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Parallel RLC Circuit Analysis

Parallel RLC Circuit Analysis

A parallel resonant circuit consists of a parallel R-L-C combination in parallel with an applied current source

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 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.

second order equation

 

The opposition to current flow in this type of AC circuit is made up of three components: XL XC and R with the combination of these three values giving the circuits impedance, Z. We know from above that the voltage has the same amplitude and phase in all the components of a parallel RLC circuit. Then the impedance across each component can also be described mathematically according to the current flowing through, and the voltage across each element as.

Impedance of a Parallel RLC Circuit

impedance of a parallel rlc circuit

 

You will notice that the final equation for a parallel RLC circuit produces complex impedance’s for each parallel branch as each element becomes the reciprocal of impedance, ( 1/Z ). The reciprocal of impedance is commonly called Admittance, symbol ( Y ).

In parallel AC circuits it is generally more convenient to use admittance to solve complex branch impedance’s especially when two or more parallel branch impedance’s are involved (helps with the math’s). The total admittance of the circuit can simply be found by the addition of the parallel admittances. Then the total impedance, ZT of the circuit will therefore be 1/YT Siemens as shown.

Admittance of a Parallel RLC Circuit

admittance of a parallel rlc circuit

 

The unit of measurement now commonly used for admittance is the Siemens, abbreviated as S, ( old unit mho’s , ohm’s in reverse ). Admittances are added together in parallel branches, whereas impedance’s are added together in series branches. But if we can have a reciprocal of impedance, we can also have a reciprocal of resistance and reactance as impedance consists of two components, R and X. Then the reciprocal of resistance is called Conductance and the reciprocal of reactance is called Susceptance.

Conductance, Admittance and Susceptance

The units used for conductance, admittance and susceptance are all the same namely Siemens ( S ), which can also be thought of as the reciprocal of Ohms or ohm-1, but the symbol used for each element is different and in a pure component this is given as:

Admittance ( Y ) :

Admittance is the reciprocal of impedance, Z and is given the symbol Y. In AC circuits admittance is defined as the ease at which a circuit composed of resistances and reactances allows current to flow when a voltage is applied taking into account the phase difference between the voltage and the current.

The admittance of a parallel circuit is the ratio of phasor current to phasor voltage with the angle of the admittance being the negative to that of impedance.

Admittance

Conductance ( G ) :

Conductance is the reciprocal of resistance, R and is given the symbol G. Conductance is defined as the ease at which a resistor (or a set of resistors) allows current to flow when a voltage, either AC or DC is applied.

Conductance

Susceptance ( B ) :

Susceptance is the reciprocal of of a pure reactance, X and is given the symbol B. In AC circuits susceptance is defined as the ease at which a reactance (or a set of reactances) allows an alternating current to flow when a voltage of a given frequency is applied.

Susceptance has the opposite sign to reactance so Capacitive susceptance BC is positive, (+ve) in value while Inductive susceptance BL is negative, (-ve) in value.

susceptance

We can therefore define inductive and capacitive susceptance as being:

susceptance definition

In AC series circuits the opposition to current flow is impedance, Z which has two components, resistance R and reactance, X and from these two components we can construct an impedance triangle. Similarly, in a parallel RLC circuit, admittance, Y also has two components, conductance, G and susceptance, B. This makes it possible to construct an admittance triangle that has a horizontal conductance axis, G and a vertical susceptance axis, jB as shown.

Admittance Triangle for a Parallel RLC Circuit

admittance triangle for a parallel rlc circuit

 

Now that we have an admittance triangle, we can use Pythagoras to calculate the magnitudes of all three sides as well as the phase angle as shown.

from Pythagoras

parallel circuit admittance

Then we can define both the admittance of the circuit and the impedance with respect to admittance as:

admittance and impedance of the circuit

Giving us a power factor angle of:

power factor for admittance

 

As the admittance, Y of a parallel RLC circuit is a complex quantity, the admittance corresponding to the general form of impedance Z = R + jX for series circuits will be written as Y = G – jB for parallel circuits where the real part G is the conductance and the imaginary part jB is the susceptance. In polar form this will be given as:

admittance in polar form

Parallel RLC Circuit Example No1

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

Impedance of a Parallel RLC Circuit

impedance of parallel rlc circuit

In an AC circuit, the resistor is unaffected by frequency therefore R = 1kΩ

Inductive Reactance, ( XL ):

inductive reactance of inductor

Capacitive Reactance, ( XC ):

capacitive reactance of capacitor

Impedance, ( Z ):

impedance of parallel rlc circuit

Supply Current, ( Is ):

supply current

Parallel RLC Circuit Example No2

A 50Ω resistor, a 20mH coil and a 5uF capacitor are all connected in parallel across a 50V, 100Hz supply. Calculate the total current drawn from the supply, the current for each branch, the total impedance of the circuit and the phase angle. Also construct the current and admittance triangles representing the circuit.

Parallel RLC Circuit

parallel rlc circuit for question 1

1). Inductive Reactance, ( XL ):

inductive reactance

2). Capacitive Reactance, ( XC ):

capacitive reactance

3). Impedance, ( Z ):

circuit impedance

4). Current through resistance, R ( IR ):

resistance current

5). Current through inductor, L ( IL ):

Inductor Current

6). Current through capacitor, C ( IC ):

capacitor current

7). Total supply current, ( IS ):

parallel circuit current

8). Conductance, ( G ):

circuit conductance

9). Inductive Susceptance, ( BL ):

inductive susceptance

10). Capacitive Susceptance, ( BC ):

capacitive susceptance

11). Admittance, ( Y ):

circuit admittance

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

circuit phase angle

Current and Admittance Triangles

current and admittance triangle

Parallel RLC Circuit Summary

In a parallel RLC circuit containing a resistor, an inductor and a capacitor the circuit current IS is the phasor sum made up of three components, IR, IL and IC with the supply voltage common to all three. Since the supply voltage is common to all three components it is used as the horizontal reference when constructing a current triangle.

Parallel RLC networks can be analysed using vector diagrams just the same as with series RLC circuits. However, the analysis of parallel RLC circuits is a little more mathematically difficult than for series RLC circuits when it contains two or more current branches. So an AC parallel circuit can be easily analysed using the reciprocal of impedance called Admittance.

Admittance is the reciprocal of impedance given the symbol, Y. Like impedance, it is a complex quantity consisting of a real part and an imaginary part. The real part is the reciprocal of resistance and is called Conductance, symbol Y. The imaginary part is the reciprocal of reactance and is called Susceptance, symbol B and expressed in complex form as: Y = G + jB  with the duality between the two complex impedance’s being defined as:

Series Circuit Parallel Circuit
Voltage, (V) Current, (I)
Resistance, (R) Conductance, (G)
Reactance, (X) Susceptance, (B)
Impedance, (Z) Admittance, (Y)

As susceptance is the reciprocal of reactance, in an inductive circuit, inductive susceptance, BL will be negative in value and in a capacitive circuit, capacitive susceptance, BC will be positive in value. The exact opposite to XL and XC respectively.

We have seen so far that series and parallel RLC circuits contain both capacitive reactance and inductive reactance within the same circuit. If we vary the frequency across these circuits there must become a point where the capacitive reactance value equals that of the inductive reactance and therefore, XC = XL.

The frequency point at which this occurs is called resonance and in the next tutorial we will look at series resonance and how its presence alters the characteristics of the circuit.

202 Comments

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  • JEAN BOSCO BUGIRUWENDA

    thank you for your content it is helpful

  • Ahmad

    Data given for Example No1: R = 500Ω, L = 7mH,= 15nF,

  • PROFESSOR BARNABAS J. KWAHA

    This write-up and the analysis contained in it, including the illustrations are quite elaborate and educative. I enjoyed using it to teach. Well done and God bless you. Many thanks and regards.

  • Abdurahaman Mohammed

    Keep up the good work

  • Saeed Ahmad

    Can we have comparison of RLC with Phasor diagram of Series RLC & Parallel RLC circuits ?
    Thanks

  • Jyotimoni Chaudang

    Very helpful

  • Tanish

    Perfection. Very very helpful.
    Can you also do one or two examples of the parallel LCR circuits where RC are in series, and overall parallel with another branch with RL in series.

  • Mary

    I love this and it’s helpful

  • wasily

    TAK

  • Raufu Kazeem

    Your tutorial is very much explanatory and interesting one .you keep it up . Thanks

  • Ndambi

    It was useful

  • Ir. JGM van der Zanden

    Example 1: Z = 24,0 Ohm should be Z = 23,0 Ohm

    Example 2: Z = 12,7 should be Z = 12,91 Ohm

    • Wayne Storr

      Well let’s look at your calculations and see if your abacus is the same as ours.

      Data given for Example No1: R = 1kΩ, L = 142mH, therefore: XL = 53.54Ω, C = 160uF, therefore: XC = 16.58Ω, as given in the tutorial.

      Circuit impedance (Z) at 60Hz is therefore:

      Z = 1/sqr-root( (1/R)2 + (1/XL – 1/Xc)2)
      = 1/sqr-root( 0.000001 + 0.001734) = 1/0.04165 = 24.01Ω

      Thus at 60Hz supply frequency, the circuit impedance Z = 24Ω (rounded to nearest integer value). Then the tutorial is correct as given.

      Data given for Example No2: R = 50Ω, L = 20mH, therefore: XL = 12.57Ω, C = 5uF, therefore: XC = 318.27Ω, as given in the tutorial.

      Circuit impedance (Z) at 100Hz is therefore:

      Z = 1/sqr-root( (1/R)2 + (1/XL – 1/Xc)2)
      = 1/sqr-root( 0.0004 + 0.005839) = 1/0.07899 = 12.66Ω

      Thus at 100Hz supply frequency, the circuit impedance Z = 12.7Ω (rounded off to the first decimal point). Then the tutorial is correct as given.

  • Adam

    Fantastic

  • Shuaib

    Very understanding

  • Mapalo mwansa

    This is very useful

  • Haiku

    Every parallel RLC circuit acts like a band-pass filter. What happens to this band if I connect two of them in series? Does it widens or tightens?

    • Wayne Storr

      Firstly, a parallel RLC circuit does not “act like a band-pass filter”, it behaves more like a band-stop circuit to current flow as the voltage across all three circuit elements R, L, and C is the same, but supply currents divides among the components in proportion to their conductance/susceptance. At the resonant frequency, (fr) the circuits complex impedance increases to equal R.

      Secondly, any number of parallel resistances and reactances can be combined together to form a parallel RLC circuit. The total equivalent resistive branch, R(t) will equal the resistive value of all the resistors in parallel. The total equivalent impedance of the inductive branch, XL(t) will be equal to all the inductive reactances, (XL). Similarly, the total capacitance will be equal to the sum of the capacitive reactances, XC(t) in parallel.

      Clearly, the resosnant frequency point will be determined by the individual values of the R, L and C components used. The Q of the inductances will determine the Q of the parallel circuit, because it is generally less than the Q of the capacitive branch. The resulting bandwidth can be calculated as: fr/Q or 1/(2piRC) Hz.

  • Ngang ransome

    Thanks sir this was helpful

    I appreciate 💯

  • raphael

    please i need a full definition of all thius phasor diagrams

  • Simeon Bisar

    Really need to understand RLC for my exams

  • Abdullahi Hassan Bello

    Transfer function of the above circuit