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:

$i_{C}=C\textstyle {dv \over dt}\,$

For

$v_{C}(t)=V_{\mathrm {p} }\cos \left(\omega t+\varphi \right)$

I_C is:

I_C=-ωCV_psin(ωt+φ) =ωCV_pcos(ωt+φ+π/2)

In terms of phasor rerpresentation:

V_C=V_pe^(jφ)

and

I_C=ωCV_pe^(j(φ+π/2)) =jωCV_pe^(jφ)

Therefore by defenition of Z_capacitor:

Z_capacitor=V_C/I_C =V_pe^(jφ)/jωCV_pe^(jφ) =1/jωC=-j/ωC

For an inductor:

By definition:

V_L=LdI_L/dt

For

I_L=I_pcos(ωt+φ)

V_L is:

V_L=-ωLI_psin(ωt+φ) =ωLI_pcos(ωt+φ+π/2)

In terms of phasor rerpresentation:

I_L=I_pe^(jφ)

and

V_L=ωLI_pe^(j(φ+π/2)) =jωLI_pe^(jφ)

Therefore by defenition of Z_inductor:

Z_inductor=V_L/I_L =jωLI_pe^(jφ)/I_pe^(jφ) =jωL