Sensitivity in potential divider circuits.

In summary, sensitivity in a potential divider circuit is greatest when the resistance of the fixed resistor is equal to the average resistance of the other resistor. This is because the sensitivity is maximized when the fixed resistor is equal to the sensor resistance, as shown through differentiation. Therefore, choosing the average resistance of the sensor as the fixed resistor will result in the highest sensitivity for the voltage divider.
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Homework Statement


Why is sensitivity in a potential divider circuit greatest when the resistance of the fixed resistor is equal to the average resistance of the other resistor (in my case a LDR)?

This isn't a homework question, but I need to include a bit on why I chose the fixed resistor I did in my coursework - I know I chose it for the reason that it has the highest sensitivity, but I need to explain why this is.


Homework Equations



V2 = (R2 / (R1 + R2)) * V1


The Attempt at a Solution



...I don't know >.>; help much appreciated.
 
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Suppose that we have a voltage divider with a fixed resistance ##R_f## and a sensor with an average resistance ##R## configured so that:

##\frac{V_{out}}{V_{in}} = \frac{R_f}{R_f +R}##

We'll call the sensitivity ##S## of the divider the magnitude of amount by which the voltage ratio changes with respect to small changes in R. So:

##S = \left| \frac{d}{dR}\left( \frac{R_f}{R_f +R} \right) \right| = \frac{R_f}{(R_f + R)^2}##

Now we'd like to maximize this sensitivity through a good choice of ##R_f##. Differentiating again and setting equal to zero:

##\frac{dS}{dR_f} = \frac{(R_f - R)}{(R_f + R)^3} = 0##

We can see that it is zero when ##R_f = R##, making the average sensor resistance value ##R## the best choice for ##R_f##.

Note that I didn't check to prove that what was found was a maxima rather than a minima; Left as an exercise for the student :smile:
 
Last edited:

1. What is a potential divider circuit?

A potential divider circuit is a circuit that divides a voltage into smaller parts using resistors. It is commonly used in electronic devices to regulate voltage and create specific voltage levels.

2. How does sensitivity affect potential divider circuits?

Sensitivity in potential divider circuits refers to how a small change in the input voltage can cause a significant change in the output voltage. A highly sensitive circuit will produce a larger change in output voltage compared to a less sensitive circuit with the same input voltage change.

3. What factors can affect the sensitivity of a potential divider circuit?

The sensitivity of a potential divider circuit can be affected by the values of the resistors used, the overall resistance in the circuit, and the input voltage. A higher overall resistance will result in a more sensitive circuit, while using resistors with lower values will decrease sensitivity.

4. How can the sensitivity of a potential divider circuit be calculated?

The sensitivity of a potential divider circuit can be calculated by dividing the change in output voltage by the change in input voltage. This is represented by the formula: sensitivity = ΔVout / ΔVin. Alternatively, the sensitivity can also be calculated by taking the inverse of the voltage division ratio (Vout / Vin).

5. How can sensitivity be controlled in a potential divider circuit?

The sensitivity of a potential divider circuit can be controlled by choosing appropriate resistor values and adjusting the overall resistance in the circuit. Higher resistance values will result in a more sensitive circuit, while lower resistance values will decrease sensitivity. Additionally, using a voltage regulator can also help to stabilize the output voltage and reduce sensitivity.

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