58 Phasors in action - Phasors TheoryandLabdata...

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10/26/09 1 Phasors for AC Circuit Analysis Theory and Lab data obtained with Sinusoidal signals © 2009 A. Ganago 1 Phasors in AcCon DC and AC Voltages and Currents DC currents and voltages are constant values AC currents and voltages are sinusoids, which differ by their amplitude A and phase shiK φ; we need to keep track of both A and φ © 2009 A. Ganago Phasors in AcCon 2 Lab example: Green=IN, Blue=OUT © 2009 A. Ganago Phasors in AcCon 3 Lab example: Opposite Sign of φ © 2009 A. Ganago Phasors in AcCon 4 Real and Complex Numbers for Circuits We calculate DC currents and voltages by using real numbers We use complex numbers for AC voltages and currents, because complex numbers have both magnitude and phase A phasor is a shorthand notaCon for the magnitude and phase of sinusoidal voltage or current at the chosen frequency ω © 2009 A. Ganago Phasors in AcCon 5 Phasor NotaCon for Sine Waves We keep in mind the frequency ω but we do not include it in the phasor notaCon Phasor describes the whole sinusoidal signal, at all moments of Cme For example, © 2009 A. Ganago Phasors in AcCon 6 X = A φ is shorthand notation for : x ( t ) = A cos( ω t + φ )
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10/26/09 2 2 types of problems with phasors (1) The frequency ω can be always the same, like in the power grid Then we calculate a phasor for the parCcular voltage or current as a constant (2) The frequency ω can vary, as in an audio signal Then we have to calculate the dependence of the current or voltage on the frequency ω © 2009 A. Ganago Phasors in AcCon 7 Phasors and Impedances Phasors are used for voltages and currents Impedances are used as resistances With phasors V for voltage and I for current, and impedance Z , we can use Ohm’s law for resistors, inductors and capacitors V = Z I © 2009 A. Ganago Phasors in AcCon 8 Formulas for Impedances (1) Impedance Z R of a resistor equals resistance R and does not depend on frequency For a capacitor: Z C = 1/(jωC), where j =√(‐1) If ω is in rad/sec, and C in farads, Z C is in ohms For example, C = 1.5 nF, f = 3 MHz; note that ω=2πf = 1.885e7 rad/sec; thus Z C = ‐j 35.37 Ω Z C decreases with the frequency ω Z C
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