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Impedance and Admittance

Impedance (ZZ) is the complex ratio of voltage to current (Z=V/IZ = V/I), accounting for resistance and reactance. Admittance (YY) is its reciprocal (Y=1/ZY = 1/Z). Z=R+jXZ = R + jX.
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The statement of the theorem

Let the time-domain voltage V(t)V(t) and current I(t)I(t) across a linear two-terminal element be represented by their Fourier transforms, V(s)V(s) and I(s)I(s), respectively, where s=jjomegas = j\text{j}\text{omega} is the complex frequency variable. The generalized relationship is defined by the complex transfer function Z(s)Z(s): \n\nV(s)=Z(s)I(s)V(s) = Z(s) I(s) \n\nwhere Z(s)Z(s) is the complex impedance, defined as the ratio of the voltage phasor to the current phasor: \n\nZ(s)=V(s)I(s)=R+jjomegaL′+1jjomegaC′+1sdds (for generalized elements)Z(s) = \frac{V(s)}{I(s)} = R + j\text{j}\text{omega}L' + \frac{1}{j\text{j}\text{omega}C'} + \frac{1}{s} \frac{d}{ds} \text{ (for generalized elements)}\n\nConversely, the complex admittance Y(s)Y(s) is defined as the reciprocal of the impedance, representing the ratio of current to voltage: \n\nY(s)=I(s)V(s)=1Z(s)Y(s) = \frac{I(s)}{V(s)} = \frac{1}{Z(s)}\n\nFor a general circuit network described by nodal analysis, the relationship between the nodal voltage vector V(s)\mathbf{V}(s) and the injected current vector I(s)\mathbf{I}(s) is given by the generalized nodal admittance matrix Y(s)\mathbf{Y}(s): \n\nI(s)=Y(s)V(s)\mathbf{I}(s) = \mathbf{Y}(s) \mathbf{V}(s) \n\nwhere Y(s)\mathbf{Y}(s) is the matrix whose elements are the admittances between nodes.