Othogonality of Discrete Sine and Cosine Series
The specific case of interest is the inner product:
\[ \label{forceEq} f = p' \Lambda q \] where \(p\) and \(q\) are vectors defining the \(m^{th}\) and \(n^{th}\) harmonic of flux around a magnetic bearing rotor and
\[ \label{lambdax} \Lambda = \mbox{diag}(\cos(\Theta) ) \] or \[ \label{lambday} \Lambda = \mbox{diag}(\sin(\Theta)) \] are diagonal matrices that define the contributions of each pole to the force in a particular force direction where pole angle \(\Theta\) is defined as:
\[ \label{poleAngle} \Theta = 360^o \left( \frac{k}{N} \right)_{k=0 \ldots N-1} \] Inner product \(\ref{forceEq}\) can be re-written as an explicit summation with a single index, since \(\Lambda\) is diagonal:
\[ \label{appEq0} f = \sum_{k=0}^{N-1} Z_{k} \] In all, there are six cases to consider, spanning all combinations of cosine or sine distributions for \(\Lambda\), \(p\) and \(q\):
\[ \label{appEq1} Z_{1,k}= \cos \left(\frac{2 \pi k}{N} \right) \cos \left(\frac{2 \pi m k }{N} \right) \cos \left(\frac{2 \pi n k }{N} \right) \] \[ \label{appEq2} Z_{2,k}= \cos \left(\frac{2 \pi k}{N} \right) \cos \left(\frac{2 \pi m k }{N} \right) \sin \left(\frac{2 \pi n k }{N} \right) \] \[ \label{appEq3} Z_{3,k}= \cos \left(\frac{2 \pi k}{N} \right) \sin \left(\frac{2 \pi m k }{N} \right) \sin \left(\frac{2 \pi n k }{N} \right) \] \[ \label{appEq4} Z_{4,k}= \sin \left(\frac{2 \pi k}{N} \right) \cos \left(\frac{2 \pi m k }{N} \right) \cos \left(\frac{2 \pi n k }{N} \right) \] \[ \label{appEq5} Z_{5,k}= \sin \left(\frac{2 \pi k}{N} \right) \cos \left(\frac{2 \pi m k }{N} \right) \sin \left(\frac{2 \pi n k }{N} \right) \] \[ \label{appEq6} Z_{6,k}= \sin \left(\frac{2 \pi k}{N} \right) \sin \left(\frac{2 \pi m k }{N} \right) \sin \left(\frac{2 \pi n k }{N} \right) \]
The goal is to show that, each case, \(\ref{forceEq}\) equals zero when \(|m-n|\neq1\).
Using trigonometric angle sum and difference identities [1], the six cases can be re-written as:
\[ \label{appEq1a} Z_{1,k}= \frac{1}{2} \cos \left(\frac{2 \pi k}{N}(m-1) \right) \cos \left(\frac{2 \pi n k }{N} \right) + \frac{1}{2} \cos \left(\frac{2 \pi k}{N}(m+1) \right) \cos \left(\frac{2 \pi n k }{N} \right) \] \[ \label{appEq2a} Z_{2,k}= \frac{1}{2} \cos \left(\frac{2 \pi k}{N}(m-1) \right) \sin \left(\frac{2 \pi n k }{N} \right) + \frac{1}{2} \cos \left(\frac{2 \pi k}{N}(m+1) \right) \sin \left(\frac{2 \pi n k }{N} \right) \] \[ \label{appEq3a} Z_{3,k}= \frac{1}{2} \sin \left(\frac{2 \pi k}{N}(m-1) \right) \sin \left(\frac{2 \pi n k }{N} \right) + \frac{1}{2} \sin \left(\frac{2 \pi k}{N}(m+1) \right) \sin \left(\frac{2 \pi n k }{N} \right) \] \[ \label{appEq4a} Z_{4,k}= \frac{1}{2} -\sin \left(\frac{2 \pi k}{N}(m-1) \right) \cos \left(\frac{2 \pi n k }{N} \right) + \frac{1}{2} \sin \left(\frac{2 \pi k}{N}(m+1) \right) \cos \left(\frac{2 \pi n k }{N} \right) \] \[ \label{appEq5a} Z_{5,k}= \frac{1}{2} -\sin \left(\frac{2 \pi k}{N}(m-1) \right) \sin \left(\frac{2 \pi n k }{N} \right) + \frac{1}{2} \sin \left(\frac{2 \pi k}{N}(m+1) \right) \sin \left(\frac{2 \pi n k }{N} \right) \] \[ \label{appEq6a} Z_{6,k}= \frac{1}{2} \cos \left(\frac{2 \pi k}{N}(m-1) \right) \sin \left(\frac{2 \pi n k }{N} \right) - \frac{1}{2} \cos \left(\frac{2 \pi k}{N}(m+1) \right) \sin \left(\frac{2 \pi n k }{N} \right) \]
Orthogonality of Digital Fourier Transform (DFT) sinusoids (see [2]) can then be invoked to understand the conditions for non-zero \(\ref{forceEq}\). Inspecting cases 2, 4, and 6, these cases always yield zero force since cosine and sine series are always orthogonal. For cases 1, 3, and 5, cosines are interacting with cosines and sines are interacting with sines. However, the frequency has to be the same for a non-zero result. Inspecting each of these cases, the frequency can only be the same if \(m-n=1\) (first term in each case yields a non-zero sum and second term yields a zero sum) or if \(n-m=1\) (first term in each case yields a zero sum and second term yields a non-zero sum).
In every case, \(f=0\) if \(|m-n| \neq 1\). This result allows the set of even-numbered harmonics or the set of odd-numbered harmonics to be selected as bias spaces because \(|m-n| \geq 2\) for each of these sets.
References
[1] "List of trigonometric identities", Wikipedia, accessed 05Oct2019.
[2] J. Smith, Mathematics of the Discrete Fourier Transform (DFT): with Audio Applications, 2nd ed., W3K Publishing, 2007.