Impedance of different devices (derivations)
Impedance of different devices or circuir elements(derivation)
editFor a resistor:
The Ohm's law for a linear constant resistor is:
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:
Thus comparing above equation with Ohm's law we have:
For a capacitor:
By definition:
For
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