Question 5:
A single ac circuit containing resistor and
inductor connected parallel across 230 V
supply, calculate current if admidance of the
circuit 0.012 mho.
2.76 A
1.76 A
4.55 A
3.55 A
Answers
Answer:
Parallel RLC Circuit Analysis
The Parallel RLC Circuit is the exact opposite to the series circuit we looked at in the previous tutorial although some of the previous concepts and equations still apply.
However, the analysis of a parallel RLC circuits can be a little more mathematically difficult than for series RLC circuits so in this tutorial about parallel RLC circuits only pure components are assumed in this tutorial to keep things simple.
This time instead of the current being common to the circuit components, the applied voltage is now common to all so we need to find the individual branch currents through each element. The total impedance, Z of a parallel RLC circuit is calculated using the current of the circuit similar to that for a DC parallel circuit, the difference this time is that admittance is used instead of impedance. Consider the parallel RLC circuit below.
Parallel RLC Circuit
parallel rlc circuit
In the above parallel RLC circuit, we can see that the supply voltage, VS is common to all three components whilst the supply current IS consists of three parts. The current flowing through the resistor, IR, the current flowing through the inductor, IL and the current through the capacitor, IC.
But the current flowing through each branch and therefore each component will be different to each other and also to the supply current, IS. The total current drawn from the supply will not be the mathematical sum of the three individual branch currents but their vector sum.
Like the series RLC circuit, we can solve this circuit using the phasor or vector method but this time the vector diagram will have the voltage as its reference with the three current vectors plotted with respect to the voltage. The phasor diagram for a parallel RLC circuit is produced by combining together the three individual phasors for each component and adding the currents vectorially.
Since the voltage across the circuit is common to all three circuit elements we can use this as the reference vector with the three current vectors drawn relative to this at their corresponding angles. The resulting vector current IS is obtained by adding together two of the vectors, IL and IC and then adding this sum to the remaining vector IR. The resulting angle obtained between V and IS will be the circuits phase angle as shown below.
Phasor Diagram for a Parallel RLC Circuit
parallel rlc circuit phasor diagram
We can see from the phasor diagram on the right hand side above that the current vectors produce a rectangular triangle, comprising of hypotenuse IS, horizontal axis IR and vertical axis IL – IC Hopefully you will notice then, that this forms a Current Triangle. We can therefore use Pythagoras’s theorem on this current triangle to mathematically obtain the individual magnitudes of the branch currents along the x-axis and y-axis which will determine the total supply current IS of these components as shown.
Current Triangle for a Parallel RLC Circuit
Current Triangle for a Parallel RLC Circuit
Since the voltage across the circuit is common to all three circuit elements, the current through each branch can be found using Kirchhoff’s Current Law, (KCL). Rember that Kirchhoff’s current law or junction law states that “the total current entering a junction or node is exactly equal to the current leaving that node”. Thus the currents entering and leaving node “A” above are given as:
kirchoffs current law
Taking the derivative, dividing through the above equation by C and then re-arranging gives us the following Second-order equation for the circuit current. It becomes a second-order equation because there are two reactive elements in the circuit, the inductor and the capacitor.