The Modulation Process The harmonic transfers through the a.c.–d.c. converter are best explained using modulation theory [17,18]. The voltage and current relationships across the converter can be expressed as follows:
vd =
Yψdcvψ (3.74)
id = Yψac· idc (3.75)
where Yψdc and Yψac are transfer functions for the voltages and current, respectively, and ψ= 0◦, 120◦, 240◦ for the three phases.
This process is illustrated in Figure 3.49, which shows the modulated output cur-rent on the a.c. side of the converter in response to a d.c. curcur-rent that contains a ripple frequency [19]. In the absence of commutation overlap, the transfer functions are rectangular, as shown in Figure 3.49.
The simplified modulation process, explained above, can be extended to the more realistic case where the commutation process is included. This extension, explained further in Chapter 8, is necessary to derive accurate quantitative information from the generalised table to be described in the following section.
t
t +1
I
Iac/phase
−1
+1
t
d.c. + ripple Switching function
Modulating function
ia ib ic
Modulated output
Figure 3.49 A.c. current modulation under ideal converter switching
Cross-Modulation Across an a.c. –d.c. –a.c. Link The frequency transfer relation-ships in the cross-modulation process of a line-commutated 12-pulse a.c.–d.c.–a.c.
link have been collected together in Figure 3.50 [20]. The link can be an HVd.c. inter-connection between two a.c. systems of frequencies f1 and f2 or the supply system for a synchronous variable speed drive.
The ‘exciting’ harmonic sources are multiples, integers or non-integer, of the fre-quency in system 1.
k1 is a current harmonic source, whereas k1− 1 and k1+ 1 are voltage harmonic sources.
The resulting harmonic orders in system 2 are related to the frequency of system 2.
The DC column refers to the d.c. side of the link; the AC1+and AC1− columns rep-resent the positive and negative sequences of system 1, and AC2+ and AC2−represent the positive and negative sequences of system 2, respectively.
When the d.c. link interconnects separate power systems, either of the same nominal (but in practice slightly different) frequency or different nominal frequencies, there will be a wide range of harmonic and non-harmonic frequency transfers. These can be divided into two groups:
(1) Frequencies at terminal 1 caused by the characteristic d.c. voltage harmonics (12nf2) and their consequential currents from terminal 2. These are represented in the expression
fac1= (12m ± 1)f1± 12nf2 (3.76)
114 HARMONIC SOURCES
AC2−AC2+DCAC1+AC1− k1 f1/f2 −1 (12n − 1) ± k1 f1/f2
k1f1/f2 +1 (12n +1) ±k1 f1/f2k1
k1 + 1 (12n + 1)± k1
k1 − 1 (12n − 1) ± k1 k1 f1/f2 −1 (12n ± k1) ± f1/f2 −1 (12m − 1) ± k1f1/f2 (12m−1) ±(12n ± k1) f1/f2
k1 f1/f2 + 1 (12n ± k1) f1/f2 + 1 (12m + 1) ± k1 f1/f2 (12m + 1) ±(12n ± k1)f1/f2
k1 12n ± k1
k1+ 1 k1+ 1 (12n ± k1) + 1 (12m + 1) ±k1 (12m + 1)± (12n ± k1)
k1 − 1 (12n ± k1) − 1 (12m − 1) ± k1 (12m − 1) ±(12n ± k1) k1 − 1 Figure3.50Harmonictransfersacrossa12-pulsea.c.–d.c.–a.c.link.Theencircledelementsindicateharmonicsourcesandm,n∈(1,2,3...)
where m, n ε (0, 1, 2, 3. . .) which can have any frequency, including frequencies below the fundamental.
The back-to-back frequency conversion schemes represent the worst condition for non-integer harmonic frequencies. In this case, with small smoothing reactors the d.c. side coupling is likely to be strong which means that the flow of harmon-ically unrelated currents on the d.c. side can be large. In six-pulse operation such schemes can produce considerable subharmonic content even under perfect a.c.
system conditions. However, 12-pulse converters do not produce subharmonic content under symmetrical and undistorted a.c. system conditions. These will produce inter-harmonic currents as defined by equation (3.76).
When the link interconnects two isolated a.c. systems of the same nominal frequency but they differ by a small increment f0, then the characteristic har-monics are different by 12nf0. A d.c. side voltage at frequency 12n(f0+ f0) is generated by one converter, and this will be modulated down again at the other converter by a characteristic frequency in the thyristor switching pattern in accordance with equation (3.76) i.e.
(12m± 1)f0± 12n(f0+ f0)
which on the a.c. side, among other frequencies, includes (for m= n):
f0± 12nf0
The latter will beat with the fundamental component at a frequency 12nf0, which at some values of n will allow flicker-inducing currents to flow.
(2) Frequencies caused in system 1 by unbalance or distortion in the supply voltage of system 2. Negative sequence voltages at frequencies (k− 1)f2 produce the following non-characteristic frequencies on the d.c. side:
fdc= (12n ± k)f2 (n= 0, 1, 2 . . .) (3.77) Cross-modulation of these current components produces the following frequen-cies in system 1
fac1= (12m ± 1)f1± (12n ± k)f2. (3.78) Let us first consider a frequency conversion scheme with a sinusoidal but negative sequence unbalanced voltage in system 2, i.e. (k− 1) = 1 (and therefore k= 2). Substituting m = n = 0 and k = 2 in equation (3.76) yields currents (and therefore voltage) at frequencies
fac1= ±f1± 2f2 (3.79)
One of these frequencies (f1− 2f2) will beat with the fundamental frequency voltage of system 1 at a frequency
f1+ (f1− 2f2)= 2(f1− f2) (3.80)
116 HARMONIC SOURCES
which for a 50–60 Hz conversion scheme becomes 20 Hz. This is a flicker-producing frequency. This same frequency will be referred to generator rotor shaft torque at 20 Hz, which may excite mechanical resonances.
Again, this type of cross-modulation effect is most likely to happen in back-to-back schemes due to the stronger coupling between the two converters, although it is also possible with any HVd.c. scheme in the presence of a suitable resonance.
Now consider two a.c. systems of the same nominal (but slightly different) frequency.
Substituting m= n = 0 and k = 2, for fundamental frequency f0 into equation (3.78), a current and resultant voltage (through the a.c. system impedance) of frequency
fac1= ±f0± 2(f0+ f0) (3.81) which leads to f0± 2f0 is induced on the a.c. side. This will either beat with the fundamental frequency f0 or produce generator/motor shaft torques at 2f0. This frequency is generally too low to produce flicker but may induce mechanical oscillations.
Substituting m= n = 1 and k = 2 in equation (3.78) gives, among others, a current (and thus voltage) at the frequency
(12+ 1)f0− (12 + 2)(f0− f )
and for f0= 50 Hz and f = 1 Hz, the resulting a.c. current (and thus voltage) in system 1 is:
13× 50 − 14 × 49 = 36 Hz
This distorting voltage will, therefore, beat with the fundamental, producing 14 Hz flicker. However, the subharmonic levels expected from this second-order effect will normally be too small to be of consequence.