Zener Diode Cases with PYQ

ece-1109 ECE-1109 Introduction to ECE The mathematical analysis of a Zener diode used as a voltage regulator involves three distinct cases, primarily to ensure the diode remains in the active Zener breakdown region (the “on” state) where it maintains a nearly constant output voltage ($V_L=V_Z$).

The fundamental circuit analyzed for these cases is the basic Zener regulator, which includes an input source ($V_i$ or $V_{PS}$), a series limiting resistor ($R$ or $R_i$), and a load resistor ($R_L$) in parallel with the Zener diode ($D_Z$).

The core analysis relies on Kirchhoff’s laws applied to the circuit in the “on” state:

• Loop Equation: The voltage drop across the series resistor is $V_R=V_i-V_L=V_i-V_Z$.
• Series Resistor Current ($I_R$): $I_R=\frac{V_i-V_L}{R}=\frac{V_i-V_Z}{R}$.
• Node Equation (KCL): The series current splits between the Zener and the load: $I_R=I_Z+I_L$.
• Load Current ($I_L$): $I_L=\frac{V_L}{R_L}=\frac{V_Z}{R_L}$ (since $V_L=V_Z$ in the “on” state).
• Zener Current ($I_Z$): $I_Z=I_R-I_L$.

The diode must satisfy two critical conditions for proper regulation:

1. It must be “on” ($V_L\ge V_Z$).
2. The maximum Zener power rating must not be exceeded: $P_Z=V_Z{I}_Z\le P_{ZM}$.

Here are the three mathematical cases, their steps, and necessary equations:

---

1. Case I: Fixed Input Voltage ($V_i$), Fixed Load Resistance ($R_L$)

This is the simplest case, where all components are fixed, and the goal is to determine the diode’s state and operating point ($V_L,I_Z,P_Z$).

Steps to Solve:

1. Test the State of the Zener Diode: To determine if the Zener is “on,” remove it and calculate the open-circuit voltage ($V$) across the resulting terminals using the voltage divider rule: $V=V_L=\frac{R_L{V}_i}{R+R_L}$
2. Determine “On” or “Off”:
   • If $V\ge V_Z$, the Zener is “on,” and the equivalent circuit is replaced by a $V_Z$ source.
   • If $V<V_Z$, the Zener is “off,” and the equivalent circuit is an open circuit ($I_Z=0\text{ A}$).
3. Calculate Parameters (If “On”): If the Zener is “on,” the load voltage is fixed:
   • $V_L=V_Z$
   • Calculate $I_L$ and $I_R$: $I_L=V_Z/R_L$ and $I_R=(V_i-V_Z)/R$.
   • Calculate $I_Z$: $I_Z=I_R-I_L$.
   • Check power dissipation: $P_Z=V_Z{I}_Z$.

---

2. Case II: Fixed Input Voltage ($V_i$), Variable Load Resistance ($R_L$)

The objective here is to determine the range of $R_L$ (and consequently $I_L$) that allows the Zener to remain in the “on” state, maintaining $V_L=V_Z$.

Steps to Solve:

1. Determine Minimum Load Resistance ($R_{L\min }$):

   • The transition point from “off” to “on” occurs when $V_L=V_Z$ and $I_Z\approx 0\text{ A}$.
   • Use the voltage divider equation at this transition point to find $R_{L\min }$: $R_{L\min }=\frac{R{V}_Z}{V_i-V_Z}$
   • Any $R_L>R_{L\min }$ ensures the Zener is “on”.

2. Determine Maximum Load Current ($I_{L\max }$):

   • $I_{L\max }$ occurs when $R_L$ is at its minimum required value, $R_{L\min }$: $I_{L\max }=\frac{V_Z}{R_{L\min }}$
   • Alternatively, determine $I_R$: $I_R=(V_i-V_Z)/R$. Since $I_Z\ge I_{ZK}\approx 0\text{ A}$ when $R_L=R_{L\min }$, $I_{L\max }\approx I_R$.

3. Determine Maximum Load Resistance ($R_{L\max }$):

   • The maximum current that the Zener can safely handle is $I_{ZM}=P_{ZM}/V_Z$.
   • The Zener operates near $R_{L\max }$ when $I_Z$ is at its maximum limit, $I_{ZM}$.
   • First, determine the series current $I_R$ (which is fixed since $V_i$ is fixed): $I_R=\frac{V_i-V_Z}{R}$
   • Find the minimum load current $I_{L\min }$ that ensures $I_Z$ does not exceed $I_{ZM}$: $I_{L\min }=I_R-I_{ZM}$
   • Finally, find the corresponding maximum load resistance: $R_{L\max }=\frac{V_Z}{I_{L\min }}$

---

3. Case III: Fixed Load Resistance ($R_L$), Variable Input Voltage ($V_i$ or $V_{PS}$)

The objective is to find the range of input voltages ($V_{i\min }$ to $V_{i\max }$) that keeps the Zener “on” and within its power rating.

Steps to Solve:

1. Determine Minimum Input Voltage ($V_{i\min }$):

   • $V_{i\min }$ is the lowest voltage required to turn the Zener “on” (i.e., when $V_L=V_Z$ and $I_Z\approx 0\text{ A}$).
   • Use the voltage divider equation at the turn-on point: $V_{i\min }=\frac{(R_L+R)V_Z}{R_L}$
   • This condition also assumes that the Zener current is at its minimum acceptable value, $I_{Z\min }$ (often set slightly above zero or $I_{ZK}$).

2. Determine Maximum Input Voltage ($V_{i\max }$):

   • $V_{i\max }$ is limited by the maximum continuous Zener current ($I_{ZM}$).
   • First, calculate the fixed load current $I_L$: $I_L=V_Z/R_L$ (since $R_L$ is fixed).
   • The maximum series current ($I_{R\max }$) is the sum of the fixed load current and the maximum Zener current: $I_{R\max }=I_{ZM}+I_L$
   • $V_{i\max }$ is found by applying Kirchhoff’s voltage law to the loop using $I_{R\max }$: $V_{i\max }=I_{R\max }R+V_Z$

Consideration of Non-Ideal Zener Resistance ($r_Z$ or $Z_{ZT}$)

The formulas above assume an ideal Zener diode where the voltage is perfectly constant ($V_L=V_Z$). If the dynamic Zener resistance ($r_Z$) is included, the circuit analysis becomes more complex:

• The equivalent circuit includes $V_Z$ in series with $r_Z$.
• When the Zener is “on,” the load voltage changes slightly with Zener current: $V_L=V_Z+I_Z{r}_Z$.
• The voltage change in $V_L$ due to changes in input voltage ($V_{PS}$) is measured by Source Regulation.
• The voltage change in $V_L$ due to changes in load current ($I_L$) is measured by Load Regulation.

The source uses the Zener resistance for design and specification purposes, particularly for:

• Source Regulation: $\frac{\Delta v_L}{\Delta v_{PS}}\times 100$

• Load Regulation: $\frac{v_{L,\text{no load}}-v_{L,\text{full load}}}{v_{L,\text{full load}}}\times 100$%.

Previous Year Questions (PYQs) on Zener Diodes

A. Conceptual Questions (Breakdown Mechanism)

Question Text	Years of Appearance
Briefly describe the process of ‘Avalanche’ and ‘Zener’ breakdowns.	2015, 2017, 2018, 2021, 2022, 2024

B. Calculation and Design Questions (Voltage Regulation)

These questions involve applying the Zener diode in regulator circuits to determine voltage ranges, currents, and power ratings. Just to get a idea of the questions:

Question Text	Years of Appearance
Design a voltage regulator using a Zener diode that will remain an output voltage of $\pm 50\text{V}$… find the maximum and minimum values of zener diode current.	2018
A zener diode used in a voltage regulator remains an output voltage of $\pm 50\text{V}$… then find out the range of input voltage that can be varied.	2019, 2021
Design a voltage regulator using a Zener diode… Determine the maximum power rating of the Zener diode required to design the circuit.	2015, 2023
Determine the range values of $V_i$ that will maintain the Zener diode of Figure 7(d) in ‘on’ state.	2015, 2018
A zener diode has the specifications $V_z=5.2\text{V}$ and $P_{zmax}=260\text{mW}$… Find the maximum allowable current $I_L$ when the zener diode is acting as a regulator.	2017
Design a Zener diode voltage regulator using the circuit in Figure 3(d)…	2022
For the network of figure below, determine the range of $V_i$ that will maintain $V_L$ at $8\text{V}$ and not exceed the maximum power rating of the Zener diode.	2024
For the Figure 6(a) [Zener regulator circuit], find the (i) voltage at point ‘a’, (ii) voltage across resistor $R_r$, (iii) voltage drop across resistor $R$, (iv) the current through zener diode…	2019
The circuit of Figure 6(d) uses two zener diodes… determine (i). the regulated output voltage, (ii) maximum load current, and (iii) minimum load current.	2019
In the circuit shown in Fig. 5(a), the Knee current of the zener diode is $10\text{mA}$… find the minimum value of $R_L$ in $\Omega$ and the minimum power rating of the zener diode in $\text{mW}$.	2020
The circuit of Figure 6(b1) is an inexpensive voltage regulator. All the diodes are ideal. Find the regulation of $V_0$…	2017