Ohmic and Non-Ohmic Circuit Elements

Resistance need not behave in the same way for every component

Resistance measures how strongly a component opposes the flow of charge. Potential difference $V$, current $I$ and resistance $R$ are related by

$V=IR$

where potential difference is measured in volts (V), current in amperes (A) and resistance in ohms (Ω). Rearranging gives

$R=\frac{V}{I}$

This equation can be used at any particular operating point: measure the potential difference across a component and the current through it, then calculate the resistance at that point. It does not mean that every component has a resistance which remains constant.

For some resistors, the ratio $V/I$ stays constant as the current changes. For other circuit elements, this ratio changes, so their resistance is not constant. A fixed resistor has a set nominal resistance rather than an adjustable one, while a variable resistor allows that setting to be changed. However, calling a resistor “fixed” does not remove the need to consider temperature: heating can still affect its resistance.

Ohmic conductors

An ohmic conductor obeys a particular relationship: at constant temperature, its current is directly proportional to the potential difference across it.

$I\propto V$

For example, if the potential difference is doubled while the temperature remains constant, the current doubles. Therefore $V/I$ remains constant, so the resistance remains constant as the current changes.

The temperature condition matters. A larger current can transfer more energy to a conductor and make it warmer. If its temperature changes enough to alter its resistance, current will no longer remain directly proportional to potential difference even if the material behaves ohmically at a fixed temperature.

Recognising a linear I–V characteristic

An I–V characteristic normally places potential difference on the horizontal axis and current on the vertical axis. An ohmic conductor at constant temperature produces a straight line through the origin. It is therefore a linear circuit element.

Passing through the origin is important: zero potential difference produces zero current, and every non-zero point has the same ratio $V/I$. Reversing the supply reverses both the potential difference and the direction of conventional current, so the line continues into the negative part of the graph with the same shape.

On a graph of current against potential difference,

$\text{gradient}=\frac{\Delta I}{\Delta V}=\frac{1}{R}$

for a straight ohmic line. A steeper line therefore represents a smaller resistance, while a shallower line represents a larger resistance. If the axes are exchanged, the gradient becomes $\Delta V/\Delta I=R$, so the axis labels must always be checked.

Recognising non-linear behaviour

A non-linear circuit element produces a curved I–V characteristic. As the operating point moves along the curve, the ratio $V/I$ changes, showing that the resistance is changing. The changing gradient is another visual sign that one constant relationship cannot describe the whole graph.

Different components produce different curves because their properties respond differently as conditions change. Current may cause heating, for example, while some specialised components respond to temperature or light. The precise shapes and uses of lamps, diodes, thermistors and light-dependent resistors are considered separately; the key idea here is that a curve records changing resistance.

Data for Linear and non-linear I–V characteristics

In the linear series, doubling the potential difference doubles the current and $V/I$ remains constant. In the curved series, the current does not change in a constant proportion to potential difference, so calculations of $V/I$ at different non-zero points give different resistances.

Reading an unfamiliar I–V graph

Use a consistent sequence when analysing a graph:

1. Read the axis labels, units and scales.
2. Decide whether the plot is a straight line or a curve.
3. If it is straight, check whether it passes through the origin.
4. Compare values of $V/I$ at two or more non-zero points.
5. State the conclusion: a constant ratio indicates constant resistance and linear behaviour; a changing ratio indicates changing resistance and non-linear behaviour.

Do not calculate $R$ from the origin itself, because $V/I$ would be $0/0$. Instead, use another point or use the gradient of the straight line. A single non-zero point gives the resistance only at that operating point; several points are needed to decide whether the resistance is constant.

Constant resistance

For an ohmic conductor at constant temperature:

• $I\propto V$;
• doubling $V$ doubles $I$;
• and therefore remain constant;