1. Topic 16
Sinusoidal Signals
(9.1)

2. Sounds
sawing
market
truck
12/3/2018
Sinusoids

3. Banjo
G=195.8 Hz T 1 2 3 4 5 6 7
35 mS
12/3/2018
Sinusoids

4. Cello
C=264 Hz T 1 2 3 4 5 6 7 8 9 10
38.3 mS
12/3/2018
Sinusoids

5. Saxaphone
G=195.8 Hz T 1 2 3 4 5 6 7
35.4 mS
12/3/2018
Sinusoids

6. Single Cycle Comparison Comparison
12/3/2018
Sinusoids

7. Jean Baptiste Joseph Fourier
extended fourier transforms into random signals ca. 1937
professor at M.I.T.
inventer of Cybernautics
mathemetician and electrical engineer
Norbert Wiener
Jean Baptiste Joseph Fourier
mathemtician and civil engineer
scientific advisor to Napolean
professor at Ecole Polytechnique
developed Fourier Transform ca.1807
12/3/2018
Sinusoids

8. Fourier Theory In a nutshell… …any time-varying signal, f(t)…
…may be represented as a sum of sinewaves of varying frequencies
sound data, similar to ours
plot of the amplitude of sine waves needed to create it
Fourier Transform
12/3/2018
Sinusoids

9. Consequences
All we need to know in order to be able to predict how a network will respond to a general signal
SoundBlaster
is to find out how it will respond to various sinusoidal frequencies
This sucks. But Bobby’s got such a gravelly voice, I might not notice
12/3/2018
Sinusoids

10. Electrical Engineers are interested in sinewaves
Sinusoidal Sources
Because
we can characterize any network just using sinewaves
and Then
use the result to figure out how the network will respond to any signal
& Because
all our ac power systems are sinusoidal
Electrical Engineers are interested in sinewaves
and how our networks respond to them
12/3/2018
Sinusoids

11. AC power as it comes form a conventional wall socket is sinusoidal
Sinusoidal Sources
AC power as it comes form a conventional wall socket is sinusoidal
Although we quite reasonably talk about power what is provided to a household plug is a (more or less) constant voltage
Vm: the amplitude in volts (nominally 177 for North American AC)
ω: the frequency, in radians per sec
φ: the phase, in radians
The current that results when we connect something to a household circuit will also be sinusoidal
the amplitude, Im, is in amps. Its value will depend upon the resistance of the load being connected.
12/3/2018
Sinusoids

12. Plotting Sinusoids But how do f & T relate to this? v
Max value is the amplitude, Vm Vm
-Vm t T
Min value is -Vm
T is the period, the time taken to sweep out 1 complete cycle of the wave
The reciprocal of the period
is the frequency, the number of cycles that occurs in 1 sec, measured in Hertz
i.e., if it takes a mS to sweep out one complete cycle…
… a thousand cycles are swept out in a sec
12/3/2018
Sinusoids

13. ω, f, φ, T & all that stuff
What angular change does the sinewave undergo in 1 period? v Vm
-Vm T
But these must
always be angles t T T
By definition
By definition, a cycle is 2π radians
for any t
That is
where
ω is the frequency in radians/sec
f is the frequency in cycles/sec (Hz)
12/3/2018
Sinusoids

14. Phase angle v Vm Phase angle 0 t -Vm v Positive phase angle Vm
Peak occurs when
Wave arrives early
positive phase angle
We say it leads
12/3/2018
Sinusoids

15. Note that if we delay a cosine wave by 90 degrees (or π/2 radians)
Phase angle Vm t v
-Vm
Negative phase angle
Peak occurs when
Wave is delayed
negative phase angle
We say it lags Vm t v
-Vm
Note that if we delay a cosine wave by 90 degrees (or π/2 radians)
90°
We get a sine wave
12/3/2018
Sinusoids

16. ω, f, φ, T & all that stuff Is it degrees or radians?
Engineers routinely use both …
… and go back and forth as needed
Do we use ω or f ?
Engineers routinely use both …
… f in the lab where T is easy to measure
… ω in calculations where it is more natural…
… and go back and forth as needed!
12/3/2018
Sinusoids

17. RMS measurement rms stands for root mean square
Given any periodic function
Its rms value is
1. Square the function
For a cosinusoidal voltage
2. Average it over one cycle
3. Take square root of result
(See problem 9.2)
North American household voltage:
12/3/2018
Sinusoids

18. End of this lecture
12/3/2018
Sinusoids

19. Truck
12/3/2018
Sinusoids

20. Truck
12/3/2018
Sinusoids

21. Truck
12/3/2018
Sinusoids

22. Truck
12/3/2018
Sinusoids

23. Marketplace
12/3/2018
Sinusoids

24. Marketplace
12/3/2018
Sinusoids

25. Marketplace
12/3/2018
Sinusoids

26. Marketplace
12/3/2018
Sinusoids

27. Sawing
12/3/2018
Sinusoids

28. Sawing
12/3/2018
Sinusoids

29. Sawing
12/3/2018
Sinusoids

30. Sawing
12/3/2018
Sinusoids