Solution: ADC Resolution and Calibration Error

Given:

• ADC resolution = 14 bits
• ADC counts = $0$ to $16383$
• Input range = 0–2.500 V
• Temperature range = 0–200°C
• Sensor output = 0–2.5 V linearly
• Measured code at 100°C = 6550

(i) Ideal LSB and Ideal Code at 100°C

Step 1: Calculate ideal LSB

A 14-bit ADC has:

$$2^{14}=16384$$

quantization intervals.

Therefore:

$$LSB=\frac{2.500}{16384}$$

$$LSB=0.0001525879\text{ V}$$

Hence:

$$\boxed{LSB\approx152.59\,\mu V}$$

Step 2: Ideal voltage at 100°C

The sensor is linear from 0 V at 0°C to 2.5 V at 200°C.

At 100°C, which is half of the temperature range:

$$V_{100}=2.5\times\frac{100}{200}=1.25\text{ V}$$

Step 3: Ideal ADC code

Using the ideal ADC transfer convention:

$$Code_{ideal}=\frac{V_{100}}{V_{FS}}(2^{14}-1)$$

$$=\frac{1.25}{2.5}\times16383$$

$$=8191.5$$

So the ideal code is approximately:

$$\boxed{8192}$$

(ii) Offset Error

Measured code:

$$Code_{measured}=6550$$

Ideal code:

$$Code_{ideal}\approx8192$$

Code error:

$$\Delta Code=6550-8192=-1642$$

Convert this code error into voltage using the LSB:

$$\Delta V=\Delta Code\times LSB$$

$$=-1642\times0.0001525879$$

$$\boxed{\Delta V\approx-0.25055\text{ V}}$$

So the measured result is approximately 0.251 V below the ideal value.

Equivalent temperature error

The sensor has a range of:

$$2.5\text{ V}\leftrightarrow200°C$$

Therefore its sensitivity is:

$$\frac{200}{2.5}=80°C/V$$

Hence:

$$\Delta T=\Delta V\times80$$

$$=-0.25055\times80$$

$$\boxed{\Delta T\approx-20.04°C}$$

Therefore, at the 100°C reference point, the calibrated reading corresponds to an error of approximately −20°C relative to the ideal transfer.

Cross-check Using Measured Code

Measured voltage corresponding to code 6550:

$$V_{measured}\approx6550\times\frac{2.5}{16384}$$

$$\approx0.99945\text{ V}$$

This corresponds to:

$$T_{measured}\approx\frac{0.99945}{2.5}\times200$$

$$\approx79.96°C$$

So a true 100°C input is being reported at approximately 79.96°C, giving an error of approximately:

$$79.96-100\approx-20.04°C$$

Final Answer

Quantity	Result
ADC resolution	14 bit
Ideal LSB	152.59 μV
Ideal code at 100°C	≈ 8192
Measured code	6550
Code error	−1642 counts
Offset error	≈ −0.2506 V
Equivalent temperature error	≈ −20.04°C
Measured temperature equivalent	≈ 79.96°C

Note: Strictly, a single calibration point reveals a combined offset/gain error unless the gain is independently known. Under the exam's stated linear-sensor assumption, the difference at the 100°C reference is treated as the offset error.