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Chapter 5

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Objective of Lecture Describe the mathematical relationships between ac voltage and ac current for a resistor, capacitor, and inductor. Discuss the phase relationship between the ac voltage and current. Explain how Ohms Law can be adapted for inductors and capacitors when an ac signal is applied to the components. Derive the mathematical formulas for the impedance and admittance of a resistor, inductor, and capacitor.

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Resistors Ohms Law v(t) = Ri(t) = R I m cos( t + ) V = RI m RI where The voltage and current through a resistor are in phase as there is no change in the phase angle between them.

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Capacitors i(t) = C dv(t)/dt where v(t) = V m cos( t) i(t) = -C V m sin( t) i(t) = CV m sin( t ) i(t) = CV m cos( t - ) i(t) = CV m cos( t )

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Capacitors V = V m I = CV m cos( t + ) V m cos( t ) = V e j = V = jV I = j CV or V = (1/j C) I = - (j/ C) I

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Capacitors 90 o phase difference between the voltage and current through a capacitor. Current needs to flow first to place charge on the electrodes of a capacitor, which then induce a voltage across the capacitor Current leads the voltage (or the voltage lags the current) in a capacitor.

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Inductors v(t) = L d i(t)/dt where i(t) = I m cos( t) v(t) = - L I m sin( t) = LI m cos( t ) V = LI m I = I m cos( t) I m cos( t ) = I e j = I = jI V = j LI or I = (1/j L) V = - (j/ L) V

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Inductors 90 o phase difference between the voltage and current through an inductor. The voltage leads the current (or the current lags the voltage).

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Impedance If we try to force all components to following Ohms Law, V = Z I, where Z is the impedance of the component. Resistor: Z R = Capacitor: Z C = Inductor:Z L =

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Admittance If we rewrite Ohms Law: I = Y V (Y = 1/Z), where Y is admittance of the component Resistor: Y R = Capacitor: Y C = Inductor:Y L =

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Impedances Value at = Admittance s Value at = 0 rad/s rad/s0 rad/s rad/s Z R = R = 1/GRRY R = 1/R = GGG Z L = j L0 Y L =-j/( L) 0 Z C = -j/( C) 0 Y C = j C0 Inductors act like short circuits under d.c. conditions and like open circuits at very high frequencies. Capacitors act like open circuits under d.c. conditions and like short circuits at very high frequencies.

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Impedance Generic component that represents a resistor, inductor, or capacitor.

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Admittance

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Summary Ohms Law can be used to determine the ac voltages and currents in a circuit. Voltage leads current through an inductor. Current leads voltage through a capacitor. ComponentImpedanceAdmittance ResistorZRZR CapacitorZCZC InductorZLZL

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