To regulate the output voltage of the unregulated DC power supply to a constant 9V using a Zener diode, we can use a simple series voltage regulator configuration.

In this configuration, the Zener diode is connected in reverse bias across the output terminals, and a series resistor is placed in between the input voltage and the Zener diode. The purpose of the series resistor is to limit the current flowing through the Zener diode and protect it from excessive power dissipation.

We can calculate the minimum series resistor value by considering the worst-case scenario, i.e., the minimum input voltage and the maximum load current. Let's assume the worst-case input voltage is 12V.

First, we need to calculate the current flowing through the Zener diode (Iz) using Ohm's law:
Iz = (Vin - Vz) / Rs
where Vin is the input voltage, Vz is the breakdown voltage of the Zener diode, and Rs is the series resistor.

In this case, Vin = 12V and Vz = 9.1V. Let's assume Iz should be around 20mA (maximum power dissipation of the Zener diode is 1W, so 1W/9.1V ≈ 0.11A or 110mA is the maximum current we should use).

So, 20mA = (12V - 9.1V) / Rs

Simplifying the equation:
0.02A = 2.9V / Rs
Rs = 2.9V / 0.02A
Rs = 145Ω

Therefore, the minimum series resistor value required to regulate the unregulated DC power supply to a constant 9V using the Zener diode is approximately 145Ω.

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