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Mathematical Methods in the Physical Sciences Mary L. Boas 2nd edition Solutions
Fourier series for output voltages of inverter waveforms.
The Fourier series for a periodic function vo( ω ωt) t ) can be expressed as
vo (ω t ) = a o +
∞
∑a
n
cos(nω t ) + bn sin( nω t )
n =1
For an odd quarter-wave symmetry waveform,
ao = 0
an = 0
and
2 4 bn = π ∫ vo sin( nω t ) d (ω t ) 0 0 π
for odd n for even n
Therefore, vo( ω ωt t ) can be written as
vo (ω t ) =
∞
∑b
n
sin( nω t )
n =odd
(i)
Square-wave π
2
4
bn =
∫ v
π
+V dc dc
sin( nω t )d (ω t )
o
0 π
π/2
=
π
2π
π
=
-V dc dc
=
(ii)
2
4
∫ V
dc
sin( nω t ) d (ω t )
0
4V dc
nπ 4V dc
π
[− con(nω t ]02
nπ
Quasi square-wave π
+V dc dc
bn =
4 π
2
∫ v
o
sin( nω t ) d (ω t )
α π
α
π/2
π
2π
=
4 π
2
∫ V
dc
sin( nω t ) d (ω t )
α
-V dc dc
= =
4V dc
nπ 4V dc nπ
π
[− con(nω t ] 2
α
cos(nα )
(iii)
Notched waveform (Harmonics Elimination PWM)
Vdc
α1 α2 α3
π/2
π
Vdc π
bn =
4 π
2
∫ v
o
sin(nω t ) d (ω t )
0
2 4 = ∫ V dc sin(nω t )d (ω t ) + ∫ V dc sin(nω t )d (ω t ) π 4V = dc [− con(nω t ] + [− con(nω t ] 2 nπ π
α 2
α 1
α 3
π
α 2 α
1
=
(iv)
4V dc
nπ
cos(nω t )
α
3
1 ,α 3
α
α 2
,
π
2
=
4V dc π cos( ) α ) + cos( nα ) − cos( nα ) − cos( n n 1 3 2 2 nπ
=
4V dc
nπ
[cos(nα 1 ) + cos(nα 3 ) − cos(nα 2 )]
Sinusoidal PWM (unipolar and bipolar)
Unipolar SPWM waveform as an example
vo (ω t ) = M aV dc sin(ω t ) +
4V dc π
sin nπ + ( k ) cos(ω t ) 2 2 n n =1 ∞
∑
= M aV dc sin(ω t ) + Bessel Function for harmonic terms The tables are required to resolve for the Bessel function for harmonic terms. The harmonics in the inverter output appear as sidebands, centered around the switching frequency, that is, around mf, 2mf , 3mf and so on. This general pattern hold true for all values of ma in the range 0 – 1 and m f >9. The unipolar SPWM switching scheme has the advantage of “effectively” doubling the switching frequency as far as the output harmonics are concerned, compared to the bipolar SPWM switching scheme. Because of that the harmonics in the inverter output of unipolar SPWM are centered around 2m f , 4mf , 6mf and so on.
Table 8.3 shows the first harmonic frequencies in the output spectrum at and around m f for the bipolar SPWM switching scheme. The harmonics at and around 2m f , 3m f , 4m f and so on are not indicated.
Table 8.5 shows the first harmonic frequencies in the output spectrum at and around 2m f for the uipolar SPWM switching scheme. The harmonics at and around 4m f , 6m f , 8m f and so on are not indicated.
Table 8.3 and 8.5 can be used to predict the THD for the ouput current of the inverter connected to RL load. Higher order harmonics are assumed to contribute little power and effect, so they can be neglected. To evaluate the THD for the output voltage of the inverter, higher order harmonics should be taken into account.
