A thermoelectric-based power meter has a response time of 20 s and sensitivity

(output voltage per unit incident power) of 90 mV/W. If the time response is reduced
to 8 s by increasing the thermal conductance between the sensor element
and heat sink, what will be the new sensitivity?

photonics & lasers page 224

I need the answer for support my studying

To determine the new sensitivity of the thermoelectric-based power meter, we need to consider the relationship between the response time, sensitivity, and thermal conductance of the sensor element and heat sink.

First, we can use the formula for the response time of a thermoelectric-based power meter:

Response time = 1 / (2πf)

Here, f represents the thermal time constant, which is the reciprocal of the thermal conductance (G). Therefore, we have:

f = 1 / G

To find the new sensitivity, we need to use the relationship between the response time, sensitivity, and thermal time constant:

Sensitivity = (dV / dP) * (1 / (2πf))

Where:
- dV / dP represents the change in output voltage per unit incident power.
- (1 / (2πf)) is the correction factor accounting for the response time.

Now, let's calculate the new sensitivity:

Given:
- Current response time (RT1) = 20 s.
- Current sensitivity (S1) = 90 mV/W.
- New response time (RT2) = 8 s.

Step 1: Calculate the initial thermal time constant (f1) based on the current response time:

f1 = 1 / RT1
= 1 / 20
= 0.05 Hz

Step 2: Calculate the initial sensitivity (S1) using the given formula:

S1 = (dV / dP) * (1 / (2πf1))

Since we know that S1 = 90 mV/W, we can rearrange the formula and solve for (dV / dP):

(dV / dP) = S1 * (2πf1)
= 90 * (2π * 0.05)
≈ 28.27 mV/W

Step 3: Calculate the new thermal conductance (G2) based on the new response time:

f2 = 1 / RT2
= 1 / 8
= 0.125 Hz

G2 = 1 / f2
= 1 / 0.125
= 8 W/Hz

Step 4: Calculate the new sensitivity (S2) using the formula:

S2 = (dV / dP) * (1 / (2πf2))

S2 = (28.27 mV/W) * (1 / (2π * 0.125))
≈ 90.14 mV/W

Therefore, the new sensitivity (S2) of the thermoelectric-based power meter, after reducing the response time to 8 s by increasing the thermal conductance, is approximately 90.14 mV/W.