(d) If the AHU fan speed of range \( 0 \% \) of \( 100 \% \) corresponds to a DDC control signal range of \( 2-10 \mathrm{~V} \). (i) Determine the fan speed if the control signal is 5.5 V . (ii) Determine DDC output voltage if the fan runs at \( 55 \% \) of the full speed.

To solve the problems, we'll assume a **linear relationship** between the DDC control signal voltage and the fan speed percentage. The control signal ranges from 2 V (corresponding to 0% fan speed) to 10 V (corresponding to 100% fan speed). ### (i) Determining the Fan Speed for a 5.5 V Control Signal **Given:** - Control signal, \( V = 5.5 \, \text{V} \) - Voltage range: \( 2 \, \text{V} \) (0%) to \( 10 \, \text{V} \) (100%) **Steps:** 1. **Calculate the voltage above the minimum:** \[ V_{\text{above}} = V - 2 = 5.5 \, \text{V} - 2 \, \text{V} = 3.5 \, \text{V} \] 2. **Determine the proportion of the voltage range:** \[ \text{Proportion} = \frac{V_{\text{above}}}{10 \, \text{V} - 2 \, \text{V}} = \frac{3.5 \, \text{V}}{8 \, \text{V}} = 0.4375 \] 3. **Convert the proportion to a percentage:** \[ \text{Fan Speed} = 0.4375 \times 100\% = 43.75\% \] **Answer:** The fan operates at **43.75 %** of its full speed when the control signal is 5.5 V. ### (ii) Determining the DDC Output Voltage for 55 % Fan Speed **Given:** - Desired fan speed, \( \text{Speed} = 55\% \) - Voltage range: \( 2 \, \text{V} \) (0%) to \( 10 \, \text{V} \) (100%) **Steps:** 1. **Determine the proportion of the full speed:** \[ \text{Proportion} = \frac{55\%}{100\%} = 0.55 \] 2. **Calculate the voltage increase corresponding to the desired speed:** \[ V_{\text{increase}} = 0.55 \times (10 \, \text{V} - 2 \, \text{V}) = 0.55 \times 8 \, \text{V} = 4.4 \, \text{V} \] 3. **Add the minimum voltage to find the total output voltage:** \[ V = 2 \, \text{V} + 4.4 \, \text{V} = 6.4 \, \text{V} \] **Answer:** To run the fan at **55 %** of full speed, the DDC output voltage should be **6.4 V**. ### Summary - **(i)** A control signal of **5.5 V** corresponds to a fan speed of **43.75 %**. - **(ii)** To achieve a fan speed of **55 %**, the DDC output voltage should be **6.4 V**.