Chapt 1_2 Fourier_THD_Terminologies

Chapt 1_2 Fourier_THD_Terminologies - 27 Periodic function:...

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27 Periodic function: For any physically realizable periodic function: f(t) = f(t+T), (period T ) Everage value: = = T X T T ave dx x v X dt t v T V 0 0 ) ( 1 ) ( 1 Applications: average voltage, average current, average power,… Root mean square value: = = T X T T rms dx x v X dt t v T V 0 2 0 2 ) ( 1 ) ( 1 Applications: Root mean square values of voltage, current (Vrms, Irms) Harmonics components- Fourier series Analysis: Switch action is periodic by design. We often have specific input frequencies, and seek specific output frequencies, but many frequencies occur together. These mean that we need to explore frequency content of our waveforms. Any physically realizable periodic function, can be written as a sum of sinusoids: where the sum is taken over n =1 to infinity, ω = 2π/ T , and the a n and b n coefficients are given by explicit integral equations, ( 29 ( 29 T t f t f + = - Using angular time: The varia ( 29 ( 29 ( 29 [ ] + + = nwt b nwt a a t f n n sin cos 0 ( 29 ( 29 ( 29 [ ] = + + = 1 0 sin cos n n n nwt b nwt a a t f
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28 - Each cosine term, c n cos( n ω t + θ n ), is called a Fourier component or a harmonic of the function f(t). We call each the n th harmonic. - The value c n is the component amplitude; θ n is the component phase. - c 0 = a 0 is the dc component, equal to the average value of f(t), c 0 = < f ( t )>. - The term c 1 cos(ω t + θ 1 ) is the fundamental of f(t), while 1/ T is the fundamental frequency . In most converters, we seek a single desired frequency (perhaps the output frequency). This is associated with a single wanted component . Others are unwanted components The change of variables θ = ω t is often useful. In many cases, the waveform shape, rather than explicit timing, is the important issue. Example: three-pulse rectifier load voltage
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29 Example: Square wave of inverter: Definition Power Assume a voltage and a current + = + = ) cos( ) ( ) cos( ) ( m m n n t m d t i t n c t v φ ϖ θ With the same base frequency T / 2 π = Power and average power: [ ] [ ] + + = = ) cos( . ) cos( ) ( ) ( ). ( ) ( m m n n t m d t n c t p t i t v t p
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30 Energy conversion: [ ] [ ] + + = T m m n n ave dt t m d t n c P 0 . ) cos( . ) cos( φ ϖ θ ∑∑ + + = T m n m n m n ave dt t m t n d c P 0 . ) cos( ) cos( Power result: -Cross-frequency terms like cos(nwt)cos(mwt) with n and m unequal do not contribute to average power. (to energy flow) -Average power is the sum of contributions at each frequency. Only Fourier components that appear in both the current and the voltage contribute to average power flow. Frequency matching condition: -To draw power from a source or - To deliver it to a load, there must be components at matching frequencies. If the source is given, we must match
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Chapt 1_2 Fourier_THD_Terminologies - 27 Periodic function:...

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