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Zener Diode as Voltage Stabilizer

A circuit sho...

Question

A circuit shown in Fig. has resistances $R_{1} = 20 Ω$ and $R_{2} = 30 Ω$ . At what value of resistance $R_{x}$ will the thermal power generated in it be practically independent of small variations of that resistance? The voltage between the points $A$ and $B$ is supposed to be constant in this case.

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Solution

The equivalent resistance between A and B
$R_{e q} = R_{1} + \frac{R_{2} R_{x}}{R_{2} R_{y}} I = \frac{V}{R_{e q}} I = \frac{V}{R_{1} + \frac{R_{2} R_{y}}{R_{2} {+ R}_{Y}}}$
By current division
$I_{x} = ⎛ ⎜ ⎜ ⎜ ⎜ ⎝ \frac{V}{R_{1} + \frac{R_{2} R_{y}}{R_{2} + R_{y}}} ⎞ ⎟ ⎟ ⎟ ⎟ ⎠ (\frac{R_{2}}{R_{2} + R_{y}}) p_{x} = I_{y}^{2} R y = \frac{(V R_{2})^{2}}{(R_{1} R_{2} + R_{1} x_{y} + R_{2} R_{y})^{2}} R_{x}$
For the maximum value,
$\frac{d ρ_{x}}{d R_{y}} = 0$
Solving Differentiation we get,
$R_{y} = 12 Ω$

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