# Worst Case Calculation and Selection of Resistors for a Voltage Divider Circuit

A voltage divider is a simple yet fundamental circuit in electronics, used to create a specific fraction of an input voltage. The basic configuration involves two resistors,

$R_1$ and $R_2$, connected in series across a voltage source $V_{in}$. The output voltage $V_{out}$ is taken from the junction of $R_1$ and $R_2$.

### Basic Theory

The output voltage $V_{out}$ of a voltage divider is given by: $V_{out} = V_{in} \cdot \frac{R_2}{R_1 + R_2}$

### Worst Case Calculation

Worst case analysis ensures that the circuit performs reliably under the most extreme conditions. For a voltage divider, the worst-case scenario occurs due to the tolerance of the resistors. The tolerance indicates the permissible deviation from the nominal value, typically expressed as a percentage.

#### Steps for Worst Case Calculation

**Determine Nominal Values and Tolerances**:- Let $R_{1_{nom}}$ and $R_{2_{nom}}$ be the nominal values of $R_1$ and $R_2$, respectively.
- Let $T_1$ and $T_2$ be the tolerances of $R_1$ and $R_2$, respectively.

**Calculate Extremes**:- Maximum $R_1$: $R_{1_{max}} = R_{1_{nom}} (1 + T_1)$
- Minimum $R_1$: $R_{1_{min}} = R_{1_{nom}} (1 - T_1)$
- Maximum $R_2$: $R_{2_{max}} = R_{2_{nom}} (1 + T_2)$
- Minimum $R_2$: $R_{2_{min}} = R_{2_{nom}} (1 - T_2)$

**Calculate Worst Case $V_{out}$**:- Calculate $V_{out}$ for the maximum and minimum possible values of $R_1$ and $R_2$.

$V_{out_{max}} = V_{in} \cdot \frac{R_{2_{max}}}{R_{1_{min}} + R_{2_{max}}}$ $V_{out_{min}} = V_{in} \cdot \frac{R_{2_{min}}}{R_{1_{max}} + R_{2_{min}}}$

#### Example Calculation

Assume:

- $V_{in} = 10V$
- $R_{1_{nom}} = 1k\Omega$, $T_1 = 5\%$
- $R_{2_{nom}} = 2k\Omega$, $T_2 = 5\%$

Calculate extremes:

- $R_{1_{max}} = 1k\Omega \times 1.05 = 1.05k\Omega$
- $R_{1_{min}} = 1k\Omega \times 0.95 = 0.95k\Omega$
- $R_{2_{max}} = 2k\Omega \times 1.05 = 2.1k\Omega$
- $R_{2_{min}} = 2k\Omega \times 0.95 = 1.9k\Omega$

Calculate worst case $V_{out}$:

- $V_{out_{max}} = 10V \cdot \frac{2.1k\Omega}{0.95k\Omega + 2.1k\Omega} = 10V \cdot \frac{2.1}{3.05} \approx 6.89V$
- $V_{out_{min}} = 10V \cdot \frac{1.9k\Omega}{1.05k\Omega + 1.9k\Omega} = 10V \cdot \frac{1.9}{2.95} \approx 6.44V$

### Selection of Resistor Combination

Selecting the best combination of resistors involves considering several factors:

**Tolerance**:- Lower tolerance resistors (e.g., 1% or 0.1%) will provide a more stable and precise $V_{out}$.

**Power Rating**:- Ensure the resistors can handle the power dissipation. The power dissipated in a resistor $P$ is given by $P = \frac{V^2}{R}$.

**Temperature Coefficient**:- Choose resistors with low temperature coefficients to minimize variation in resistance due to temperature changes.

**Availability and Cost**:- Balance between precision and cost. High precision resistors are typically more expensive.

### Example of Selecting Resistors

For a specific application, suppose you need $V_{out} = 5V$ from $V_{in} = 10V$. The ideal ratio is: $\frac{R_2}{R_1 + R_2} = 0.5$

Choosing $R_1 = 1k\Omega$ and $R_2 = 1k\Omega$:

- Nominal $V_{out} = 10V \cdot \frac{1k\Omega}{1k\Omega + 1k\Omega} = 5V$

If we select resistors with 1% tolerance:

- $R_{1_{max}} = 1k\Omega \times 1.01 = 1.01k\Omega$
- $R_{1_{min}} = 1k\Omega \times 0.99 = 0.99k\Omega$
- $R_{2_{max}} = 1k\Omega \times 1.01 = 1.01k\Omega$
- $R_{2_{min}} = 1k\Omega \times 0.99 = 0.99k\Omega$

Worst case $V_{out}$:

- $V_{out_{max}} = 10V \cdot \frac{1.01k\Omega}{0.99k\Omega + 1.01k\Omega} = 10V \cdot \frac{1.01}{2} = 5.05V$
- $V_{out_{min}} = 10V \cdot \frac{0.99k\Omega}{1.01k\Omega + 0.99k\Omega} = 10V \cdot \frac{0.99}{2} = 4.95V$

This combination provides a $V_{out}$ very close to the desired value, with minimal deviation.

### Conclusion

Performing worst-case calculations for a voltage divider involves determining the maximum and minimum possible values of the output voltage considering resistor tolerances. Selecting the best resistor combination requires balancing precision, power rating, temperature stability, availability, and cost. By carefully considering these factors, you can design a robust and reliable voltage divider circuit.

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