Electric Circuits II...rms value The period of the waveform is T=2π. Over a period, the voltage waveform is Example: The waveform shown is a half-wave rectified sine wave. Find the

Electric Circuits II Sinusoids and phasors

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Dr. Firas Obeidat

Dr. Firas Obeidat – Philadelphia University

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Table of Contents

1. • Periodic Function

2. • The Sinusoidal Source

3. • Lagging and Leading

4. • Converting Sines to Cosines

5. • Average value

6. • rms value


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Periodic Function

The term alternating indicates only that the

waveform alternates between two prescribed

levels in a set time sequence.

A DC voltage or current has a fixed magnitude

(amplitude) and a definite direction associated

with it. And do not change their values with

regards to time, they are a constant values

flowing in a continuous steady state direction.

• The Period, (T) is the length of time in seconds that the

waveform takes to repeat itself from start to finish. This can

also be called the Periodic Time of the waveform.

• The Frequency, (ƒ) is the number of times the waveform repeats

itself within a one second time period. Frequency is the

reciprocal of the time period, ( ƒ = 1/T ) with the unit of

frequency being the Hertz, (Hz).

• The Amplitude (A) is the magnitude or intensity of the signal

waveform.

A periodic function is one that satisfies

f(t)=f(t+nT), for all t and for all integers n.


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The Sinusoidal Source

A sinusoidal voltage source (independent or dependent) produces a voltage that

varies sinusoidally with time. A sinusoidal current source (independent or

dependent) produces a current that varies sinusoidally with time.

The common relationship between frequency and

angular frequency.

The function repeats itself every 2π radians, and its

period is therefore 2π radians. A sine wave having a

period T must execute 1/T periods each second; its

frequency f is 1/T hertz, abbreviated Hz. Thus,


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Lagging and Leading

A more general form of the sinusoid.

The phase angle appears as the number of

radians by which the original sine wave

(shown in green color in the sketch) is shifted

to the left, or earlier in time.

We say that Vmsin(ωt + θ) leads Vmsinωt by θ rad. Therefore, it is correct to describe

sinωt as lagging sin(ωt + θ) by θ rad, as leading sin(ωt − θ) by θ rad.

In either case, leading or lagging, we say that the sinusoids are out of phase. If the

phase angles are equal, the sinusoids are said to be in phase.


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Converting Sines to Cosines


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Examples

Example: Find the amplitude, phase, period, and frequency of the sinusoid

Example: Calculate the phase angle between v1=-10cos(ωt+50o) and

v2=12sin(ωt-10o) State which sinusoid is leading.


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Average value

Average value is defined as the area under the curve divided by the baseline of the

curve.

Example: find the average value for the curve shown in the figure?

The area under this curve can be computed as

Area=(80×1)+(60×2)+(95×1)+(75×1)

Now divide this by the length of the base, namely 5.

Average=(80×1)+(60×2)+(95×1)+(75×1)/5=74


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Average value

The average value of a waveform: divide the area under the waveform by the

length of its base. Areas above the axis are counted as positive, while areas below

the axis are counted as negative. This approach is valid regardless of wave shape.

Example:

1. Compute the average for the current waveform shown in the figure.

2. If the negative portion of the figure is ( 3 A ) instead of (1.5 A), what is the average?

3. If the current is measured by a dc ammeter, what will the ammeter indicate for each

case?

Average values are also called dc values, because dc meters indicate average

values rather than instantaneous values. Thus, if you measure a non-dc

quantity with a dc meter, the meter will read the average of the waveform

1. The waveform repeats itself after 7ms. Thus, T=7 ms and

the average is

𝐼𝐴𝑣𝑔 =2𝐴 × 3𝑚𝑠 − (1.5𝐴 × 4𝑚𝑠)

7𝑚𝑠=6 − 6

7= 0𝐴


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Average value 2. The waveform repeats itself after 7ms. Thus, T=7 ms and the

average is

𝐼𝐴𝑣𝑔 =2𝐴 × 3𝑚𝑠 − (3𝐴 × 4𝑚𝑠)

7𝑚𝑠=−6

7𝐴 = −0.857𝐴

3. A dc ammeter measuring (a) will indicate zero, while for (b)

it will indicate 0.857 A.

Example: Compute the average value for the waveforms of the figures. Sketch the

averages for each.

1- For the first waveform, T=6 s. Thus,

𝐼𝐴𝑣𝑔 =10𝑉 × 2𝑠 + 20𝑉 × 1𝑠 + (30𝑉 × 2𝑠) + (0𝑉 × 1𝑠)

6𝑠=100 𝑉. 𝑠

6𝑠= 16.7𝑉


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Average value

1- For the first waveform, T=8s. Thus,

𝐼𝐴𝑣𝑔 =0.5 40𝑚𝐴 × 3𝑠 − 20𝑚𝐴 × 2𝑠 − (40𝑚𝐴 × 2𝑠)

8𝑠=−60𝑚𝐴

8𝑚𝐴 = −7.5𝑚𝐴


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Sine wave average value

Because a sine wave is symmetrical, its area below the horizontal axis is the same

as its area above the axis; thus, over a full cycle its net area is zero, independent

of frequency and phase angle. Thus, the average of sinωt, sin(ωt±θ), sin2ωt,

cosωt, cos(ωt±θ), cos2ωt, and so on are each zero.

The average of half a sine wave, however, is not zero.

The area under the half-cycle can be found as

Two cases are important in electronics; full-wave average and

half-wave average. The area for full-wave case from 0 to 2π is

2(2Im) and the base is 2π. Thus, the average is

The area for half-wave case from 0 to 2π is (2Im) and the base is

2π. Thus, the average is


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rms value

rms value is defined as the square root of

the area under the squared curve divided

by the baseline of the curve.

To compute rms values using this equation, do the following:

Step 1: Square the voltage (or current ) curve.

Step 2: Find the area under the squared curve.

Step 3: Divide the area by the length of the curve.

Step 4: Find the square root of the value from Step 3.

Example: One cycle of a voltage waveform is shown in the figure. Determine its (rms)

value.

𝑣𝑟𝑚𝑠 =400 × 4 + 900 × 2 + 100 × 2 + (0 × 2)

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𝑣𝑟𝑚𝑠 =3600

10= 19 𝑉

An effective (rms) value is an equivalent dc value: it tells you how many volts or

amps of DC that a time-varying waveform is equal to in terms of its ability to

produce average power.

𝑽𝒓𝒎𝒔 =𝑨𝒓𝒆𝒂 𝒖𝒏𝒅𝒆𝒓 𝒗𝟐 𝒄𝒖𝒓𝒗𝒆

𝑩𝒂𝒔𝒆


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rms value

The rms value of a periodic function is defined as the square root of the mean value

of the squared function.

𝑰𝒓𝒎𝒔 =𝟏

𝑻 𝑰𝒎

𝟐𝒄𝒐𝒔𝟐 𝝎𝒕 + 𝒅𝒕𝒕𝒐+𝑻

𝒕𝒐

For sinusoidal

current/voltage 𝑰𝒓𝒎𝒔 =

𝑰𝒎

𝟐

𝑰𝒓𝒎𝒔 = 𝑰𝒎ω

𝟐π

𝟏

𝟐+𝟏

𝟐𝒄𝒐𝒔(𝟐𝝎𝒕 + 𝟐 𝒅𝒕

𝟐π/ω

𝟎

= 𝑰𝒎ω

𝟒π×𝟐π

ω

𝑽𝒓𝒎𝒔 =𝑽𝒎

𝟐

For any periodic function x(t) , the rms value is given by

For the sinusoid i(t) = Imcos(ωt + ), T=2π/ω, the rms value of i(t) is.


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rms value

The period of the waveform is T=4. Over a period,

the current waveform is

Example: Determine the rms value of the current

waveform. If the current is passed through a

resistor, find the average power absorbed by the

resistor.


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rms value

The period of the waveform is T=2π. Over a

period, the voltage waveform is

Example: The waveform shown is a half-wave rectified sine

wave. Find the rms value and the amount of average power

dissipated in a 10 Ω resistor.

Q: Find the rms value of the

full-wave rectified sine wave.

Calculate the average power

dissipated in a 6 Ω resistor.


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Example 1:

Example 2:

rms value

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Electric Circuits II...rms value The period of the waveform is T=2π. Over a period, the voltage waveform is Example: The waveform shown is a half-wave rectified sine wave. Find the