Internal resistance (\(r\)) is an inherent characteristic of any real-world voltage source, such as a battery or generator, representing the opposition to current flow within the source itself. This property is responsible for energy loss, typically dissipated as heat during operation. All practical sources possess this measurable internal resistance. Quantifying this resistance is necessary to accurately predict a battery’s performance and the actual power it can deliver to an external circuit.
Defining Internal Resistance and Electromotive Force
Internal resistance is the opposition to the movement of charge within the battery’s chemical components, including the electrolyte, electrodes, and separators. This resistance arises from the physical and chemical processes occurring inside the cell, such as the conductivity of the materials and the speed of the electrochemical reactions. The presence of this internal opposition causes the battery’s voltage to drop when a current is drawn.
The Electromotive Force (EMF, \(\epsilon\)) is the maximum potential difference a battery can provide. This value represents the total energy converted from chemical to electrical form per unit of charge. The EMF is measured when the battery is in an open circuit condition, meaning no current (\(I\)) is flowing. Since no voltage is lost across the internal resistance in this state, the open circuit voltage reading equals the EMF.
Terminal Voltage (\(V\)) is the actual potential difference measured across the battery’s terminals when current is flowing to an external circuit. This measured voltage is always less than the EMF because energy is lost as the current passes through the internal resistance. The difference between the EMF and the Terminal Voltage is the voltage drop that occurs solely across the internal resistance, often called the “lost volts.”
The Core Mathematical Relationship
The relationship between EMF (\(\epsilon\)), Terminal Voltage (\(V\)), and internal resistance (\(r\)) is governed by energy conservation. When the battery supplies current (\(I\)), the EMF must equal the sum of the Terminal Voltage (\(V\)) delivered to the external circuit plus the voltage drop across the internal resistance. This internal voltage drop is calculated using Ohm’s Law as \(Ir\).
This relationship is expressed by the fundamental equation: \(\epsilon = V + Ir\). The term \(Ir\) represents the voltage lost inside the battery due to its internal resistance. This equation confirms that the maximum voltage (\(\epsilon\)) is only available when the current (\(I\)) is zero, which is the open circuit condition.
To calculate the internal resistance, the fundamental equation is rearranged to isolate \(r\), yielding the direct calculation formula: \(r = (\epsilon – V) / I\). For example, if a battery’s EMF (\(\epsilon\)) is \(1.5\) volts, and under load, the Terminal Voltage (\(V\)) drops to \(1.4\) volts while drawing \(0.5\) amps (\(I\)), the internal resistance is \(0.2\) ohms.
Practical Measurement Techniques
Two-Measurement Load Test
One straightforward experimental method is the two-measurement load test, which directly applies the derived formula. First, measure the open circuit voltage using a voltmeter to determine the EMF (\(\epsilon\)). Next, connect the battery in series with a known external load resistor (\(R\)) and an ammeter to measure the current (\(I\)). Reconnect the voltmeter in parallel across the battery terminals to measure the Terminal Voltage (\(V\)) under load.
The measured values for \(\epsilon\), \(V\), and \(I\) are then substituted into the formula \(r = (\epsilon – V) / I\) to find the internal resistance. The circuit should be closed only briefly to prevent the battery from discharging significantly and altering its internal resistance. This method provides a quick snapshot of the internal resistance for that specific load condition.
Graphical Analysis
A more accurate technique involves graphical analysis, which averages out experimental errors using multiple data points. This method requires connecting the battery in series with an ammeter and a variable resistor, with a voltmeter connected in parallel across the terminals. By adjusting the variable resistor, a series of different current (\(I\)) and terminal voltage (\(V\)) pairs are recorded across a wide range of loads.
The collected data is plotted on a graph with Terminal Voltage (\(V\)) on the y-axis and Current (\(I\)) on the x-axis. The equation \(V = \epsilon – Ir\) is recognized as the equation for a straight line. The y-intercept represents the EMF (\(\epsilon\)), and the slope (\(m\)) is the negative of the internal resistance (\(-r\)). Calculating the absolute value of the slope of the line of best fit provides a reliable measurement of the battery’s internal resistance.