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V=V 0 sin(ωt) V=V 0 cos(ωt) V=V 0 sin(ωt+φ) AC Circuit Theory Amplitude Phase 1

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R Apply AC current to a resistor… I I=I 0 sin(ωt) V R =IR=I 0 Rsin(ωt) C I …capacitor… Voltage lags current by /2 I L …and inductor Voltage leads current by /2 2

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Adding voltages R1R1 R2R2 R3R3 I 0 sin(ωt) V=I 0 R 1 sin(ωt)+I 0 R 2 sin(ωt)+I 0 R 3 sin(ωt) =I 0 (R 1 +R 2 +R 3 )sin(ωt) + - V(t)? Kirchoffs Voltage Law and Ohms law 3

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Adding voltages R1R1 C1C1 I 0 sin(ωt) voltages out of phase… V(t) =V R +V C + + 4

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Complex numbers in AC circuit theory Re(z) Im(z) θ j 2 =–1 Reason: i is already taken |z|cos(θ) |z|sin(θ) 5

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Complex Impedance The ratio of the voltage across a component to the current through it when both are expressed in complex notation I=I 0 sin(ωt) I=I 0 e jωt 6

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Complex Impedance Real part: resistance (R) Imaginary part: reactance jωLjωL R Ohms law 7

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Series / parallel impedances Z 1 =RZ 2 =jωL Impedances in series: Z Total =Z 1 +Z 2 +Z 3 … Z1Z1 Z2Z2 Z3Z3 Impedances in parallel L R C 8

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RC low pass filter R 1/(jC) V IN V OUT ZRZR ZCZC

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RL high pass filter R jL V IN V OUT Here is a slight trick: Get comfortable with the complex result. 10

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Bode plot (-3dB) 11

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Decibels Logarithm of power ratio V OUT /V IN dB – – –60 12

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Bode plot 13

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Something Interesting Capacitor –At low frequencies (like DC) Open circuit Not a surprise, its got a gap! –At high frequencies (fast) Short circuit! Inductor –At low frequencies Short circuit Not a surprise, its just a wire really –At high frequencies Open circuit! Sometimes this can help you with your intuition on the circuits behaviour. 14

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EXPLAIN PHASORS Go to black Board and 15

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L R C I0ejωtI0ejωt LRC series circuit Purely resistive at φ=0 Z=R 16

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L LRC parallel circuit R C I0ejωtI0ejωt 17

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Bandpass filter R C V IN V OUT L

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Bandpass filter build a radio filter f 0 =455kHz Δf=20kHz C=10nF L=12.2μH R=389Ω 19

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I(t)=I 0 e jωt LCR series circuit – current driven L R C I(t) 20 Freq. Wild Stuff going on! NOT a Very Good Idea

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L R C V(t)=V 0 e jωt LCR series circuit – voltage driven V(t) 21

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R/L=10 6 R/L=

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Q value No longer on CP2 syllabus (moved to CP3) High Q value – narrow resonance 24 Frequency at resonance Diff. in Freq. to FWHM Not always helpful

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Q value For what we did last time. High Q value – narrow resonance 25

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LC circuit – power dissipation Power is only dissipated in the resistor proof – power dissipated inductor and capacitor – none reactance L R C I(t ) 26

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LRC Power dissipation L R C V(t) Z=Z 0 e jφ I V(t) = Instantaneous power 27

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Power dissipation cosφ = power factor 28

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POWER IN COMPLEX CIRCUIT ANALYSIS On black board But then go ahead and flip to next slide now anyway. 29

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Power factor Resistive load Z=R cosφ=1 Reactive load Z=X cosφ=0 30

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Bridge circuits V(t) V Z1Z1 Z3Z3 Z2Z2 Z4Z4 To determine an unknown impedance Bridge balanced when V AV B =0 VAVA VBVB 31

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CP2 September

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BACKUP SLIDES Phasors and stuff 36

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R Phasors I I=I 0 sin(ωt) V R =IR=I 0 Rsin(ωt) C I I L 37

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VTVT Phasors ωtωt V 0 sin(ωt) V 0 cos(ωt) R C I 0 sin(ωt) 38

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RC phasors R C I 0 sin(ωt) V T =V 0 sin(ωt+φ) I 0 sin(ωt) V 0 sin(ωt+φ) 39

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Phasors – mathematics V T =V 0 sin(ωt+φ) Asin(ωt) + Bcos(ωt) = Rsin(ωt + φ) Rsin(ωt + φ) = Rsin(ωt)cos(φ) + Rcos(ωt)sin(φ) A=Rcos(φ) B=Rsin(φ) A 2 + B 2 = R 2 B/A = tan(φ) 40

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RL phasors R L I 0 sin(ωt) V T =V 0 sin(ωt+φ) I 0 sin(ωt) V 0 sin(ωt+φ) 41

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RL filter R L V IN V OUT high pass filter 42

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Phasors: RC filter R C V IN V OUT low pass filter 43

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CP2 June

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