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

2. 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

3. 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)

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

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

6. 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

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

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

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

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

11. Bode plot
(-3dB)

12. 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

13. Bode plot

14. 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.

15. Go to black Board and
Explain Phasors

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

17. LRC parallel circuit
I0ejωt R L C

18. Bandpass filter R + +
VIN
VOUT C L

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

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

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

22. R/L=106
R/L=105

24. 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

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

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

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

28. Power dissipation
cosφ = power factor

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

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

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

32. CP2 September 2003

36. Phasors and stuff
Backup Slides

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

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

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

40. 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)

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

42. RL filter R
VIN L
VOUT
high pass filter

43. Phasors: RC filter R
VIN C
VOUT
low pass filter

44. CP2 June 2003