Let's tackle each part step by step: --- ## **(a) Determine the thermocouple sensitivity, ADC resolution, and number of bits required.** ### **Given Data** - **Thermocouple type:** K - **Temperature range:** 0 to 1000°C (Full scale) - **Desired temperature resolution:** 0.20°C at 20°C #### **Step 1: Find the Sensitivity of Type K Thermocouple at 20°C** From standard tables: - **Type K thermocouple sensitivity at 20°C:** Approximately **41 μV/°C** (source: NIST ITS-90 Thermocouple Database). #### **Step 2: Find the Output Voltage Range** - **At 0°C:** 0 mV (reference) - **At 1000°C:** From tables, type K at 1000°C ≈ **41.276 mV** So, \[ V_{FS} = V_{1000°C} - V_{0°C} \approx 41.276\,\text{mV} - 0\,\text{mV} = 41.276\,\text{mV} \] #### **Step 3: Find the Voltage Increment Corresponding to 0.20°C at 20°C** - **Sensitivity at 20°C:** 41 μV/°C - **Voltage for 0.20°C:** \[ \Delta V = \text{Sensitivity} \times \text{Desired Temp Resolution} \] \[ \Delta V = 41\,\mu\text{V/°C} \times 0.20\,\text{°C} = 8.2\,\mu\text{V} \] #### **Step 4: ADC Resolution Needed** - **ADC needs to resolve** at least **8.2 μV** per step. #### **Step 5: Number of Bits Required** - **Full scale (FS) =** 41.276 mV = 41,276 μV - **Resolution required =** 8.2 μV The number of steps, \( N \): \[ N = \frac{V_{FS}}{\text{Resolution}} = \frac{41,276\,\mu\text{V}}{8.2\,\mu\text{V}} \approx 5,034 \] So, number of bits \( n \) needed: \[ 2^n \geq N \implies n \geq \log_2(5,034) \approx 12.3 \] So, you need **at least 13 bits**. --- ### **Summary for (a):** - **Type K sensitivity at 20°C:** 41 μV/°C - **ADC voltage resolution needed:** 8.2 μV - **Number of bits required:** **13 bits** --- ## **(b) If the ADC is used with type R thermocouples, what is the resulting temperature resolution at 20°C?** ### **Step 1: Type R Sensitivity at 20°C** From tables: - **Type R thermocouple sensitivity at 20°C:** ≈ 10 μV/°C ### **Step 2: ADC Resolution From (a)** - ADC voltage resolution = 8.2 μV per step ### **Step 3: Temperature Resolution with Type R at 20°C** \[ \text{Temp Resolution} = \frac{\text{ADC Resolution}}{\text{Type R Sensitivity}} \] \[ = \frac{8.2\,\mu\text{V}}{10\,\mu\text{V/°C}} = 0.82\,°C \] ### **Comments:** - The temperature resolution is much worse with a less sensitive thermocouple. For Type R, it is **0.82°C** at 20°C (vs 0.20°C for Type K). - This is because the Type R thermocouple produces less voltage per degree, so the same voltage step represents a larger temperature change. --- ### **Summary for (b):** - **Resulting temperature resolution at 20°C (Type R):** **0.82°C** --- ## **(c) A 16-bit ADC is used with type S thermocouple over the same temperature range. What is the temperature resolution at 20°C?** ### **Step 1: Type S Sensitivity at 20°C** From tables: - **Type S thermocouple sensitivity at 20°C:** ≈ 10 μV/°C ### **Step 2: Full-Scale Voltage for Type S (0–1000°C)** - From NIST tables, type S at 1000°C ≈ **10.123 mV** (relative to 0°C) - So, \( V_{FS} = 10.123\,\text{mV} = 10,123\,\mu\text{V} \) ### **Step 3: 16-bit ADC Step Size** - Number of steps: \( 2^{16} = 65,536 \) - Step size: \[ \text{Resolution} = \frac{V_{FS}}{2^{16}} = \frac{10,123\,\mu\text{V}}{65,536} \approx 0.1545\,\mu\text{V} \] ### **Step 4: Temperature Resolution at 20°C** \[ \text{Temp Resolution} = \frac{0.1545\,\mu\text{V}}{10\,\mu\text{V/°C}} = 0.01545\,°C \approx 0.015\,°C \] --- ### **Summary for (c):** - **Temperature resolution at 20°C (Type S, 16-bit ADC, 0–1000°C):** **0.015°C** --- # **Final Answers (with Explanations)** ### **(a)** - **Type K sensitivity at 20°C:** **41 μV/°C** - **ADC voltage resolution required:** **8.2 μV** - **Number of bits required:** **13 bits** ### **(b)** - **Using the same ADC with Type R thermocouples, the temperature resolution at 20°C is:** **0.82°C** (because Type R is less sensitive than Type K) ### **(c)** - **With a 16-bit ADC and Type S thermocouple (0–1000°C), the temperature resolution at 20°C is:** **0.015°C**