Electrical circuits wyklad 8

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Electrical Circuits

Dr inż. Agnieszka Wardzińska Room: 105 Polanka agnieszka.wardzinska@put.poznan.pl cygnus.et.put.poznan.pl/~award Advisor hours: Monday: 9.30-10.15 Wednesday: 10.15-11.00

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Remarks about phasor

diagrams

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Phasor Relationships for Resistor

I V I R e RI t RI iR v e I t I i j m m j m m            









) cos( law, s Ohm' By ) cos( is resistor rough the current th the If

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Phasor Relationships for Inductor

Time domain Phasor domain

Phasor diagram

I V I L j t LI dt di L v e I t I i m j m m











            ) 90 cos( is inductor the across voltage The ) cos( is inductor rough the current th the If

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Phasor Relationships for Capacitor

Phasor diagram

Time domain Phasor domain

V I V C j t CV dt dv C i e V t V v m j m m











            ) 90 cos( is capacitor rough the current th The ) cos( is capacitor the across voltage the If

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Mixed Frequency signals

 _{A sinusoidal waveform is one shaped exactly}

like a sine wave.

 _{A non-sinusoidal waveform can be anything}

from a distorted sine-wave shape to

something completely different like a square wave.

 _{Mixed-frequency waveforms can be}

accidently created, purposely created, or simply exist out of necessity. Most musical

tones, for instance, are not composed of a single frequency sine-wave, but are rich blends of different frequencies.

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Mixed Frequency signals

 _{When multiple sine waveforms are mixed together (as is}

often the case in music), the lowest frequency sine-wave is called the fundamental, and the other sine-waves

whose frequencies are whole-number multiples of the fundamental wave are called harmonics.

 _{An overtone is a harmonic produced by a particular}

device. The “ﬁrst” overtone is the ﬁrst frequency greater than the fundamental, while the “second”

overtone is the next greater frequency produced.

Successive overtones may or may not correspond to incremental harmonics, depending on the device

producing the mixed frequencies. Some devices and systems do not permit the establishment of certain harmonics, and so their overtones would only include some (not all) harmonic frequencies.

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 _{Any regular (repeating), non-sinusoidal waveform is}

equivalent to a particular series of sine/cosine waves of different frequencies, phases, and amplitudes, plus a DC

offset voltage if necessary. The mathematical process for

determining the sinusoidal waveform equivalent for any waveform is called Fourier analysis.

 _{Multiple-frequency voltage sources can be simulated} for analysis by connecting several single-frequency voltage sources in series. Analysis of voltages and

currents is accomplished by using the superposition

theorem. NOTE: superimposed voltages and currents of

different frequencies cannot be added together in

complex number form, since complex numbers only account for amplitude and phase shift, not frequency!

 _{Harmonics can cause problems by impressing unwanted}

(“noise”) voltage signals upon nearby circuits. These

unwanted signals may come by way of capacitive coupling, inductive coupling, electromagnetic radiation, or a

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 _{Any waveform at all, so long as it is}

repetitive, can be reduced to a series of sinusoidal waveforms added together.

Different waveshapes consist of different blends of sine-wave harmonics.

 _{Rectiﬁcation of AC to DC is a very common}

source of harmonics within industrial power systems.

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Square wave signals

1 V (peak) repeating square wave at 50 Hz is equivalent to:

(1 V peak sine wave at 50 Hz) + 4 π

(1/3 V peak sine wave at 150 Hz) + 4 π

(1/9 V peak sine wave at 450 Hz) + . . . ad inﬁnitum

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Parseval’s Theorem

 _{The integral of the square of a function is equal}

with the integral of the squared components od its spectrum. This means that the total energy of a waveform can be found in total energy of the waveform’s components.

 _{As each signal forming the arbitrary signals can}

be docomposed in iths spectrum componennts, all these components contribute to the total

energy of the arbitrary waveform and the RMS value is the square root of the sum of swuares of each spectrum component.

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Power in mixed signals

 _{Active Power in Watts are equal to sum of}

active power of all harmonics and power of DC component

 _{Reactive power in var are equal to sum of}

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Power in mixed signals

 _NOTE

 _{The sum of squares of active and reactive}

power of mixed signal IS NOT equal of the square of apparent power

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