1. Complex Numbers: Phasors and Capacitors
Spin My World Right Round

2. Frequency sensitive circuits
LPF
HPF

3. Applications: Audio/Speech!
I’m all about the bass

4. LPF example

5. Application: Gastric Electrical Activity
60-70 million people suffer from GI disorder
Electrically active organ

6. Mapping Colon Activity
frequency (cpm)

7. Band Pass Filter Example

8. Analog Filter Summary
Image credit:

9. Leonhard Euler ( )
𝑒 𝑖𝜃 =𝑐𝑜𝑠𝜃+𝑖 𝑠𝑖𝑛𝜃

10. Trig vs. complex exponential form
Note: In circuits, we use j instead of i

11. …and now your joke(s) of the day

12. Magnitude & Phase  PHASOR

13. Phasor = Magnitude and Phase over time sweeps out sinusoid

14. Phasor = Rotating vector in complex plane

15. Merry Go-Round
1) Plot the complex number z= 2 𝑒 𝑗 2𝜋 𝑡 for values of t = {0, 1/6, 1/3, 1/2, 2/3, 5/6, 1, 7/6, 4/3, 3/2, 5/3, 11/6, 2 }.
2) Do the same as above but for the complex number z=4 𝑒 𝑗 2𝜋 2 𝑡
3) Do the same as above, but for the complex number z=4 𝑒 𝑗 (2𝜋 2 𝑡+ 𝜋 2 )
4) Plot the projection onto the real axis for your answers in 2 and 3.

16. Describing a cosine wave with phasors5
4.33
2.5

17. How about a sine wave?

18. Phasors! Rotates in time
cos(wt)
-cos(wt)
sin(wt)
-sin(wt)
Rotates in time
But we just read off the magnitude and phase information

19. Red vs. blue: What’s different?
1. Magnitude
2. Phase

20. Graphical Representation of Cosinesw
f = 60; %Hz
w = 2*pi*f;
R = 1e4; % 1Mohm
C = 1e-6; % = 1 uF
Vout = 1/(sqrt(1+(w*R*C)^2))
phi = -atan(-w*R*C) = 75 deg.
fc = 1/(2*pi*R*C) = 15.9 Hz.
Blue = Vin(t)
Red = VC(t)
Green = VR(t)

21. Fourier Series Example

22. Recording Studio
A 110 (Hz)

23. Capacitors: Typical ones you’ll find in the lab
Some big and some small…

24. These are capacitors too!
Cell membrane
Icky cells (bacteria)

25. And so is this!
PC board
Red = top layer
Green = bottom layer

26. More Applications of Capacitors

27. Applications: Audio/Speech!
I’m all about the bass

28. LPF: Magnitude Response
Pass Band
Attenuation Band
𝑓 𝑜 = 1 2𝜋𝑅𝐶 = 234 Hz
1 kW
0.68 mF fo

29. LPF: Phase Response
Pass Band
Attenuation Band
1 kW
0.68 mF fo

30. HPF: Magnitude Response
Attenuation Band
Pass Band
𝑓 𝑜 = 1 2𝜋𝑅𝐶 = 234 Hz
0.68 mF
1 kW

31. HPF: Phase Response
Attenuation Band
Pass Band

32. HPF vs. LPF: Magnitude Response

33. HPF vs. LPF: Phase Responses

34. Implementing a low pass filter (LPF)
Strongly attenuated 60Hz
Barely attenuated 0.05Hz
R = 10e4; % 100k
C = 1e-6; % = 1 uF
Vout =
0.0265
phi =
fc =
1.5915
0.9995

35. 60 Hz signal zoomed in f

36. Raw vs. Filtered Signal

37. Low Pass Filter Example: Vout and Vin

38. Vout, Vc, and VR