The second approximation includes a

**zener resistance**\((R_Z)\) in series with an**ideal battery**.Total voltage across the zener diode: \[V_{total} = V_Z + V_{R_Z}\]

\(V_Z\): Breakdown voltage.

\(V_{R_Z}\): Voltage drop across zener resistance.

**Effect on Load Voltage**:**Ideal Case**: Load voltage \(V_L\) is equal to the breakdown voltage \(V_Z\).**Second Approximation**: Includes the zener resistance. \[V_L = V_Z + I_Z R_Z\]\(I_Z\): Zener current flowing through zener resistance.

**Change in Load Voltage**:The change in load voltage from the ideal case: \[\Delta V_L = I_Z R_Z\]

Usually, \(R_Z\) is small, resulting in a minor voltage change (in tenths of a volt).

**Example Calculation**:If \(I_Z = 10 \, ~\mathrm{mA}\) and \(R_Z = 10 \, \Omega\): \[\Delta V_L = 0.1 \, V\]

**Ripple**: AC voltage variations superimposed DC output voltage.Equivalent circuit to analyze ripple

Three resistances affect the ripple:

\(R_Z\) (Zener resistance), \(R_S\) (Series resistance), \(R_L\) (Load resistance)

**Simplification**: Typically, \(R_Z << R_L\).

**Voltage Divider Equation**:The output ripple voltage can be calculated as: \[V_{R(\text{out})} = \frac{R_Z}{R_S + R_Z} V_{R(\text{in})}\]

**Approximation for Troubleshooting**:In most designs, \(R_S\) is significantly greater than \(R_Z\): \[V_{R(\text{out})} \approx \frac{R_Z}{R_S} V_{R(\text{in})}\]

The zener diode has a breakdown voltage of 10 V and a zener resistance of 8.5 \(\Omega\). Apply the second approximation to determine the load voltage when the zener current is 20 mA.

\[\begin{aligned} \text{change in load voltage}~\Delta V_{L} & =I_{Z}R_{Z}=(20 \mathrm{mA})(8.5 \Omega)=0.17 \mathrm{V} \\ \text{load voltage (second approx.)}~V_{L} &=10 \mathrm{V}+0.17 \mathrm{V}=10.17 \mathrm{V} \end{aligned}\]

What is the approximate ripple voltage across the load, if \(R_s = 270~\Omega,~R_Z = 8.5~\Omega, V_{R(in)}=2~\mathrm{V}\) \[V_{R(\mathrm{out})}\approx\frac{8.5 \Omega}{270 \Omega} \times 2 \mathrm{V}=63 \mathrm{mV}\]

**Zener Regulation Condition**:For proper regulation, the

**Zener diode**must remain in the**breakdown region**.There must be sufficient

**Zener current**for all source voltages and load currents.

**Worst-Case Scenarios**:**Decreasing Source Voltage**:Example: For \(V_S = 20\ V\) and \(R_S = 200\ \Omega\): \[I_S = \frac{V_S - V_Z}{R_S} = 50\ \mathrm{mA}, \quad I_L = 10\ \mathrm{mA}, \quad I_Z = 40\ \mathrm{mA}\]

As \(V_S\) decreases (e.g., from 20V to 12V), \(I_S\) reduces to 10 mA and \(I_Z = 0\) mA.

If \(V_S\) drops further,

**regulation is lost**.

**Decreasing Load Resistance**:Example: \(R_L\) decreases from \(1\ \mathrm{k}\Omega\) to \(200\ \Omega\).

Load current \(I_L\) increases to 50 mA, causing \(I_Z\) to drop to zero.

Regulation fails when load resistance is too low.

**Increasing Series Resistance**:If \(R_S\) increases (e.g., from \(200\ \Omega\) to \(1\ \mathrm{k}\Omega\)), \(I_S\) reduces, and regulation is lost.

**Formula for Maximum Series Resistance**:To avoid failure under worst-case conditions: \[R_{S(\max)} = \left( \frac{V_{S(\min)}}{V_Z} - 1 \right) R_{L(\min)}\]

Alternatively: \[R_{S(\max)} = \frac{V_{S(\min)} - V_Z}{I_{L(\max)}}\]

\[\boxed{ R_{S(\max)} = \left( \frac{V_{S(\min)}}{V_Z} - 1 \right) R_{L(\min)}} \qquad \boxed{R_{S(\max)} = \frac{V_{S(\min)} - V_Z}{I_{L(\max)}}}\]

A zener regulator has an input voltage that can range from 22 to 30 V. Given that the regulated output voltage is 12 V and the load resistance varies from 140 \(\Omega\) to 10 k\(\Omega\), what is the maximum permissible series resistance?

\[R_{\mathrm{S(max)}}=\left(\frac{22 \mathrm{V}}{12 \mathrm{V}}-1\right)140 \Omega=117~ \Omega\]

A zener regulator has an input voltage ranging from 15 to 20 V and a load current varying from 5 to 20 mA. If the zener voltage is 6.8 V, what is the maximum allowable series resistance?

\[R_S(\max)=\frac{15 \mathrm{V}-6.8 \mathrm{V}}{20 \mathrm{mA}}=410~ \Omega\]

**Maximum Power Dissipation**:Power dissipated by the Zener diode: \[P_Z = V_Z \cdot I_Z\]

Ensure \(P_Z\) does not exceed the Zener diode’s power rating (0.25 W to 50 W).

**Maximum Current (\(I_{ZM}\))**:Maximum current without exceeding power rating: \[I_{ZM} = \dfrac{P_{ZM}}{V_Z}\]

Zener current: \[I_Z = \frac{V_S - V_Z}{R_S}\]

Example: For \(V_S = 20\ \text{V}\) and \(R_S = 1\ \text{k}\Omega\): \[I_Z = \frac{20 - V_Z}{1000}\]

**Saturation Point**: \(V_Z = 0\), \(I_Z = 20\ \text{mA}\)**Cutoff Point**: \(I_Z = 0\), \(V_Z = 20\ \text{V}\)

For \(V_S = 20\ \text{V}\), \(R_S = 1\ \text{k}\Omega\), and \(V_Z = 12\ \text{V}\):

Intersection point at

**Q1**with approximately 12 V.Changing \(V_S\) to 30 V: \[I_Z = \frac{30 - V_Z}{1000} = 18\ \text{mA}\]

Intersection point shifts to

**Q2**, but \(V_Z\) remains approximately 12 V.

**Conclusion**: Zener diodes maintain nearly constant voltage despite changes in source voltage.