半導体工学その4


## ■ Why Verification Under Worst-Case Conditions Is Essential

In analog IC design, **process parameters and device characteristics inherently vary** due to manufacturing tolerances. These variations must be taken into account during the design phase to ensure **robust performance across all possible use cases**.

The power supply voltage and ambient temperature conditions at the time of IC operation are also **not fixed**. Therefore, designers must evaluate their circuits under **worst-case conditions**, where the combined influence of **process, voltage, and temperature (PVT) variations** is most detrimental.

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## ◉ Typical Causes of Variation

* **Process variations**:

  * Gate oxide thickness (Tox)
  * Gate length (L)
  * Substrate doping concentration (Nsub)

→ These affect key transistor characteristics such as:

* Threshold voltage (Vth)

* Transconductance (gm)

* Parasitic capacitance (C)

* **Environmental variations**:

  * Power supply fluctuations (VDD ± tolerance)
  * Temperature changes (e.g., -40°C to +85°C)

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## ◉ What Is Worst-Case Verification?

### ● Combinatorial Explosion of Conditions

If there are 6 parameters, each with 3 conditions (e.g., Low, Typical, High), then:

**3⁶ = 729 combinations**
Running simulations for all of these and manually checking results is **impractical and time-consuming**.

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## ◉ Efficient Worst-Case Strategy

Instead of brute-force verification, designers use **analytical reasoning** to narrow down the critical combinations:

* For example:

  * The **maximum output voltage** may occur when resistance is low and Vth is high.
  * The **minimum current** may occur when resistance is high and Vth is low.

By **identifying the extreme (limiting) conditions** based on circuit topology, one can **selectively simulate** the worst cases rather than exhaustively covering all.

However, if **worst-case combinations cannot be predicted confidently**, full-coverage simulation becomes necessary.

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## ◉ Use of Design Performance Tables

A **Design Performance Table** (often called a "design verification matrix") is created to compare expected vs. actual results across key test conditions.

### Example Parameters & Ranges:

| Parameter           | Unit | Low   | Typ     | High  |
| ------------------- | ---- | ----- | ------- | ----- |
| NMOS Vth (VTN)      | V    | -0.15 | 0       | +0.15 |
| PMOS Vth (VTP)      | V    | -0.15 | 0       | +0.15 |
| Resistance (R)      | Ω    | -20%  | Nominal | +20%  |
| Capacitance (C)     | F    | -10%  | Nominal | +10%  |
| Supply Voltage (V+) | V    | Vmin  | Typ     | Vmax  |
| Temperature (Ta)    | °C   | -40   | 27      | +85   |

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## ◉ Final Verification Step

All simulation results and test data (e.g., from **TEG chips**) are **summarized in a comparison table** that checks whether the design still meets its **development specifications under all relevant worst-case conditions**.

This includes:

* Absolute and relative device variation
* Environmental variation (voltage and temperature)
* Power supply limits
* Combined stress conditions

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## ◉ Key Takeaways

* ICs must be **designed and verified** to operate correctly not only in typical conditions but also in **worst-case combinations** of process, voltage, and temperature.
* Worst-case simulation ensures **reliability, yield, and robustness**.
* Overlooking worst-case analysis can lead to **field failures**, costly recalls, or inconsistent performance.
* Intelligent selection of worst-case scenarios based on **circuit behavior** allows for efficient and effective verification.

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