Current Division Rule
When current flows through more than one parallel path, each of the paths shares a definite portion of the total current depending upon the impedance of that path.
The definite portion of the total current shared by any of the parallel paths can easily be calculated if the impedance of that path and the equivalent impedance of the parallel system are known to us.
The rule or formula derived from these known impedances to know the portion of total current through any parallel path is known as the current divider rule. This rule is very important and widely used in the field of electric engineering in different applications.
Actually, this rule finds application when we have to find the current passing through each impedance when these are connected in parallel.
Let us say, two impedances Z1 and Z2 are connected in parallel as shown below.
A current I passes and is being divided into I1 and I2 at the junction of these two impedances as shown. I1 and I2 pass through Z1 and Z2 respectively. Our aim is to determine I1 and I2 in terms of I, Z1, and Z2.
As Z1 and Z2 are connected in parallel, the voltage drop across each will be the same. Hence, we can write
Also applying Kirchoff’s current law at the junction, we get
We have two equations and can determine I1 and I2.
From (1), we have
Putting this in (2), we get
Putting the value of I1, we get
Thus, we have determined I1 and I2 in terms of I, Z1, and Z2.
This rule is applied as follows.
Suppose we have to determine I1. We proceed as
Applying the above rule, we will get
Let us apply this rule to some problems.
Applying the current division rule, we will have
Where I1 = current passing through Z1.
Putting given numerical values, we get
The other way to find I2 is as
This is how we can apply the current division rule.
Voltage Division Rule
The voltage division rule is applied when we have to find the voltage across some impedance. Let us assume that the impedances Z1, Z2, Z3,…..Zn are connected in series, and the voltage source (V) is connected across them.
This is shown in the voltage divider circuit below:
Our aim is to find the voltage across some impedance, say, Z3. We see that Z1, Z2, Z3 …. Zn are connected in series. Hence, effective impedance Zeff as seen by the voltage is given by
The current passing the circuit is given by
This current is passing through all the impedances connected in series. Hence, the voltage across Z3 is given by
Similarly, the voltage across Z1 will be given by
In general, we can write
Where, k = 1, 2, 3,….n and impedances Z1, Z2, Z3 ,…….Zn should be connected in series.
This is called the voltage division rule and frequently used to determine the voltage across some impedance. We can write this rule in words as given below.
The voltage across some impedance
We will solve one problem of finding voltages across impedances using the voltage division rule.
Voltage Division Rule Example Problem
The following impedances are connected in series:
Across this impedance connected in series, a voltage source of 100V is connected as shown below. Determine the voltage across each impedance.
Applying the voltage division rule, we get
We can also determine Vz3 as follows.
Actually, we can determine the voltage across any impedance in this way if voltages across all other remaining impedances are known.
With the impedances equal to:
The voltage across each impedance is given by:
Thus voltage will be the same across each impedance and it equals V/n, that is, source voltage divided by the number of impedances connected in series.