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 0.001 –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

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

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