The idea is based on modulating the rotor field current with one or several AC components that targets specific magnetic flux density harmonics to either eliminate certain voltage harmonics or exciting force harmonics. The magnetic flux density in the airgap of an electric machine can be expressed as
$$\begin{aligned} B_{s} (\theta ,t) = \bigg [\sum _{n=1}^\infty A_n \sin \left[ p\big (n\theta + \omega _\mathrm{m} t\big ) \right] \bigg ]B_\mathrm{r}(t), \end{aligned}$$
(2)
where
\(A_n\) is a scaling factor for the different space harmonics,
\(\omega _\mathrm{m}\) is the mechanical angular frequency,
p is the number of pole pairs in the machine, and
\(B_\mathrm{r}\) is the amplitude of the radial magnetic flux density in the airgap produced by the rotor. The field current,
\(I_\mathrm{f}\), in the machine is changed by adding harmonics
$$\begin{aligned} I_\mathrm{f}(t) = I_{\mathrm{f},0} + \sum _{k=1} \hat{I}_{\mathrm{f},k} \sin (\omega _{k} t+ \alpha _k), \end{aligned}$$
(3)
where
\(\omega _k\) and
\(\alpha _k\) are the angular frequency and the phase of the added current harmonic. The contribution from the rotor to the magnetic flux density in the airgap depend on the field current according to
$$\begin{aligned} B_\mathrm{r}(t) = B_{\mathrm{r},0}(I_{\mathrm{f},0}) + \sum _{k=1} \hat{B}_{\mathrm{r},k}(I_{\mathrm{f},k}) \sin (\omega _{k} t+ \alpha _k). \end{aligned}$$
(4)
By inserting (
4) into (
2) the complete expression for the magnetic field in the airgap of the machine when the rotor field current is modulated can be obtained as
$$\begin{aligned} B_{s} (\theta ,t)&= \bigg [\sum _{n=1}^\infty A_n \sin \left[ p\big (n\theta + \omega _\mathrm{m} t\big ) \right] \bigg ]\bigg [B_{\mathrm{r},0}(I_{\mathrm{f},0}) \nonumber \\&\qquad {}+ \sum _{k=1} \hat{B}_{\mathrm{r},k}(I_{\mathrm{f},k}) \sin (\omega _{k} t+ \alpha _k)\bigg ]. \end{aligned}$$
(5)
In order to show the effect on stator output voltage (dependent on B directly) and exciting magnetic force (proportional to
\(B^2\)), use
\(\omega _k = kp\omega _\mathrm{m}\), and consider adding one current harmonic
k to the DC-rotor current. Only the fundamental of the zeroth order field current,
\(A_n = 1, n = 1\) was considered for clarity. The magnetic flux density for this simplified case can then, from (
5), be expressed as
$$\begin{aligned} B_s(\theta ,t)&= B_{\mathrm{r},0}(I_{\mathrm{f},0})\sin (p\omega _\mathrm{m} t) +\frac{B_k(\hat{I}_{\mathrm{f},k})}{2} \Big (\cos \big [(k+1)\nonumber \\&\qquad {} p\omega _\mathrm{m} t - p\theta + \alpha _k\big ] + \cos \big [(k-1)p\omega _\mathrm{m} t + p\theta \nonumber \\&\qquad {} + \alpha _k\big ]\Big ). \end{aligned}$$
(6)
The stator output voltage of the machine can be obtained by integrating
\(\theta \) for one coil span and the number of poles. This means that a “
\(k+1\)”, or “
\(k-1\)”, component added in the field current will affect the voltage component “
k”. The method can easily be generalized to include higher harmonics.
Equation (
6) was squared to get an expression that is directly proportional to the exciting force, since the Maxwell stress tensor depends partly on the square of the flux density. One point per pole on the stator side was of interest,
\(\theta = 0^\circ \) was chosen for simplicity. The amplitude and order of the
\(B^2\) harmonics can be extracted from
$$\begin{aligned} B_s^2(t)&= \Bigg [B_{\mathrm{r},0}(I_{\mathrm{f},0})\cos (p\omega _\mathrm{m} t) + \frac{B_k(\hat{I}_{\mathrm{f},k})}{2} \bigg (\cos \big [(k+1) \nonumber \\&\qquad {} p\omega _\mathrm{m} t)\big ] + \cos \big [(k-1)p\omega _\mathrm{m} t)\big ]\bigg )\Bigg ]^2 \nonumber \\ {}&= \frac{B_{0}^2(I_{\mathrm{f},0})}{2}\bigg [1 + \cos (2p\omega _\mathrm{m}t)\bigg ] + \frac{B_k(I_{\mathrm{f},k}) B_{0}(I_{\mathrm{f},0})}{2} \nonumber \\&\qquad {} \bigg [\cos \big [(k+2)p\omega _\mathrm{m}t\big ] + \cos \big [(k-2)p\omega _\mathrm{m}t\big ] + 2\cos \nonumber \\&\qquad {}\big [kp\omega _\mathrm{m}t\big ]\bigg ]+\frac{B_k^2(I_{\mathrm{f},k})}{8}\bigg [2+\cos \big [2(k+1)p\omega _\mathrm{m} t\big ] \nonumber \\&\qquad {}+ \cos \big [2(k-1)p\omega _\mathrm{m} t\big ] + 2\cos (2kp\omega _\mathrm{m} t) \nonumber \\&\qquad {}+ 2\cos (2p\omega _\mathrm{m} t)\bigg ], \end{aligned}$$
(7)
where
\(\alpha _k\) is set to
\(0^\circ \). The expression (
7) contains only harmonic waves of double order (e.g. for a 50Hz system the fundamental is 100 Hz). To affect a harmonic of order
k, whose amplitude is
\(B_k(I_{\mathrm{f},k}) B_{0}(I_{\mathrm{f},0})\), a current harmonic of the same order,
k, should be added, and the phase adjusted accordingly. However, there is a drawback as adding a
k-th component will also affect the amplitude of harmonics
\(k\pm 2\).