Electrical systems that use alternating current (AC) introduce complexities. AC circuits contain elements like inductors and capacitors that store and release energy, causing the current and voltage to constantly shift relative to each other.
To accurately analyze how an AC circuit behaves, electrical engineers use specialized metrics that account for both the magnitude and the phase relationship of the current flow. These metrics move beyond simple resistance to provide a complete picture of how easily a circuit facilitates current flow.
Admittance is one such metric developed to simplify the analysis of these dynamic circuit conditions.
Defining Admittance and Reciprocal Relationship with Impedance
Admittance (\(Y\)) measures how easily a circuit allows alternating current to flow when a voltage is applied. It is the inverse of impedance (\(Z\)), which measures the total opposition to current flow in an AC circuit. The formal relationship between the two metrics is expressed as \(Y = 1/Z\).
The unit of measurement for admittance is the siemens (S). This unit replaced the older, synonymous unit known as the mho. Since impedance is a complex number comprising resistance and reactance, admittance must also be treated as a complex number to fully account for both the energy-dissipating and energy-storing properties of the circuit.
The phase angle of admittance is the negative of the phase angle of impedance. A highly conductive circuit will have a large admittance value and a low impedance value. Conversely, a circuit that strongly opposes current flow will have a low admittance and a high impedance. This inverse relationship is fundamental to understanding AC flow.
The Components of Admittance: Conductance and Susceptance
Since admittance is a complex number, it is composed of two distinct parts: a real part and an imaginary part, represented by the formula \(Y = G + jB\). The real component is called conductance (\(G\)), and the imaginary component is called susceptance (\(B\)). The operator \(j\) is the standard symbol used in electrical engineering to denote the imaginary unit \(\sqrt{-1}\).
Conductance (\(G\))
Conductance is the real part of admittance and relates directly to energy dissipation within the circuit, primarily as heat. Conductance quantifies the ease with which current flows in phase with the applied voltage, representing the useful work done by the circuit. A higher conductance value indicates a greater ability of the circuit to dissipate real power.
Susceptance (\(B\))
Susceptance is the imaginary part of admittance and represents the ability of the circuit to store and return energy to the source. This energy storage occurs in the form of electric fields within capacitors or magnetic fields within inductors. Susceptance accounts for the current component that is 90 degrees out of phase with the voltage, which is known as reactive power.
The sign of susceptance differentiates between the two energy-storing components: capacitive susceptance is conventionally positive, while inductive susceptance is negative. This sign convention ensures that the opposing effects on the total reactive power can be easily summed and analyzed. Separating admittance into these two components allows engineers to clearly distinguish between the energy that is consumed (conductance) and the energy that is exchanged (susceptance).
Utility of Admittance in Parallel AC Circuits
The primary practical application of admittance is simplifying the analysis of parallel AC circuits. While calculating the total impedance of parallel components requires complex reciprocal calculations, admittance allows for straightforward linear addition.
In a parallel AC circuit, the total admittance is found by summing the individual branch admittances: \(Y_{total} = Y_1 + Y_2 + …\). This makes calculating the overall current flow in complex parallel configurations significantly easier than using impedance.
To find the total current flowing into a parallel network, engineers calculate the total admittance and use a rearranged version of Ohm’s Law for AC circuits, \(I = V \cdot Y_{total}\). This method is favored in disciplines like power systems engineering, as it quickly determines the combined effect of multiple parallel elements.