Phasor Relationships; ImpedanceIntroductionIntroduction (cont.)The Good News!PhasorsImpedancePhasor Relationships for Circuit ElementsI-V Relationship for a ResistorI-V Relationship for a CapacitorI-V Relationship for an InductorImpedance SummaryClass ExamplesLect17 EEE 202 1Phasor Relationships; ImpedanceDr. HolbertApril 2, 2008Lect17 EEE 202 2Introduction•Any steady-state voltage or current in a linear circuit with a sinusoidal source is also a sinusoid–This is a consequence of the nature of particular solutions for sinusoidal forcing functions–All steady-state voltages and currents have the same frequency as the sourceLect17 EEE 202 3Introduction (cont.)•In order to find a steady-state voltage or current, all we need to know is its magnitude and its phase relative to the source (we already know its frequency)•Usually, an AC steady-state voltage or current is given by the particular solution to a differential equationLect17 EEE 202 4The Good News!•We do not have to find this differential equation from the circuit, nor do we have to solve it•Instead, we use the concepts of phasors and complex impedances•Phasors and complex impedances convert problems involving differential equations into simple circuit analysis problemsLect17 EEE 202 5Phasors•Recall that a phasor is a complex number that represents the magnitude and phase of a sinusoidal voltage or currentx(t) = XM cos(ωt+θ) ↔ X = XM θTime domain Frequency Domain•For AC steady-state analysis, this is all we need---we already know the frequency of any voltage or currentLect17 EEE 202 6Impedance•AC steady-state analysis using phasors allows us to express the relationship between current and voltage using a formula that looks like Ohm’s law:V = I Z•Z is called impedance (units of ohms, )•Impedance is (often) a complex number, but is not technically a phasor•Impedance depends on frequency, ωLect17 EEE 202 7Phasor Relationships for Circuit Elements •Phasors allow us to express current-voltage relationships for inductors and capacitors much like we express the current-voltage relationship for a resistor•A complex exponential is the mathematical tool needed to obtain this relationshipLect17 EEE 202 8I-V Relationship for a ResistorRR ZZIVRv(t)+–i(t))()( tiRtv RIV Lect17 EEE 202 9I-V Relationship for a CapacitorCj1IV Cv(t)+–i(t)dttdvCti)()( CjC1 ZZIVLect17 EEE 202 10I-V Relationship for an InductorLv(t)+–i(t)dttdiLtv)()( IV LjLjL ZZIVLect17 EEE 202 11Impedance SummaryLect17 EEE 202 12Class Examples•Drill Problems P8-4, P8-7, P8-5 (and P8-1, if time permits)•Remember: sin(ωt) =
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