1. ENGS2613 Intro Electrical Science Week 12 Dr. George Scheets
Read: 7.2, 9.1, & 9.2
Problems: 7.1, 14, 22, 29, & 30

2. Capacitors
Capacitor Open Circuit if DC in ↕ Short Circuit if Very Hi Freq in
i = C dv/dt
v(t) - v(0-) = i dt C i i + + - - ∫ t
Absorbing Power
(Voltage is
Increasing)
Releasing Power
(Voltage is
Decreasing) + -
i →
Note Reference
Current Direction

3. Inductors
Inductor Short Circuit if DC in ↕ Open Circuit if Very Hi Freq in
v = L di/dt
i(t) - i(0-) = v dt L i i + - ∫ t - +
Absorbing Power
(Current is
Increasing)
Releasing Power
(Current is
Decreasing) + -
i →
Note Reference
Current Direction

4. Electrical Material
Chapters 1 – 5 R's & Circuit Behavior at DC (Constant I & V)
Chapter 6 – 7 L's & C's (Time Varying i & v)
i = i(t) and v = v(t)
Chapter 9 Sinusoids
Any time domain waveform = sum of sinusoids
Resistors
Treat all sinusoids the same
L's & C's
Don't

5. Cycles/Second Heinrich Hertz 1857 - 1894 German Physicist
Around 1887 Proved existance of EM waves
University of Karlsruhe
Units of Hertz = Sinusoid Cycles/Second
Source: Wikipedia

6. Frequency a(t)sin[2πft + θ(t)] or a(t)cos[2πft + θ(t)]
Frequency = value of f, units of Hertz (cycles/sec)
U.S. Electricity mostly uses 60 Hz sinusoids
Wall socket ≈ 120 Vrms ≈ 170 Vpeak = 340 Vp-p
Smart Phone radios are up around 1 GHz
Any signal can be constructed by adding together the appropriate sinusoids

7. 5 Hertz Square Wave... 1 volt peak, 2 volts peak-to-peak, 0 V average
1.5
-1.5
1.0
1 volt peak, 2 volts peak-to-peak, 0 V average

8. Generating a Square Wave...
1.5
-1.5
1.0
1 vp
5 Hz
1.5
-1.5
1.0
1/3 vp
15 Hz

9. Generating a Square Wave...
1.5
5 Hz +
15 Hz
-1.5
1.0
1.5
-1.5
1.0
1/5 vp
25 Hz

10. Generating a Square Wave...
5 Hz +
15 Hz +
25 Hz
1.5
-1.5
1.0
1.5
-1.5
1.0
1/7 vp
35 Hz

11. Generating a Square Wave...
5 Hz +
15 Hz +
25 Hz
35 Hz
1.5
-1.5
1.0
cos2*pi*5t - (1/3)cos2*pi*15t
+ (1/5)cos2*pi*25t - (1/7)cos2*pi*35t)

12. Generating a Square Wave...
1.5
-1.5
1.0
5 cycle per second square wave generated using first 50 cosines, Absolute Bandwidth = 495 Hertz.

13. Generating a Square Wave...
1.5
-1.5
1.0
5 cycle per second square wave generated using first 100 cosines, Absolute Bandwidth = 995 Hertz.

14. Time Varying Waveforms
High Frequencies
Rapidly Changing Waveforms
Low Frequency content
Slowly Changing Wavefom
Could be a single sinusoid
Could be a arbitrary waveform shape = ∑ (bunch of sinusoids)
Note: DC = 0 Hertz sinusoid
5 V DC = 5cos(2π0t) volts

15. Sinusoidal Behavior Can construct any time domain signal
Add together appropriate sine waves & cosine waves
Appropriate frequencies & amplitudes
How do sinusoids react with
Resistors?
Capacitors?
Inductors?

16. Resistor: v = iR
Suppose time varying current thru resistor i = 2cos(2πft) amps
If R = 10 Ω, then voltage across resistor v = 20cos(2πft) volts
Note: Voltage & Current are time aligned Sinusoid frequency doesn't affect v/i ratio
If one freq has v/i ratio = 10, they all do

17. 100 Hz Cosine thru Resistor

18. 200 Hz Cosine thru Resistor

19. Square-ish Wave i(t) = cos2*pi*5t - (1/3)cos2*pi*15t
5 Hz +
15 Hz +
25 Hz
35 Hz
1.5
amps
-1.5
1.0
i(t) = cos2*pi*5t - (1/3)cos2*pi*15t
+ (1/5)cos2*pi*25t - (1/7)cos2*pi*35t)
Run this current into a resistor network?
Voltage shape will be…
identical.

20. Inductor: v = L di/dt
Suppose time varying current thru inductor i = 2cos(2πft) amps
If L = 0.1 H, then voltage across inductor v = -0.4πf sin(2πft) volts
Note: Voltage & Current are 90º out of phase Sinusoid frequency does affect v/i ratio
Voltage drop at 200 Hz is 2x drop at 100 Hz
Inductor impedance at 200 Hz = 2x that at 100 Hz
Higher frequencies attenuated more than lower freqs

21. 100 Hz Cosine thru Inductor
Voltage leads Current by 90º
=1/4 wavelength
= (1/4)(1/100)
= 1/400 second

22. 200 Hz Cosine thru Inductor
Voltage leads Current by 90º
=1/4 wavelength
= (1/4)(1/200)
= 1/800 second

23. Square-ish Wave i(t) = cos2*pi*5t - (1/3)cos2*pi*15t
5 Hz +
15 Hz +
25 Hz
35 Hz
1.5
amps
-1.5
1.0
i(t) = cos2*pi*5t - (1/3)cos2*pi*15t
+ (1/5)cos2*pi*25t - (1/7)cos2*pi*35t)
Run this thru an inductor & shape will…
… change.
Amplitude attenuation & time shift differs
with freq.

24. Square-ish Wave Current 1.5 thru 0.1 H Inductor v = L di/dt amps -1.5
-1.5
1.0
Voltage
across
0.1 H
Inductor

25. ∫ Capacitor: v = (1/C) i dtt
Capacitor: v = (1/C) i dt
Suppose time varying current thru inductor i = 2cos(2πft) amps
If C = 0.1 F, then voltage across inductor v = 10sin(2πft)/πf volts
Note: Voltage & Current are 90º out of phase Sinusoid frequency does affect v/i ratio
Voltage drop at 200 Hz is 1/2 drop at 100 Hz
Capacitor impedance at 200 Hz = 1/2 that at 100 Hz
Higher frequencies attenuated less than lower freqs

26. 100 Hz Cosine thru Capacitor
Current leads Voltage by 90º
=1/4 wavelength
= (1/4)(1/100)
= 1/400 second

27. 200 Hz Cosine thru Capacitor
Current leads Voltage by 90º
=1/4 wavelength
= (1/4)(1/200)
= 1/800 second

28. Square-ish Wave i(t) = cos2*pi*5t - (1/3)cos2*pi*15t
5 Hz +
15 Hz +
25 Hz
35 Hz
1.5
amps
-1.5
1.0
i(t) = cos2*pi*5t - (1/3)cos2*pi*15t
+ (1/5)cos2*pi*25t - (1/7)cos2*pi*35t)
Run this thru an capacitor & shape will…
… change.
Attenuation & time shift differs with freq.

29. Square-ish Wave Current 1.5 thru 0.1 F Capacitor v = amps (1/C) ∫ i dt
-1.5
1.0
Voltage
across
0.1 F
Capacitor

30. Passive Devices Resistors Inductors Capacitors
Treat all electrical frequencies identically
v & i show no phase shift
Inductors
High freqs show larger voltage drop than low freqs
Attenuate high frequencies more than low freqs
v leads i by 90° (or v lags i by 270°)
Capacitors
Low freqs show larger voltage drop than hi freqs
Attenuate low frequencies more than high freqs
i leads v by 90° (or i lags v by 270°)
NOTE: Output frequency f = input frequency

31. Inductors & Capacitors
Are frequency selective
Provide opportunity to change signal's shape
Unwanted noise has different freqs than desired signal?
Can filter out noise
RF signal received distorted?
Can undo much of this distortion (if characteristics known)
These are examples of frequency selective filtering
How do we analyze this?
Phase shift
Frequency selectivity

32. Leonhard Euler 1707 – 1783 Swiss Mathematician
Considered one of History's Greatest Mathematicians
Published & proved ejθ = cos(θ) + jsin(θ) in 1748
source: Wikipedia

33. Phasor Representation of a Time Domain Cosine
Useful for analyzing steady state response
Time Domain Waveform: Vp cos(ωt + θ)
Got a sine wave? Convert to a cosine via sin(ωt + θº) = cos(ωt + θº - 90º)
Phasor Representation = Vp ∟θ = Vp ejθ = Vp
Contains amplitude and cosine phase info
Does not directly contain frequency info
Phasors are vectors
Can be plotted on real [cosθ)] & imaginary [jsin(θ)] axis via Vp [cos(θ) + jsin(θ)].
Vector tail at origin? Tip is at coordinates (Vpcos(θ), Vpsin(θ)).
θ = angle in radians, θº = angle in degrees

34. Phasor Addition Example