up is a depletion NMOS with gate tied to source (VGS,pu = 0).
- Assuming the pull-up is in saturation (VDS > VGS - Vt,pu, i.e., \(V_{DD} - V_{INV} > 0 - V_{td} = -V_{td}\)), its current is \(I_{pu} = \frac{1}{2} k'_n (\frac{W}{L})_{pu} (0 - V_{td})^2 = \frac{1}{2} k'_n (\frac{W}{L})_{pu} (-V_{td})^2\).
Equating the currents \(I_{pd} = I_{pu}\) at VINV:
\(\frac{1}{2} k'_n (\frac{W}{L})_{pd} (V_{INV} - V_t)^2 = \frac{1}{2} k'_n (\frac{W}{L})_{pu} (-V_{td})^2\)
Canceling \(\frac{1}{2} k'_n\) from both sides, we get:
\((\frac{W}{L})_{pd} (V_{INV} - V_t)^2 = (\frac{W}{L})_{pu} (-V_{td})^2\)
This exactly matches option (C). This equation relates the \(W/L\) ratios of the pull-down and pull-up transistors to the voltages VINV, Vt (pull-down threshold), and Vtd (pull-up depletion threshold). Since the pull up to pull down ratio is directly related to the ratio of these \(W/L\) values, this equation can indeed be used to evaluate that ratio.
Analyzing Option (A)
Option (A) states: \(V_{inv} = V_t - \frac{V_{td}}{\sqrt{Z_{pu}/Z_{pd}}}\)
Let's rearrange the equation from option (C) to solve for VINV. From \((\frac{W}{L})_{pd} (V_{INV} - V_t)^2 = (\frac{W}{L})_{pu} (-V_{td})^2\), we can write:
\((V_{INV} - V_t)^2 = \frac{(W/L)_{pu}}{(W/L)_{pd}} (-V_{td})^2\)
\(|V_{INV} - V_t| = \sqrt{\frac{(W/L)_{pu}}{(W/L)_{pd}}} |-V_{td}|\)
Assuming VINV > Vt (which is typical for the inverter threshold of a depletion-load NMOS inverter) and |-Vtd| = -Vtd (since Vtd is a negative voltage for depletion mode), we get:
\(V_{INV} - V_t = \sqrt{\frac{(W/L)_{pu}}{(W/L)_{pd}}} (-V_{td})\)
\(V_{INV} = V_t - \sqrt{\frac{(W/L)_{pu}}{(W/L)_{pd}}} V_{td}\)
The term \(\frac{(W/L)_{pu}}{(W/L)_{pd}}\) represents the ratio of the pull-up transistor's \(W/L\) to the pull-down transistor's \(W/L\). The impedance \(Z\) of a MOSFET is inversely proportional to its \(k(W/L)\) value. Thus, the ratio of pull-up impedance to pull-down impedance, \(Z_{pu}/Z_{pd}\), is proportional to \(\frac{(L/W)_{pu}}{(L/W)_{pd}} = \frac{(W/L)_{pd}}{(W/L)_{pu}}\), which is the inverse of the term under the square root derived above.
Let's assume the definition of the pull up to pull down ratio \(Z_{pu}/Z_{pd}\) in option (A) is actually proportional to \(\frac{(L/W)_{pu}}{(L/W)_{pd}}\) or \(\frac{(W/L)_{pd}}{(W/L)_{pu}}\). If we take \(Z_{pu}/Z_{pd} = \frac{(W/L)_{pd}}{(W/L)_{pu}}\) (the ratio \(k_R\)), then \(\sqrt{\frac{(W/L)_{pu}}{(W/L)_{pd}}} = \frac{1}{\sqrt{(W/L)_{pd}/(W/L)_{pu}}} = \frac{1}{\sqrt{Z_{pu}/Z_{pd}}}\).
Substituting this into the derived equation:
\(V_{INV} = V_t - \frac{1}{\sqrt{Z_{pu}/Z_{pd}}} V_{td} = V_t - \frac{V_{td}}{\sqrt{Z_{pu}/Z_{pd}}}\)
This matches option (A). This equation gives the inverter threshold voltage VINV in terms of the threshold voltages Vt and Vtd and the pull up to pull down ratio \(Z_{pu}/Z_{pd}\) (defined here as \((W/L)_{pd}/(W/L)_{pu}\)). If VINV, Vt, and Vtd are known, this equation can also be used to evaluate the ratio \(Z_{pu}/Z_{pd}\).
Conclusion for Options (A) and (C)
Both option (A) and option (C) represent valid equations relating the operating point VINV and the transistor parameters (dimensions ratios and threshold voltages) of an NMOS depletion-load inverter. Both equations involve the pull up to pull down ratio and can be used to evaluate it.
Analyzing Options (B) and (D)
- Option (B) \(V_{inv} = - \frac{V_{td}}{\sqrt{Z_{pu}/Z_{pd}}}\) does not include the pull-down threshold voltage \(V_t\), which is essential for determining VINV.
- Option (D) \((\frac{W}{L})_{p d}\) (V inv − V t ) = \(\rm\frac{W_{p u}}{L_{p u}}\) (−V td )2 does not equate the currents correctly. The current is proportional to the square of \((V_{GS} - V_t)\) in saturation.
Therefore, only options (A) and (C) provide valid equations that can be used to evaluate the pull up to pull down ratio for an NMOS inverter based on its operating characteristics and transistor parameters.
The correct answer is the one that includes both (A) and (C).
| Option | Equation | Relevance to Pull-Up/Pull-Down Ratio Evaluation |
|---|---|---|
| (A) | \(V_{inv} = V_t - \frac{V_{td}}{\sqrt{Z_{pu}/Z_{pd}}}\) | Equation for VINV involving the ratio. Can be rearranged to find the ratio. |
| (B) | \(V_{inv} = - \frac{V_{td}}{\sqrt{Z_{pu}/Z_{pd}}}\) | Incorrect form for VINV. |
| (C) | \((\frac{W}{L})_{pd} (V_{inv} - V_t)^2 = (\frac{W}{L})_{pu} (-V_{td})^2\) | Current equality at VINV. Directly relates W/L ratios (the ratio) to voltages. |
| (D) | \((\frac{W}{L})_{p d}\) (V inv − V t ) = \(\rm\frac{W_{p u}}{L_{p u}}\) (−V td )2 | Incorrect form for current equation. |
Revision Table: Key Concepts
| Concept | Description | Significance for NMOS Inverter |
|---|---|---|
| Pull-Up Network (PU) | Transistor(s) connected between VDD and the output. Pulls output towards VDD. | Determines output high level, charging speed, and influences VTC. |
| Pull-Down Network (PD) | Transistor(s) connected between the output and GND. Pulls output towards GND. | Determines output low level, discharging speed, and influences VTC. |
| Pull-Up to Pull-Down Ratio | Ratio related to the relative current driving strengths or impedances of the PU and PD networks. Often expressed as a ratio of \(W/L\) values. | Crucially affects the VTC shape, switching threshold (VINV), and noise margins. |
| Inverter Threshold Voltage (VINV) | The input voltage at which Vout = Vin. | A key point on the VTC used for analyzing gate performance and noise margins. At this point, Ipd = Ipu. |
| NMOS Depletion Load | A depletion-mode NMOS transistor used as a pull-up. Can act like a current source or resistor. | Simpler process than CMOS, but lower performance and higher static power consumption compared to CMOS. |
Additional Information: NMOS Inverter Analysis
Analyzing the Voltage Transfer Characteristic (VTC) of an NMOS inverter helps understand its behavior. The VTC is a plot of output voltage (Vout) versus input voltage (Vin).
- When Vin is low (Vin < Vt), the pull-down NMOS is off (or in cut-off). The pull-up charges the output towards VDD. Vout is high (VOH).
- As Vin increases beyond Vt, the pull-down NMOS turns on and starts conducting. It pulls the output voltage down.
- The VTC has critical points: VOL (output low voltage), VOH (output high voltage), VIL (input low voltage - max Vin considered a logical 0), VIH (input high voltage - min Vin considered a logical 1), and VINV (inverter threshold).
- For a depletion-load NMOS inverter, VOH is typically VDD and VOL is close to 0V.
- The pull up to pull down ratio significantly affects VINV. A larger \((W/L)_{pd} / (W/L)_{pu}\) ratio (stronger pull-down relative to pull-up) shifts VINV towards VDD, while a smaller ratio shifts it towards GND. Ideally, VINV should be near VDD/2 for symmetrical noise margins.
- The equations in options (A) and (C) are derived based on the assumption that both transistors are in the saturation region at the VINV point, which is a common analysis simplification for depletion-load inverters with sufficient supply voltage. In reality, one transistor might be in saturation and the other in the linear region depending on the voltages. However, the current equality always holds at VINV.
Important Questions from Types of FET
A JFET is also called ________ device.
In JFET, the Pinch‐off Voltage can be defined as:
In JFET, the current density in the x-direction is:
A. σ(x)E x
B. qN DμE x
C. \(\rm \frac{q}{2 \in_s}N_D\mu\)
D. \(\rm \frac{N_D\mu}{2 \in_s}\)
Choose the correct answer from the options given below:
The layers of Nos involves a subset of layers. For NMOS we require:
(A) Contact, (Black or Brown)
(B) Polysilicon (Green)
(C) Metal (Blue)
(D) Implant (Yellow)
Choose the correct answer from the options given below:
The layout of NMOS involves a subset of the layers and features set out. For NMOS we require
(A) N-diffusion (Green)
(B) Implant (Yellow)
(C) Polysilicon (Red)
(D) N-diffusion (Blue)
Choose the correct answer from the options given below:
