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The reciprocal of electrical impedance, relating electric current and electric potential difference phasors by $\tilde{I}=Y\tilde{V}$. Its unit is the siemens, $\mathrm{S}={\Omega }^{-1}$. Electrical admittances add for branches in a parallel circuit.

A series circuit comprising electrical resistance $R>0$ and inductance $L>0$, placed in a parallel circuit with capacitance $C>0$, has electrical admittance $Y=(R+i\omega L{)}^{-1}+i\omega C$. Vanishing imaginary part at nonzero angular frequency gives the displayed condition, requiring $L>R^{2}C$. It defines zero voltage-current phase difference, which need not coincide exactly with the maximum of impedance magnitude when loss is present. At this frequency the electrical impedance is real and equals $L/(RC)$.

A connection in which branches share both terminal nodes and therefore the same electric potential difference. Branch electric currents add. For sinusoidal signals, the total electrical admittance is the sum of branch electrical admittances; ideal capacitances add.

For electric potential difference and electric current varying as sinusoids, choose the phasor conventionThe electrical impedance isIts modulus gives the voltage-to-current amplitude ratio, and its complex argument gives the voltage-minus-current phase difference. For ideal components,These follow from Ohm's law $V=RI$, the capacitor law $I=CdV/dt$, and the inductor law $V=LdI/dt$. In phasor notation a time derivative becomes multiplication by $i\omega$, turning the linear ordinary differential equations of the electrical circuit into algebraic equations. Electrical impedances add in a series circuit; their reciprocals, the electrical admittances, add in a parallel circuit. This computes both amplitudes and phase differences in the steady response to sinusoids.

The original diagram has a series circuit containing $R$ and $L$, in parallel with $C$; it is not three separate parallel components. The electrical admittance is thereforeFor $R>0$, voltage and current are in phase when the imaginary part of $Y$ vanishes. Excluding the zero-frequency solution, division by $\omega$ givesA nonzero lossy parallel coil resonance exists only if $L>R^{2}C$. With $R=10\Omega$, $L=0.150\mathrm{H}$ and $C=1.00\times 10^{-5}\mathrm{F}$,For zero electrical resistance, the ideal parallel inductor-capacitor cancellation occurs atThe fractional reduction relative to the zero-resistance coil isAt exactly $R=0$ and $\omega ={\omega }_0$, $Y=0$, so finite applied voltage draws zero net current in the ideal model. The quoted frequency is the limiting parallel-resonance value; the phase of an exactly zero current is undefined.