(III) You are designing a wire resistance heater to heat an...

(III) You are designing a wire resistance heater to heat an enclosed volume of gas. For the apparatus to function properly, this heater must transfer heat to the gas at a very constant rate. While in operation, the resistance of the heater will always be close to the value $R=R_{0},$ but may fluctuate slightly causing its resistance to vary a small amount $\Delta R\left(\ll R_{0}\right) .$ To maintain the heater at constant power, you design the circuit shown in Fig. $26-44,$ which includes two resistors, each of resistance $r$. Determine the value for $r$ so that the heater power will remain constant even if its resistance $R$ fluctuates by a small amount. [Hint: If $\Delta R \ll R_{0},$ then $\left.\left.\Delta P \approx \Delta R \frac{d P}{d R}\right|_{R=R_{0}^{0}}\right]$

(III) You are designing a wire resistance heater to heat an enclosed volume of gas. For the apparatus to function properly, this heater must transfer heat to the gas at a very constant rate. While in operation, the resistance of the heater will always be close to the value $R=R_{0},$ but may fluctuate slightly causing its resistance to vary a small amount $\Delta R\left(\ll R_{0}\right) .$ To maintain the heater at constant power, you design the circuit shown in Fig. $26-44,$ which includes two resistors, each of resistance $r$. Determine the value for $r$ so that the heater power will remain constant even if its resistance $R$ fluctuates by a small amount. [Hint:
If $\Delta R \ll R_{0},$ then $\left.\left.\Delta P \approx \Delta R \frac{d P}{d R}\right|_{R=R_{0}^{0}}\right]$

Key Concepts

Sensitivity Analysis

Sensitivity analysis in circuit design examines how variations in component values affect the overall performance of the system. Here, the heater’s power output is analyzed with respect to small changes in its resistance. By computing the derivative of power with respect to resistance and setting it to zero, the circuit can be designed to be insensitive to small resistive fluctuations. This approach is crucial for ensuring constant power delivery despite component tolerances or operational variations.

Resistor Network Design for Constant Power

This concept focuses on the arrangement of additional resistors to stabilize the circuit's power output. By strategically selecting the value of extra resistors in the network (in this case, the resistors of value r), the circuit can compensate for the small variations in the heater’s resistance. The proper network design ensures that the power delivered remains constant, which is essential for applications like a resistance heater where a steady heat output is required.

Small-Signal Approximation

This concept involves linearizing functions around an operating point when changes are small compared to the nominal value. In the context of electrical circuits, when a parameter such as resistance varies by a very small amount, the resulting change in a dependent quantity (such as power) can be approximated using the derivative at the nominal point. This simplification is useful in design and analysis as it reduces complex nonlinear behavior to a manageable linear form.

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