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

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

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

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)

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

Complex numbers in AC circuit theory
Im(z) |z|sin(θ) θ j2=–1 Reason: ‘i’ is already taken |z|cos(θ) Re(z)

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

Complex Impedance Real part: resistance (R) Imaginary part: reactance
Ohm’s law jωL R

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

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

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

Bode plot (-3dB)

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

Bode plot

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.

Go to black Board and Explain Phasors

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

LRC parallel circuit I0ejωt R L C

Bandpass filter R + + VIN VOUT C L

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

LCR series circuit – current driven
I(t)=I0ejωt L R C I(t) Freq.Wild Stuff going on! NOT a Very Good Idea

LCR series circuit – voltage driven
V(t)=V0ejωt R V(t) L C

R/L=106 R/L=105

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

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

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

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

Power dissipation cosφ = power factor

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

Bridge circuits To determine an unknown impedance Z1 Z3 VA VB V(t) V
Bridge balanced when VAVB=0

CP2 September 2003

Phasors and stuff Backup Slides

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

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

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

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)

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

RL filter R VIN L VOUT high pass filter

Phasors: RC filter R VIN C VOUT low pass filter

CP2 June 2003