Parallel RLC Circuit Analysis

Parallel RLC Circuit is the opposite of the serial circuit that we looked at in the previous tutorial, but some of the previous concepts and equations still apply.

However, the analysis of a parallel RLC circuit can be somewhat more mathematically difficult than serial RLC circuits, so in this tutorial about parallel RLC circuits, we will assume the circuit elements used to keep things simple pure and ideal.

This time, instead of being a partner for current circuit components, the applied voltage is now common to everyone, so we need to find branch currents separate from each element. The total impedance of a parallel RLC circuit, Z, is calculated using the current of the circuit, as for a DC parallel circuit, this time the difference is the use of admittans instead of impedance.

Admittans =Admittans refers to complex conductivity in electrical engineering. The impedance product with admittans is 1. Admittans is indicated by Y. Its unit is siemens in the MKS system.

Parallel RLC Circuit

parallel rlc circuit
Parallel RLC Circuit

In the parallel RLC circuit above, we can see that the feed voltage VSis common to all three components, while the feed current consists of three parts. Current flowing from resistance, IR,current flowing from the inductor, IL and current passing through the capacitor, IC.

However, the current flowing from each branch and therefore each component will differ from each other and at the same time from the feed current. The total current drawn from the feed will be the vector sum, not the mathematical sum of three separate branch currents.

Like the serial RLC circuit, we can solve this circuit using the phaser or vector method, but this time the vector diagram will have voltage as a reference with three current vectors drawn according to the voltage. The phaser diagram for a parallel RLC circuit is produced by combining three separate phasers for each component and adding currents vectorly.

Since the voltage in the circuit is common to all three circuit elements, we can use it as a reference vector with three current vectors drawn accordingly at the corresponding angles. The resulting vector current is achieved by combining two of the vectors, IL and IC,and then adding this sum to the remaining vector IR. The resulting angle between V and IS will be the phase angle of the circuits, as shown below.

Phaser Diagram for Parallel RLC Circuit

parallel rlc circuit

From the phaser diagram on the right side above, we can see that the current vectors produce a rectangular triangle consisting of hypotenuse IS,horizontal axis IR and vertical axis IL – IC. Therefore, we can use pythagorean theorem on this current triangle to mathematically obtain the individual sizes of branch currents along the x-axis and y-axis, which, as shown, will determine the total feed current of these components.

Current Triangle for Parallel RLC Circuit

parallel rlc circuit

Since the voltage in the circuit is common to all three circuit elements, the current passing through each branch can be found using Kirchhoff's Current Act (KCL).Note that Kirchhoff's current law or the law of connection states that "the total current entering a connection or node is exactly equal to the current that leaves that node."Thus, the currents entering and exiting the "A" node above are given as follows:

parallel rlc circuit

Taking the derivative, dividing the above equation into C and then rearranging it gives us the equation from the second order below for circuit current.It becomes a quadratic equation because there are two reactive elements in the circuit, inductor and capacitor.

parallel rlc circuit

Opposition to current flow in this type of AC circuit consists of three components: XL XC and R, the combination of these three values, and the circuit impedance, give Z.We know from above that voltage has the same amplitude and phase in all components of a parallel RLC circuit.Then, the impedance on each component can also be mathematically defined according to the current passing through it and the voltage on each element.

Impedance of parallel RLC Circuit

parallel rlc circuit

You may notice that the final equation for a parallel RLC circuit produces complex impedances for each parallel branch, since each element is equivalent to impedance (1/Z).The opposite of impedance is often referred to as admittans(admittance), ( Y ).

In parallel AC circuits, it is usually more convenient to use admittans to solve complex branch impedance, especially when it comes to two or more parallel branch impedances (it helps mathematics).The total admittan of the circuit can be easily found with the addition of parallel admittans.Then the total impedance of the circuit will be Z T , 1/Y T Siemens as shown.

Admittan of parallel RLC Circuit

parallel rlc circuit

The unit of measure now widely used for admittans is Siemens, abbreviated as S (the old unit mho ℧ , the opposite of ohm).Admittans are added to each other in parallel branches, while impedances are collected in series branches.But if we can have a response to impedance, we can also have resistance and reassurance, since the impedance consists of two components, R and X.Then the reciprocity of resistance is called conductivity, and the reciprocity of recess is called susceptance.

Conductivity, Admittans and Susceptance

The units used for conductivity, admittans and susceptance are all the same, which means that siemens ( S ), Ohm or ohm -1can also be considered as equivalent, but the symbol used for each element is different and in a pure component it is given as follows:

Conductivity

parallel rlc circuit

Conductivity is the opposite of resistance, indicated by the symbol R and G.Conductivity is defined as the ease of allowing a resistance to flow when a voltage is applied to ac or DC.

Admittans

parallel rlc circuit

Admittans is the opposite of impedance, indicated by the symbol Z and Y. In AC circuits, admittans are defined as the ease of allowing the current to flow when a voltage is applied, taking into account the phase difference between voltage and current.

The admittany of the parallel circuit is the ratio of the phaser current to the phaser voltage, and the admittans angle is equal to the negative of the impedance.Also, Admittans is the real part of susceptance.

Susceptance

parallel rlc circuit

Susceptance is the opposite of a pure recess, indicated by the symbol X and B. In AC circuits, susceptance is defined as the ease of allowing an alternate current to flow when a voltage of a particular frequency is applied to a recess (or a series of reacques).

Susceptance has the opposite mark of recess, so capacitive sensitivity is positive for C, (+and) is negative, (and) inductive sensitivity B L is negative. In addition, Susceptanceis the virtual part of admittans.

Therefore, we can define inductive and capacitive susceptance as follows:

parallel rlc circuit

Opposition to current flow in AC series circuits is impedance, with two components Z , resistance R and reactase X, and we can create an impedance triangle from these two components.Similarly, in a parallel RLC circuit, admittans Y also has two components, conductivity G and susceptance B. This makes it possible to create an admittans triangle with a horizontal axis of conductivity, G, and Jb, a vertical axis of sensitivity, as shown.

Admittans Triangle for Parallel RLC Circuit

parallel rlc circuit

Now that we have a admittans triangle, we can use Pythagoras to calculate the phase angle as well as the size of all three sides, as shown.

parallel rlc circuit

Then we can define both the input of the circuit and the impedance according to the input as follows:

parallel rlc circuit

Using the power factor angle we can use these trigonometric angles:

parallel rlc circuit

Because the Y-admittan of a parallel RLC circuit is a complex amount, the general impedance for serial circuits corresponds to the Z = R + jX form. For parallel circuits, it will be written as Y = G – jB; Here the real part is G conductivity, and the imaginary part is JB susceptance. In polar form, this will be given as follows:

parallel rlc circuit

Parallel RLC Circuit Question Example 1

A 1kΩ resistance, a 142mH coil and a 160uF capacitor are connected in parallel throughout the 240V, 60Hz feed. Calculate the impedance of the parallel RLC circuit and the current drawn from the feed.

Impedance of parallel RLC Circuit

parallel rlc circuit

Resistance in an AC circuit is not affected by frequency, so R = 1kΩ

Inductive Reactance, ( X L ):

parallel rlc circuit

Capacitive Reactance, ( X C ):

parallel rlc circuit

Impedance, ( Z ):

parallel rlc circuit

Feed Current, ( Is ):

parallel rlc circuit

Parallel RLC Circuit Question Sample 2

A 50Ω resistance, a 20mH coil and a 5uF capacitor are connected in parallel throughout the 50V, 100Hz feed. Calculate the total current drawn from the feed, the current for each branch, the total impedance of the circuit and the phase angle. Also create current and input triangles that represent the circuit.

Parallel RLC Circuit

parallel rlc circuit

1).Inductive Reactance, ( X L ):

parallel rlc circuit

2).Capacitive Reactance, ( X C ):

parallel rlc circuit

3).Impedance, ( Z ):

parallel rlc circuit

4).Current passing through resistance, R ( I R ):

parallel rlc circuit

5).Current passing through the inductor, L ( I L ):

parallel rlc circuit

6).Current passing through the capacitor, C ( I C ):

parallel rlc circuit

7).Total feed current, ( I S ):

parallel rlc circuit

8).Conductivity, ( G ):

parallel rlc circuit

9).Inductive Susceptance, ( B L ):

parallel rlc circuit

10).Capacitive Susceptance, ( B C ):

parallel rlc circuit

11).Admittance, ( Y ):

parallel rlc circuit

12).Phase Angle, (φ) difference between the resulting current and the supply voltage:

parallel rlc circuit

Current and Admittance Triangles

parallel rlc circuit

Parallel RLC Circuit Summary

In a parallel RLC circuit that includes a resistance, an inductor and a capacitor, the circuit current is the sum of phasers consisting of three components, IR, IL and IC, each with a common feeding voltage for all three. Since the supply voltage is common to all three components, it is used as a horizontal reference when creating a current triangle.

Parallel RLC networks can be analyzed using vector diagrams, as in serial RLC circuits.However, the analysis of parallel RLC circuits is slightly more mathematically difficult than serial RLC circuits when they contain two or more current branches.Therefore, an AC parallel circuit can be easily analyzed using the opposite of the impedance called Admittans.

Admittans is the opposite of the impedance given the Y symbol.Like impedance, it is a complex quantity consisting of a real part and a virtual part.The actual part is the opposite of resistance and is represented by the Y symbol, which is called conductivity, while the imaginary part is the opposite of recess and susceptance,represented by the symbol B and expressed in complex form as follows: Y = G + jB is defined as follows: Y = G + jB with duality between the two complexes:

Serial circuitParallel circuit
Voltage, (V)Current, (I)
Resistance, (R)Conductivity, (G)
Reassurance, (X)Susceptance, (B)
Impedance, (Z)Admittans, (H)

Since the voltage is the opposite of recess, in an inductive circuit, the value of B L in inductive sensitivity will be negative, capacitive sensitivity in the capacitive circuit, the value of B C will be positive.The opposite of X L and X C,respectively.

So far, we have seen that serial and parallel RLC circuits contain both capacitive and inductive reacques in the same circuit.If we change the frequency between these circuits, there should be a point where the capacitive reacquance value is equal to the value of the inductive rectal, and therefore X C = X L.

The frequency point at which this occurs is called resonance, and in the next lesson we will look at serial resonance and how its presence changes the properties of the circuit.