CHAPTER 2. ALTERNATING CURRENT CIRCUITS29v(ω)= Z(ω)i(ω)(2.32)and hence you see the power of the complex notation. For a physically quantity we take theamplitude of the real signal|v(ω)| =|Z(ω)||i(ω)|.(2.33)We will now examine each circuit element in turn with a voltage source to deduce itsimpedance.2.3.1Resistive ImpedanceKirchoﬀ’s voltage law for a voltage source and resistor isv(t)− Ri(t)=0.(2.34)Trying the solutionsi(t)= iejωtandv(t)= vejωt(2.35)leads tov = Ri⇒ ZR = R.(2.36)The impedance is equal to the resistance, as expected.2.3.2Capacitive ImpedanceKirchoﬀ’s voltage law for a voltage source and capacitor isv(t)−q(t)C=0.(2.37)Ordv(t)dt−i(t)C=0.(2.38)Solving this equation givesjωv =iC⇒ ZC =1jωC(2.39)For DC circuits ω = 0 and hence ZC→∞. The capacitor acts like an open circuit(inﬁnite resistance) in a DC circuit.