**Resistance, Inductance, and Capacitance in Parallel Circuit**

The characteristic of a **parallel circuit** is that the same voltage appears across all parallel branches. We use this common voltage as the reference phasor in phasor diagrams for any parallel AC circuits. **Ohm’s law** then gives the current through each branch of the circuit in **Figure 1**.

**Figure 1** A parallel circuit containing resistance, inductance, and capacitance

For the resistance,

\[{{\text{I}}_{\text{R}}}=\frac{E\angle {{0}^{o}}}{R\angle {{0}^{o}}}=\frac{E}{R}\angle {{0}^{o}}\]

Therefore, **I**_{R} is in phase with the applied voltage.

For the inductance,

\[{{\text{I}}_{\text{L}}}=\frac{E\angle {{0}^{o}}}{{{X}_{L}}\angle +{{90}^{o}}}=\frac{E}{{{X}_{L}}}\angle -{{90}^{o}}\]

And **I**_{L} lags behind the reference voltage by 90°.

For the capacitance,

\[{{\text{I}}_{\text{C}}}=\frac{E\angle {{0}^{o}}}{{{X}_{C}}\angle -{{90}^{o}}}=\frac{E}{{{X}_{C}}}\angle +{{90}^{o}}\]

And **I**_{C} leads the reference voltage by 90°.

The total current in the parallel circuit is the phasor sum of the branch currents:

\[\begin{matrix}{{I}_{T}}={{I}_{R}}-j{{I}_{L}}+j{{I}_{C}}={{I}_{R}}+j\left( {{I}_{C}}-{{I}_{L}} \right) & {} & \left( 1 \right) \\\end{matrix}\]

Converting **Equation 1** to polar coordinates gives

$\begin{matrix}{{\text{I}}_{\text{T}}}=\sqrt{I_{R}^{2}+I_{X}^{2}}\angle {{\tan }^{-1}}\frac{{{I}_{X}}}{{{I}_{R}}} & {} & \left( 2 \right) \\\end{matrix}$

Where I_{X} is the net reactive current I_{C} – I_{L}.

We can apply the following equation to determine the equivalent impedance of the parallel circuit:

\[\begin{matrix}{{\text{Z}}_{\text{eq}}}\text{=}\frac{\text{E}}{{{\text{I}}_{\text{T}}}} & {} & \left( 3 \right) \\\end{matrix}\]

This method of solving for the equivalent impedance of a parallel circuit is called the total-current method. If the applied voltage is not known, we can assume any convenient value in order to find the equivalent impedance.

**Conductance, Susceptance, and Admittance**

For parallel DC circuits, it is convenient to think in terms of conductance, the reciprocal of resistance. Conductance is a measure of how easily a DC circuit passes current. We can use a similar approach for AC circuits. Since **I = E/Z**, we can rewrite **Equation 1** as

\[\frac{E\angle {{0}^{o}}}{\text{Z}}=\frac{E}{\text{Z}}=\frac{E}{R}-j\frac{E\angle {{0}^{o}}}{{{X}_{L}}}+j\frac{E\angle {{0}^{o}}}{{{X}_{C}}}\]

Dividing by E gives

\[\begin{matrix}\frac{1}{\text{Z}}=\frac{1}{R}-j\frac{1}{{{X}_{L}}}+j\frac{1}{{{X}_{C}}} & {} & \left( 4 \right) \\\end{matrix}\]

In this equation, we can replace 1/R with conductance, G. There are similar reciprocals for impedance and reactance.

Admittanceis the overall ability of an electric circuit to pass alternating current. The symbol for admittance isY. Admittance is the reciprocal of impedance:Y = 1/Z

Susceptanceis the ability of inductance or capacitance to pass alternating current. The letter symbol for susceptance isB. Susceptance is the reciprocal of reactance:B = 1/X

The SI unit for both admittance and susceptance is the Siemens, the same as for conductance.

When we divide 1∠0° by a phasor quantity with a +90° angle, the quotient has a -90° angle. Hence, the reciprocal of +jX_{L} is -jB_{L}. Similarly, the reciprocal of -jX_{C} is +jB_{C}. Thus, substituting admittance, conductance, and susceptance into **Equation 4** gives

$\begin{matrix}Y=G-j{{B}_{L}}+j{{B}_{C}}=G+j\left( {{B}_{C}}-{{B}_{L}} \right) & {} & \left( 5 \right) \\\end{matrix}$

The polar coordinates of admittance are

\[\begin{matrix}Y=\sqrt{{{G}^{2}}+B_{eq}^{2}}\angle {{\tan }^{-1}}\frac{{{B}_{eq}}}{G} & {} & \left( 6 \right) \\\end{matrix}\]

Where B_{eq }is the net equivalent susceptance, B_{C} – B_{L}.

Since B_{L} = 1/X_{L},

\[\begin{matrix}{{B}_{L}}=\frac{1}{2\pi fL}=\frac{1}{\omega L} & {} & \left( 7 \right) \\\end{matrix}\]

Similarly,

$\begin{matrix}{{B}_{C}}=2\pi fC=\omega C & {} & \left( 8 \right) \\\end{matrix}$

**Summary**

• In a parallel AC circuit, the total current is the phasor sum of all the branch currents.

• Admittance is the overall ability of an electric circuit to pass alternating current.

• Susceptance is a measure of the ability of inductance or capacitance to pass alternating current.

• Admittance is the reciprocal of impedance.

• The rectangular coordinates of impedance are resistance and reactance.

•The rectangular coordinates of admittance are conductance and susceptance