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Published byJayce Loveall Modified over 2 years ago

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**AC Circuit Theory Amplitude Phase V=V0sin(ωt) V=V0cos(ωt)**

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**Apply AC current to a resistor…**

I=I0sin(ωt) VR=IR=I0Rsin(ωt) R C I …capacitor… Voltage lags current by /2 I L …and inductor Voltage leads current by /2

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**Adding voltages V(t)? + R1 I0sin(ωt) R2**

Kirchoff’s Voltage Law and Ohm’s law R3 - V=I0R1sin(ωt)+I0R2sin(ωt)+I0R3sin(ωt) =I0(R1+R2+R3)sin(ωt)

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Adding voltages + R1 I0sin(ωt) V(t) =VR+VC + C1 voltages out of phase…

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**Complex numbers in AC circuit theory**

Im(z) |z|sin(θ) θ j2=–1 Reason: ‘i’ is already taken |z|cos(θ) Re(z)

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**Complex Impedance I=I0sin(ωt) I=I0ejωt**

The ratio of the voltage across a component to the current through it when both are expressed in complex notation I=I0sin(ωt) I=I0ejωt

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**Complex Impedance Real part: resistance (R) Imaginary part: reactance**

Ohm’s law jωL R

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**Series / parallel impedances**

Z1=R Z2=jωL Impedances in series: ZTotal=Z1+Z2+Z3… Impedances in parallel Z1 Z2 Z3

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RC low pass filter R + + VIN VOUT 1/(jwC) - - ZR ZC

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**RL high pass filter Here is a slight trick:**

+ + VIN jwL VOUT Here is a slight trick: Get comfortable with the complex result. - -

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Bode plot (-3dB)

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**Decibels Logarithm of power ratio VOUT/VIN dB 10 20 1 0 0.1 –20**

10 20 1 0 0.1 –20 0.01 –40 –60

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Bode plot

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**Something Interesting**

Capacitor At low frequencies (like DC) Open circuit Not a surprise, it’s got a gap! At high frequencies (“fast”) Short circuit! Inductor At low frequencies Short circuit Not a surprise, it’s just a wire really At high frequencies Open circuit! Sometimes this can help you with your intuition on the circuit’s behaviour.

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Go to black Board and Explain Phasors

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LRC series circuit R L C I0ejωt Purely resistive at φ=0 Z=R

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LRC parallel circuit I0ejωt R L C

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Bandpass filter R + + VIN VOUT C L

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**Bandpass filter build a radio filter C=10nF f0=455kHz L=12.2μH**

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**LCR series circuit – current driven**

I(t)=I0ejωt L R C I(t) Freq.Wild Stuff going on! NOT a Very Good Idea

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**LCR series circuit – voltage driven**

V(t)=V0ejωt R V(t) L C

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R/L=106 R/L=105

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**Q value No longer on CP2 syllabus (moved to CP3)**

Not always helpful Frequency at resonance Diff. in Freq. to FWHM High Q value – narrow resonance

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**Q value For what we did last time. Dw𝐹𝑊𝐻𝑀**

High Q value – narrow resonance

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**LC circuit – power dissipation**

Power is only dissipated in the resistor proof – power dissipated inductor and capacitor – none reactance

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LRC Power dissipation R I V(t) = L V(t) Z=Z0ejφ C Instantaneous power

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Power dissipation cosφ = power factor

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**Power in complex circuit analysis**

On black board But then go ahead and flip to next slide now anyway. Power in complex circuit analysis

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

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**Bridge circuits To determine an unknown impedance Z1 Z3 VA VB V(t) V**

Bridge balanced when VAVB=0

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CP2 September 2003

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Phasors and stuff Backup Slides

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Phasors I I=I0sin(ωt) VR=IR=I0Rsin(ωt) R C I I L

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VT Phasors V0cos(ωt) V0sin(ωt) ωt R C I0sin(ωt)

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RC phasors R C I0sin(ωt) VT=V0sin(ωt+φ) I0sin(ωt) V0sin(ωt+φ)

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**Phasors – mathematics VT=V0sin(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) A2 + B2 = R2 B/A = tan(φ)

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RL phasors R L I0sin(ωt) VT=V0sin(ωt+φ) V0sin(ωt+φ) I0sin(ωt)

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RL filter R VIN L VOUT high pass filter

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Phasors: RC filter R VIN C VOUT low pass filter

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CP2 June 2003

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