CHAPTER 2. ALTERNATING CURRENT CIRCUITS302.3.3Inductive ImpedanceKirchoﬀ’s voltage law for a voltage source and inductor isv(t)− Ldi(t)dt=0.(2.40)Solving this equation givesv = jωLi⇒ ZL = jωL.(2.41)For DC circuits ω = 0 and hence ZL = 0. There is no voltage drop across an inductor inDC (zero resistance).2.3.4Combined ImpedancesWe now know the impedance for each of our passive circuit elements:ZR = R;ZL = jωL;ZC =−j/(ωC).(2.42)The equivalent impedance of a circuit can be obtained by using the following rules forcombining impedances.In seriesZeq =iZi.(2.43)In parallelZeq =i Ziij=i Zj.(2.44)Appealing to the complex notation we can writeZeq = R + jX(ω),(2.45)where R is the resistance and X is called the reactance (always a function of ω).For a series combination of R, L and CZeq =R + jωL +1jωC,(2.46)=R + jωL−1ωC.(2.47)(ωL− 1/ωC) = 0 gives a special frequency, ω =1/√LC.Example: An inductor and capacitor in parallel form the tank circuit shown inﬁgure 2.5.