I have a quick question regarding how the complex impedance of an AC circuit is derived.(adsbygoogle = window.adsbygoogle || []).push({});

The voltage and current in an AC circuit are given by the equations

[tex]

V=V_{m}cos(\omega t) = V_{m}Re(e^{i\omega t})

[/tex]

[tex]

I=I_{m}cos(\omega t-\phi) = I_{m}Re(e^{i(\omega t-\phi)})

[/tex]

The impedance is given by

[tex]Some online resources state that

Z = \frac{V}{I}} = \frac{V_{m}Re(e^{i\omega t})}{I_{m}Re(e^{i(\omega t - \phi)})}

[/tex]

[tex]

Z=\frac{V_{m}}{I_{m}}e^{i\phi}

[/tex]

Do they actually mean that

[tex]

Z=\frac{V_{m}}{I_{m}}Re(e^{i\phi})

[/tex]?

if so...

[tex]

\frac{Re(e^{i\omega t})}{Re(e^{i(\omega t - \phi)})}\neq Re(\frac{e^{i\omega t}}{e^{i(\omega t - \phi)}})

[/tex]

What am I failing to see?

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# Complex Impedance Derivation

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