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Phasors v3

# Phasors v3 - Using Phasors to Analyze AC Circuits...

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Unformatted text preview: Using Phasors to Analyze AC Circuits in Steady State DePiero EE 201 Learning [email protected] •  Convert sin & cos signals and device impedances to/from phasor form •  Incorporate phasors when analyzing circuits by previous analysis methods (Mesh I, Node V…) for AC steady ­state [email protected] Note: sin() and cos() are suﬃcient to support analysis of any signal. Merci Fourier! Signals Diﬀer in Amplitude and Phase Only In Steady State •  R limits charging rate of cap, delaying peak of Vout •  R ­C form V divider –  Eﬀects amplitude •  Compare Vout vs Vin –  Frequency same –  Amplitude & phase diﬀer •  Transients: –  Wave shape may diﬀer [email protected] due to stored charge on cap. –  Transients gone in steady ­state Amplitude and Phase Changes For an Inductor •  Compare v(t) vs i(t) v ( t ) = A cos( wt ) 1 i( t ) = L 1 i( t ) = L ∫ € A cos( wt ) dt = ∫ v (t ) dt A A π sin( wt ) = cos( wt − ) wL wL 2 € v (€ = A cos( wt ) t) i( t ) = A π cos( wt − ) wL 2 These are the specific amplitude and phase changes for an inductor € € Need to Compute Ohm’s Law Given Amplitude and Phase Changes •  Compare v(t) vs i(t) for basic elements V(t) i(t) Inductor v ( t ) = A cos( wt ) Capacitor v ( t ) = A cos( wt ) Resistor € € v ( t ) = A cos( wt ) A π cos( wt − ) wL 2 π i( t ) = AwC cos( wt + ) 2 A i( t ) = cos( wt ) R i( t ) = € € •  Need a ‘conversion’ for Ohm’s Law in each case € € •  Note prior analysis boils down to Ohm’s Law Ⱥ R1 Ⱥ ȺR3 € R2 ȺȺ I1 Ⱥ ȺV1 Ⱥ ȺȺ Ⱥ = Ⱥ Ⱥ R4 ȺȺ I2 Ⱥ ȺV2 Ⱥ € Ⱥ G1 Ⱥ ȺG3 G2 ȺȺV1 Ⱥ Ⱥ I1 Ⱥ ȺȺ Ⱥ = Ⱥ Ⱥ G4 ȺȺV2 Ⱥ Ⱥ I2 Ⱥ Avoid d/dt, Avoid Trig – Use Phasors! •  Consider phasor ­domain [email protected] v ( t ) = A cos( wt + 0) = Re{ Ae j ( wt + 0 )} j = −1 •  Ohm’s Law in phasor ­domain € € –  [email protected] by device impedance, Z: –  For coil, [email protected] i(t) to v(t) € e jθ = cos θ + j sin θ ~ ~~ v ( t ) = Re{V } = Re{I Z} A π A j ( wt −π ) 2 i( t ) = cos( wt − ) = Re{ e }€ wL 2 wL π j A j ( wt −π ) j ( wt + 0 ) 2 v ( t ) = A cos( wt + 0) = Re{ Ae } = Re{ e wL e 2 } wL ~ Re{V } = Re{I Z} € € ~~ Compare! This yields definition of a phasor for the impedance of a coil ZL and for a signal Coil ‘Impedance:’ Z L = wL e € j π 2 Signal Representation: A cos( wt + α ) ⇔ Ae jα (Drop w for simplicity) Phasors Are Great! •  [email protected] in phasor ­domain simple: +  ­ * / –  (Albeit with complex numbers!) •  Phasor [email protected] determine amp & phase change of sinusoids in @me domain. •  Summary: Device Impedances: –  All phasors at same frequency w 0 = 2π f 0 –  [email protected] for w1 w2 w3… € –  Use sin() or cos() consistently € •  FYI: cos(θ ) = − sin(θ − 90) sin(θ ) = cos(θ € 90) − –  Impedances are f(w) € € € € j π 2 Z L = wL e = jwL 1 − jπ 1 ZC = e 2= wC jwC ZR = R e j 0 = R Signal Representation: A cos( wt + α ) ⇔ Ae jα ...
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