Slides 9 - Lecture notes 1 PDF

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Chapter 9.1...

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ECE 233 circuit theory

©UWECE TC Chen, ST Dunham

EE 233 circuit theory Lecture 2

Scott Dunham University of Washington

©UWECE TC Chen, ST Dunham

EE 233 Circuit Theory Sinusoidal Steady-State Analysis (9.1-3)

Scott Dunham University of Washington

©UWECE TC Chen, ST Dunham

ANNOUNCEMENTS • Mastering Engineering intro due Wednesday. • Homework #0 due Monday

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The sinusoidal source (9.1)

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SINUSOIDAL STEADY-STATE ANALYSIS Why sinusoids: • It is an AC source • It is a form of natural phenomena, such as the motion of a pendulum, the vibration of a string, the ripple on the ocean surface, and the natural response of underdamped second order systems… • It is easier to generate and transmit. • A sum of sinusoids can represent any practical periodic signal through Fourier analysis.

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SINUSOIDAL STEADY-STATE ANALYSIS Why sinusoidal steady-state: • Generation, transmission, distribution and consumption of electric energy occur under essentially sinusoidal steady-state conditions. • Make it possible to predict the behavior of circuits with nonsinusoidal sources. • Steady-state sinusoidal behavior simplifies the design of electric systems.

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SINUSOIDAL SOURCE • A sinusoidal voltage source produces a voltage that varies sinusoidally with time. • It can be expressed as either a sine or cosine function; in this class, we use cosine functions. 



is the magnitude of the source.

The sinusoidal function is periodic and its period is denoted . The reciprocal of gives frequency (in units of Hz if is in seconds),

©UWECE TC Chen, ST Dunham

SINUSOIDAL SOURCE • A sinusoidal voltage source produces a voltage that varies sinusoidally with time. • It can be expressed either sine or cosine function, in this class, we use cosine functions to express sinusoidal sources. 

represents the angular frequency of the sinusoidal function, or The angle is phase angle of the sinusoidal function. It determines the value of the sinusoidal function at

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.

SINUSOIDAL SOURCE Changing the phase angle shifts the sinusoidal function in time but has no effect on either amplitude or angular frequency. Note also that if is positive, the sinusoidal function shifts to the left. The example shows 

𝑉 cos 𝜔𝑡 𝑉 cos 𝜔𝑡 + 𝜙

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which is



shifted to the left by

 

time units.

Can convert degrees to radians and back:

SINUSOIDAL SOURCE We say that

Lagging and leading:



by



If

is -90, then

leads rad.



 



Also 

Cosine lags 90 = sine 𝑉 sin 𝜔𝑡 𝑉 sin 𝜔𝑡 − 𝜃

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Sine advances 90 = cosine

EXERCISE Given

and

Fill the blanks, a. b.

leads

c.

lags

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by by

EXERCISE Given

and

Fill the blanks, a. b.

leads

c.

lags

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by by

SINUSOIDAL SOURCE/ RMS VALUE RMS value: “root mean square” •

An important characteristic of the sinusoidal function

•

The RMS value is the amplitude of DC source that would have the same average power output as the sinusoid. 1st operation: square

 -0.5

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0

0.5

1

SINUSOIDAL SOURCE/ RMS VALUE 2nd operation: mean   

 

Integrate and divide over one period to obtain mean which yields:  

 



 

   

 

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

 

SINUSOIDAL SOURCE/ RMS VALUE 3rd operation: root  

 

 

Sinusoidal source RMS value

• The RMS value of the sinusoidal voltage (current) depends only on the maximum amplitude of v(t). The RMS is not a function of either the frequency or phase angle. • We can completely describe a specific sinusoidal signal if we know its frequency, phase angle, and amplitude (either the maximum or the RMS value).

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SINUSOIDAL SOURCE Example: A sinusoidal current has passes through one complete cycle in the current at is . (a) What is the frequency? (in Hertz) (b) What is the angular frequency? (c) Write the expression for (d) RMS value?

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. The current , the magnitude of

SINUSOIDAL SOURCE . Complete cycle in



a) What is the frequency? (in Hertz)

(b) What is the angular frequency? 6283 (c) Write the expression for 

(d) RMS value?  

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)=

....