When current flows through more than one parallel paths, each of the paths shares a definite porion of the total current depending upon the impedance of that path. The definite portion of 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 current division 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, voltage drop across each will be same. Hence, we can write
Also applying Kirchoff’s current law at junction, we get
We have two equations and can determine I1 and I2.
From (1), we have
Putting this in (2), we get
Putting 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 above rule, we will get
Let us apply this rule to some problems.
Applying current divison 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 current division rule.
Voltage Division Rule
Voltage division rule is applied when we have to find voltage across some impedance. Let us assume that the impedances Z1, Z2, Z3,…..Zn are connected in series and voltage source V is connected across them as shown below.
Our aim is to find 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
Current passing the circuit is given by
This current is passing through all the impedances connected in series. Hence, voltage across Z3 is given by
Similarly, 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 voltage division rule and frequently used to determine the voltage across some impedance. We can write this rule in words as given below.
Voltage across some impedance
We will solve one problem of finding voltages across impedances using voltage divisio rule. Problem
impedance are connected in series. Across these impedance connected in series, a voltage source of 100V is connected as shown below. Determine the voltage across each impedance.
Applying voltage division rule, we get
We can also determine Vz3 as follows.
Actually, we can determine voltage across any impedance in this way if voltages across all other remaining impedances are known.
When we , voltage across each impedance is given by
Thus voltage will be same across each impedance and it equals V/n, that is, source voltage divided number of impedances connected in series.