Current Divider Rule

For series circuits we have the powerful voltage divider rule for finding the voltage across a resistor in a series circuit. We now introduce the equally powerful current divider rule (CDR) for finding the current through a resistor in a parallel circuit.

What is Current Divider Rule?

In general, In the previous section, it was pointed out that current will always seek the path of least resistance. In Fig. 1, for example, the current of $9 A$ is faced with splitting between the three parallel resistors. Based on the previous sections, it should now be clear without a single calculation that the majority of the current will pass through the smallest resistor of $10 Ω$, and the least current will pass through the $1 kΩ$ resistor. In fact, the current through the $100 Ω$ resistor will also exceed that through the $1 kΩ$ resistor. We can take it one step further by recognizing that the resistance of the $100 Ω$ resistor is 10 times that of the $10 Ω$ resistor. The result is a current through the $10 Ω$ resistor that is 10 times that of the 100 Ω resistor. Similarly, the current through the $100 Ω$ resistor is 10 times that through the $1 kΩ$ resistor. Ratio Rule: Each of the panel statements above is supported by the ratio rule, which states that for parallel resistors the current will divide as the inverse of their resistor values. In equation form: The next example will demonstrate how quickly currents can be determined using this important relationship. Although the above discussions and example allowed us to determine the relative magnitude of a current based on a known level, they do not provide the magnitude of a current through a branch of a parallel network if only the total entering current is known. The result is a need for the current divider rule, which will be derived using the parallel configuration in Fig. 3(a). The current $I_T$ (using the subscript T to indicate the total entering current) splits between the N parallel resistors and then gathers itself together again at the bottom of the configuration. In Fig. 3(b), the parallel combination of resistors has been replaced by a single resistor equal to the total resistance of the parallel combination as determined in the previous sections. The current $I_T$ can then be determined using Ohm's law: Since the voltage V is the same across parallel elements, the following is true: where the product $I_x R_x$ refers to any combination in the series. Substituting for $V$ in the above equation for $I_T$, we have Solving for $I_x$, the final result is the current divider rule: which states that

Special Case: Two Parallel Resistors

For the case of two parallel resistors, the total resistance is determined by Substituting $R_T$ into Eq. (2) for current $I_1$ results in and Similarly, for $I_2$,