Impedance of different devices (derivations)

Impedance of different devices or circuir elements(derivation)
For a resistor:

The Ohm's law for a linear constant resistor is:



V_\mathrm{R} = RI_\mathrm{R} \, $$

If V_\mathrm{R} is sinusoidal then I_\mathrm{R} will also be sinusoidal. So Ohm's law holds if both V_\mathrm{R} and V_\mathrm{R} are phasors. By definition:

Z_\mathrm{Resistor} = \frac{V_\mathrm{R}}{I_\mathrm{R}} $$

Thus comparing above equation with Ohm's law we have:

Z_{Resistor}=R \, $$ For a capacitor:

By definition:



i_C=C\textstyle \, $$

For

v_C(t) = V_\mathrm{p} \cos \left( \omega t + \varphi \right) $$ IC is:

IC=-ωCVpsin(ωt+φ) =ωCVpcos(ωt+φ+π/2)

In terms of phasor rerpresentation:

VC=Vpe^(jφ)

and

IC=ωCVpe^(j(φ+π/2)) =jωCVpe^(jφ)

Therefore by defenition of Zcapacitor:

Zcapacitor=VC/IC =Vpe^(jφ)/jωCVpe^(jφ) =1/jωC=-j/ωC

For an inductor:

By definition:

VL=LdIL/dt

For

IL=Ipcos(ωt+φ)

VL is:

VL=-ωLIpsin(ωt+φ) =ωLIpcos(ωt+φ+π/2)

In terms of phasor rerpresentation:

IL=Ipe^(jφ)

and

VL=ωLIpe^(j(φ+π/2)) =jωLIpe^(jφ)

Therefore by defenition of Zinductor:

Zinductor=VL/IL =jωLIpe^(jφ)/Ipe^(jφ) =jωL