draw admittance triangle for inductive circuit
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Circuit GlobeCircuit TheoryAdmittance Method
Admittance Method
Admittance method is used for solving parallel AC circuits. The admittance shows the reliability of the electrical circuit to allow the electric current to pass through it. First of all, we must know the meanings of some terms used in the Admittance Method.
Admittance
The reciprocal of the impedance of an AC circuit is known as Admittance of the circuit. Since impedance is the total opposition offered to the flow of alternating current in an AC circuit. Therefore, Admittance is defined as the effective ability of the circuit due to which it allows the alternating current to flow through it. It is represented by (Y). The old unit of admittance is mho (Ʊ).Its new unit is Siemens. The circuit having an impedance of one ohm has an admittance of one Siemens. The old unit was mho.
admittance-eq1
Contents:
Admittance
Application of Admittance Method
Steps for Solving Circuit by Admittance Method
Admittance Triangle
Conductance
Susceptance
Application of Admittance Method
Consider the 3-branched circuit shown in the figure below. Total conductance is found by merely adding the conductance of three branches. Similarly, total susceptance is found by algebraically adding the individual susceptance of different branches.
application-of-admittanceTotal conductance G = g1 + g2 + g3 +…..
Total susceptance B = (-b1) + (-b2) + b3….
Total admittance Y = ( G2 + B2)
Total current I = VY ; Power Factor cosΦ = G / Y
Steps for Solving Circuit by Admittance Method
Consider a parallel AC circuit having resistance and capacitance connected in series and resistance and inductance also connected in series as shown in the figure below.
admittance-method-figureStep 1 – Draw the circuit as per the given problem.
Step 2 – Find impedance and phase angle of each branch.
admittance-eq2
Step 3 – Now, find Conductance, Susceptance and Admittance of each branch.
admittance-eq3
Step 4 – Find the algebraic sum of conductance and susceptance.
admittance-eq4
Step 5 – Find the total Admittance (Y) of the circuit.
admittance-eq5
Step 6 – Find the various branch currents of the circuit.
admittance-eq6
Step 7 – Now, find the total current I of the circuit.
admittance-eq7
Step 8 – Find the phase angle of the whole circuit.
admittance-eq8
Phase angle will be lagging if B is negative.
Step 9 – Now, find the power factor of the circuit.
admittance-eq9
Admittance Triangle
Admittance triangle is also represented similarly to impedance triangle. As the impedance (Z) of the circuit has two rectangular components, resistance (R) and reactance (X). Similarly, the admittance (Y) also has two components, conductance (g) and susceptance (b). The Admittance triangle is shown below.
Admittance-triangle-phasor-diagram
Conductance
Answer:
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Explanation:
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 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 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 and we can therefore use Pythagoras’s theorem on this current triangle to mathematically obtain the magnitude of the branch currents along the x-axis and y-axis and then determine the total 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 Kirchoff’s Current Law, (KCL). Kirchoff’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”, so 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 rearranging 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 ) with the reciprocal of impedance being called Admittance.
In parallel AC circuits it is more convenient to use admittance, symbol ( Y ) 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 new unit 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.
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