Experiment 18 - Formula Internal Resistance

In the experiment 12 a formula for calculating the internal resistance of a battery was introduced. But where does the formula come from? Here you derive it.

For the calculation of the internal resistance \(R_i\) you used two measured values... the voltage of the battery without load \(U_0\) and the current flowing in the load circuit \(I\). Moreover you know the resistance of the load resistor \(R_L\)...

Illustration of the voltage and current in the load circuit.

According to the Kirchhoff's Voltage Law (KVL) in the load circuit... $$U_0 = U_i + U_L$$ and according to the Ohm's Law you can replace \(U_i\) and \(U_L\)... $$U_0 = R_iI_i + R_LI_L.$$ Since this is a simple circuit with no nodes, furthermore the Kirchhoff's Current Law (KCL) applies... $$I = I_i = I_L$$ and so you get... $$U_0 = R_iI + R_LI.$$ Rewriting the formula you get an expression for the internal resistance \(R_i\)... $$R_i = {U_0 \over I} - R_L.$$ All values ​​on the right side are known to you and therefore you can use this formula to calculate the internal resistance of the battery.