The Admittance Method is a technique used to analyze parallel AC circuits. Admittance (Y) is the reciprocal of impedance (Z), and it is a complex quantity that includes both magnitude and phase information. Just like impedance, admittance is used to describe how a circuit responds to AC signals.
To solve a parallel AC circuit using the Admittance Method, follow these steps:
Identify the Components: Determine the components in the parallel AC circuit. These could include resistors (R), capacitors (C), and inductors (L).
Convert Components to Admittance: Convert each component's impedance to its corresponding admittance. Use the following formulas:
For a resistor:
=
1
Y
R
â
=
R
1
â
For a capacitor:
=
Y
C
â
=jĎC
For an inductor:
=
1
Y
L
â
=
jĎL
1
â
Where
j is the imaginary unit,
Ď is the angular frequency (equal to
2
2Ď times the frequency),
C is the capacitance, and
L is the inductance.
Write Kirchhoff's Current Law (KCL): Write down the KCL equation for the parallel circuit. The sum of all currents entering a node must be zero.
Express Currents in Terms of Admittances: Replace the currents in the KCL equation with the corresponding admittances. This will give you an equation in terms of admittances.
Solve for Total Admittance: Sum up all the admittances to find the total admittance (
total
Y
total
â
) of the parallel circuit.
Find Total Impedance: Take the reciprocal of the total admittance to find the total impedance (
total
Z
total
â
) of the circuit:
total
=
1
total
Z
total
â
=
Y
total
â
1
â
.
Find Total Current: If a voltage source is connected to the circuit, use Ohm's Law (
=
total
I=
Z
total
â
V
â
) to find the total current (
I) flowing through the circuit.
Calculate Branch Currents: Use the total current and the individual admittances to calculate the currents through each branch of the parallel circuit.
Find Voltages and Phasors: Calculate the voltage across each component using Ohm's Law or the appropriate impedance/admittance relationship. These voltages will be complex quantities with both magnitude and phase.
Verify KCL and KVL: Check that Kirchhoff's Current Law and Kirchhoff's Voltage Law are satisfied in the circuit. The sum of the currents at any node should be zero, and the sum of the voltages around any closed loop should be zero.
Remember that when working with complex quantities, it's important to consider both the magnitude and phase of voltages, currents, and impedances/admittances. You can represent these quantities in either rectangular (Cartesian) or polar (phasor) form.
The Admittance Method simplifies the analysis of parallel AC circuits by using admittances, which are easier to manipulate in algebraic calculations compared to impedances.