Question

A zener of breakdown voltage V_Z = 8 V and maximum Zener current, I_ZM = 10 mA is subjected to an input voltage V_i = 10 V with series resistance R = 100 Ω. In the given circuit R_L represents the variable load resistance. The ratio of maximum and minimum value of R_L is __________.

Fig.

Answer 1

Correct Answer: 2

To solve this problem, we need to determine the maximum and minimum values of load resistance \(R_L\) while ensuring the Zener diode operates within its specified range, then find the ratio of these values.

Step 1: Determine Maximum \(R_L\)

The Zener diode starts conducting when the input voltage exceeds the Zener voltage, \(V_Z = 8 \text{ V}\). The maximum current through the Zener diode, \(I_{ZM}\), is 10 mA. We first calculate the total series current when the diode conducts at its maximum:

\( I_{total} = I_{ZM} \)

Using Ohm’s Law, the maximum current in the circuit:

\( I_{total} = \frac{V_i - V_Z}{R} = \frac{10 \, \text{V} - 8 \, \text{V}}{100 \, \Omega} = 20 \, \text{mA} \)

Since \(I_{total} = I_{Zmax} = 10 \, \text{mA}\), we find:

\( I_L = I_{total} - I_{ZM} = 20 \, \text{mA} - 10 \, \text{mA} = 10 \, \text{mA} \)

The maximum \(R_L\) is:

\( R_{L_{\text{max}}} = \frac{V_Z}{I_L} = \frac{8 \, \text{V}}{10 \, \text{mA}} = 800 \, \Omega \)

Step 2: Determine Minimum \(R_L\)

At the minimum, the Zener diode conducts zero current, so:

\( I_{total} = I_L = \frac{V_i - V_Z}{R} = \frac{10 \, \text{V} - 8 \, \text{V}}{100 \, \Omega} = 20 \, \text{mA} \)

The minimum \(R_L\) is:

\( R_{L_{\text{min}}} = \frac{V_Z}{I_L} = \frac{8 \, \text{V}}{20 \, \text{mA}} = 400 \, \Omega \)

Step 3: Calculate Ratio

The ratio of maximum to minimum \(R_L\) values is:

\( \frac{R_{L_{\text{max}}}}{R_{L_{\text{min}}}} = \frac{800 \, \Omega}{400 \, \Omega} = 2 \)

Conclusion: The ratio of the maximum and minimum values of \(R_L\) is 2, which fits within the expected range (2, 2).