understand the principle of a potential divider circuit

Potential Dividers

Learning Objective

By the end of this lesson you should be able to:

• Explain the principle of a potential divider circuit.
• Derive the expression for the output voltage of a two‑resistor divider.
• Apply the formula to calculate resistor values for a required output voltage.
• Identify the effect of loading the divider and how to minimise it.

1. Introduction

A potential divider (also called a voltage divider) is a simple linear circuit that produces a fraction of an input voltage. It is widely used in measurement, biasing of transistors, and as a reference voltage source.

2. Theory

Consider two resistors \$R1\$ and \$R2\$ connected in series across a source voltage \$V_{\text{s}}\$. The same current \$I\$ flows through both resistors because they are in series.

\$I = \frac{V{\text{s}}}{R1 + R_2}\$

The voltage drop across each resistor is given by Ohm’s law:

\$V{R1} = I R1,\qquad V{R2} = I R2\$

The output voltage \$V{\text{out}}\$ is taken across \$R2\$ (or \$R_1\$ depending on the application). Substituting for \$I\$ gives the classic divider formula:

\$V{\text{out}} = V{\text{s}} \frac{R2}{R1 + R_2}\$

Similarly, if the output is taken across \$R_1\$:

\$V{\text{out}} = V{\text{s}} \frac{R1}{R1 + R_2}\$

3. Derivation Step‑by‑Step

1. Write the total resistance of the series combination: \$R{\text{total}} = R1 + R_2\$.
2. Apply Kirchhoff’s voltage law (K \cdot L) around the loop: \$V{\text{s}} = V{R1} + V{R_2}\$.
3. Express each voltage drop using Ohm’s law: \$V{R1}=IR1\$, \$V{R2}=IR2\$.
4. Solve for the current: \$I = V{\text{s}}/(R1+R_2)\$.
5. Substitute \$I\$ back into \$V{R2}=IR2\$ to obtain \$V{\text{out}} = V{\text{s}}R2/(R1+R2)\$.

4. Example Calculations

Suppose \$V{\text{s}} = 12\ \text{V}\$ and we require \$V{\text{out}} = 5\ \text{V}\$ across \$R2\$. Choose \$R1 = 1.0\ \text{k}\Omega\$. Find \$R_2\$.

\$5 = 12 \frac{R2}{1.0\text{k} + R2}\$

Rearranging:

\$5(1.0\text{k}+R2)=12R2\$

\$5\,000 + 5R2 = 12R2\$

\$5\,000 = 7R_2\$

\$R_2 \approx 714\ \Omega\$

Resulting divider:

5. Loading Effect

If a load resistance \$R{\text{L}}\$ is connected across the output, the effective resistance across \$R2\$ becomes the parallel combination:

\$R{\text{eq}} = \frac{R2 R{\text{L}}}{R2 + R_{\text{L}}}\$

The output voltage then becomes:

\$V{\text{out}} = V{\text{s}} \frac{R{\text{eq}}}{R1 + R_{\text{eq}}}\$

To minimise loading, design the divider so that \$R{\text{L}} \gg R2\$ (typically at least ten times larger).

6. Practical Considerations

• Use resistors with a tolerance of 1 % or better for precise voltage references.
• Consider power dissipation: \$P = I^2 R\$ for each resistor.
• For high‑impedance loads, buffer the divider with an op‑amp voltage follower.
• Temperature coefficients can cause drift; select low‑TC resistors for stable applications.

7. Common Mistakes

• Assuming the output voltage is independent of the load – it is not unless the load is much larger than \$R_2\$.
• Using resistors that are too small, leading to excessive power loss and heating.
• Confusing the positions of \$R1\$ and \$R2\$ in the formula; remember \$V_{\text{out}}\$ is taken across the resistor that is *below* the output node.

8. Summary

The potential divider is a fundamental circuit that produces a predictable fraction of an input voltage. Its output is given by \$V{\text{out}} = V{\text{s}}\,R2/(R1+R_2)\$. The accuracy of the output depends on resistor values, tolerance, and the effect of any load connected to the output. Proper design ensures the divider provides a stable reference voltage for a wide range of A‑Level physics experiments and electronic applications.

9. Practice Questions

1. Design a divider that gives \$3.3\ \text{V}\$ from a \$9\ \text{V}\$ supply using standard resistor values. Show your calculations.
2. A divider uses \$R1 = 2.2\ \text{k}\Omega\$ and \$R2 = 1.0\ \text{k}\Omega\$. What is the output voltage? If a \$5\ \text{k}\Omega\$ load is connected, what is the new output voltage?
3. Explain why a voltage follower (buffer) is often placed after a potential divider in precision circuits.