9.2: Winding Inductance Calculation

The purpose of this section is to show how the inductances of windings in round- rotor machines with narrow air gaps may be calculated. We deal only with the idealized air- gap magnetic fields, and do not consider slot, end winding, peripheral or skew reactances. We do, however, consider the space harmonics of winding magneto-motive force (MMF).

To start, consider the MMF of a full- pitch, concentrated winding. Assuming that the winding has a total of \(\ N\) turns over \(\ p\) pole- pairs, the MMF is:

\(\ \begin{aligned}
F=& \sum_{n=1}^{\infty} \frac{4}{n \pi} \frac{N I}{2 p} \sin n p \phi \\
& \text { nodd }
\end{aligned}\)

This leads directly to magnetic flux density in the air- gap:

\(\ \begin{aligned}
B_{r}=&\sum_{n=1}^{\infty} \frac{\mu_{0}}{g} \frac{4}{n \pi} \frac{N I}{2 p} \sin n p \phi\\
&\text { nodd }
\end{aligned}\)

Note that a real winding, which will most likely not be full- pitched and concentrated, will have a winding factor which is the product of pitch and breadth factors, to be discussed later.

Now, suppose that there is a polyphase winding, consisting of more than one phase (we will use three phases), driven with one of two types of current. The first of these is balanced, current:

\(\ \begin{aligned}
I_{a} &=I \cos (\omega t) \\
I_{b} &=I \cos \left(\omega t-\frac{2 \pi}{3}\right) \\
I_{c} &=I \cos \left(\omega t+\frac{2 \pi}{3}\right)
\end{aligned}\label{1}\)

Conversely, we might consider Zero Sequence currents:

\(\ I_{a}=I_{b}=I_{c}=I \cos \omega t\)

Then it is possible to express magnetic flux density for the two distinct cases. For the balanced case:

\(\ B_{r}=\sum_{n=1}^{\infty} B_{r n} \sin (n p \phi \mp \omega t)\)

where

• The upper sign holds for \(\ n=1,7, \ldots\)
• The lower sign holds for \(\ n=5,11, \ldots\)
• all other terms are zero

and

\(\ B_{r n}=\frac{3}{2} \frac{\mu_{0}}{g} \frac{4}{n \pi} \frac{N I}{2 p}\)

The zero- sequence case is simpler: it is nonzero only for the triplen harmonics:

\(\ B_{r}=\sum_{n=3,9, \ldots}^{\infty} \frac{\mu_{0}}{g} \frac{4}{n \pi} \frac{N I}{2 p} \frac{3}{2}(\sin (n p \phi-\omega t)+\sin (n p \phi+\omega t))\)

Next, consider the flux from a winding on the rotor: that will have the same form as the flux produced by a single armature winding, but will be referred to the rotor position:

\(\ \begin{aligned}
B_{r f}=& \sum_{n=1}^{\infty} \frac{\mu_{0}}{g} \frac{4}{n \pi} \frac{N I}{2 p} \sin n p \phi^{\prime} \\
& \text { nodd }
\end{aligned}\)

which is, substituting \(\ \phi^{\prime}=\phi-\frac{\omega t}{p}\),

\(\ \begin{array}{cl}
B_{r f}= & \sum_{n=1}^{\infty} \frac{\mu_{0}}{g} \frac{4}{n \pi} \frac{N I}{2 p} \sin n(p \phi-\omega t) \\
& \text { nodd }
\end{array}\)

The next step here is to find the flux linked if we have some air- gap flux density of the form:

\(\ B_{r}=\sum_{n=1}^{\infty} B_{r n} \sin (n p \phi \pm \omega t)\)

Now, it is possible to calculate flux linked by a single- turn, full- pitched winding by:

\(\ \phi=\int_{0}^{\frac{\pi}{p}} B_{r} R l d \phi\)

and this is:

\(\ \phi=2 R l \sum_{n=1}^{\infty} \frac{B_{r n}}{n p} \cos (\omega t)\)

This allows us to compute self- and mutual- inductances, since winding flux is:

\(\ \lambda=N \phi\)

The end of this is a set of expressions for various inductances. It should be noted that, in the real world, most windings are not full- pitched nor concentrated. Fortunately, these shortcomings can be accommodated by the use of winding factors.

The simplest and perhaps best definition of a winding factor is the ratio of flux linked by an actual winding to flux that would have been linked by a full- pitch, concentrated winding with the same number of turns. That is:

\(\ k_{w}=\frac{\lambda_{\text {actual }}}{\lambda_{\text {full-pitch }}}\)

It is relatively easy to show, using reciprocity arguments, that the winding factors are also the ratio of effective MMF produced by an actual winding to the MMF that would have been produced by the same winding were it to be full- pitched and concentrated. The argument goes as follows: mutual inductance between any pair of windings is reciprocal. That is, if the windings are designated one and two, the mutual inductance is flux induced in winding one by current in winding two, and it is also flux induced in winding two by current in winding one. Since each winding has a winding factor that influences its linking flux, and since the mutual inductance must be reciprocal, the same winding factor must influence the MMF produced by the winding.

The winding factors are often expressed for each space harmonic, although sometimes when a winding factor is referred to without reference to a harmonic number, what is meant is the space factor for the space fundamental.

Two winding factors are commonly specified for ordinary, regular windings. These are usually called pitch and breadth factors, reflecting the fact that often windings are not full pitched, which means that individual turns do not span a full π electrical radians and that the windings occupy a range or breadth of slots within a phase belt. The breadth factors are ratios of flux linked by a given winding to the flux that would be linked by that winding were it full- pitched and concentrated. These two winding factors are discussed in a little more detail below. What is interesting to note, although we do not prove it here, is that the winding factor of any given winding is the product of the pitch and breadth factors:

\(\ k_{w}=k_{p} k_{b}\)

With winding factors as defined here and in the sections below, it is possible to define winding inductances. For example, the synchronous inductance of a winding will be the apparent inductance of one phase when the polyphase winding is driven by a balanced set of currents. This is, approximately:

\(\ L_{d}=\sum_{n=1,5,7, \ldots}^{\infty} \frac{3}{2} \frac{4}{\pi} \frac{\mu_{0} N^{2} R l k_{w n}^{2}}{p^{2} g n^{2}}\)

This expression is approximate because it ignores the asynchronous interactions between higher order harmonics and the rotor of the machine. These are beyond the scope of this note.

Zero- sequence inductance is the ratio of flux to current if a winding is excited by zero sequence currents:

\(\ L_{0}=\sum_{n=3,9, \ldots}^{\infty} 3 \frac{4}{\pi} \frac{\mu_{0} N^{2} R l k_{w n}^{2}}{p^{2} g n^{2}}\)

And then mutual inductance, as between a field winding \(\ (f)\) and an armature winding \(\ \text { (a) }\), is:

\(\ \begin{aligned}
M(\theta) =&\sum_{n=1}^{\infty} \frac{4}{\pi} \frac{\mu_{0} N_{f} N_{a} k_{f n} k_{a n} R l}{p^{2} g n^{2}} \cos (n p \theta) \\
& \text { nodd }
\end{aligned}\)

Now we turn out attention to computing the winding factors for simple, regular winding patterns. We do not prove but only state that the winding factor can, for regular winding patterns, be expressed as the product of a pitch factor and a breadth factor, each of which can be estimated separately.

Pitch factor is found by considering the flux linked by a less- than- full pitched winding. Consider the situation in which radial magnetic flux density is:

\(\ B_{r}=B_{n} \sin (n p \phi-\omega t)\)

A winding with pitch α will link flux:

\(\ \lambda=N l \int_{\frac{\pi}{2 p}-\frac{\alpha}{2 p}}^{\frac{\pi}{2 p}+\frac{\alpha}{2 p}} B_{n} \sin (n p \phi-\omega t) R d \phi\)

Pitch \(\ \alpha\) refers to the angular displacement between sides of the coil, expressed in electrical radians. For a full- pitch coil \(\ \alpha=\pi\).

The flux linked is:

\(\ \lambda=\frac{2 N l R B_{n}}{n p} \sin \left(\frac{n \pi}{2}\right) \sin \left(\frac{n \alpha}{2}\right)\)

The pitch factor is seen to be:

\(\ k_{p n}=\sin \frac{n \alpha}{2}\)

Now for breadth factor. This describes the fact that a winding may consist of a number of coils, each linking flux slightly out of phase with the others. A regular winding will have a number (say m) coil elements, separated by electrical angle \(\ \gamma\).

A full- pitch coil with one side at angle \(\ \xi\) will, in the presence of sinusoidal magnetic flux density, link flux:

\(\ \lambda=N l \int_{\frac{\xi}{p}}^{\frac{\pi}{p}-\frac{\xi}{p}} B_{n} \sin (n p \phi-\omega t) R d \phi\)

This is readily evaluated to be:

\(\ \lambda=\frac{2 N l R B_{n}}{n p} \operatorname{Re}\left(e^{j(\omega t-n \xi)}\right)\)

where complex number notation has been used for convenience in carrying out the rest of this derivation.

Now: if the winding is distributed into m sets of slots and the slots are evenly spaced, the angular position of each slot will be:

\(\ \xi_{i}=i \gamma-\frac{m-1}{2} \gamma\)

and the number of turns in each slot will be \(\ \frac{N}{m p}\), so that actual flux linked will be:

\(\ \lambda=\frac{2 N l R B_{n}}{n p} \frac{1}{m} \sum_{i=0}^{m-1} R e\left(e^{j\left(\omega t-n \xi_{i}\right)}\right)\)

The breadth factor is then simply:

\(\ k_{b}=\frac{1}{m} \sum_{i=0}^{m-1} e^{-j n\left(i \gamma-\frac{m-1}{2} \gamma\right)}\)

Note that this can be written as:

\(\ k_{b}=\frac{e^{j n \gamma \frac{m-1}{2}}}{m} \sum_{i=0}^{m} e^{-j n i \gamma}\)

Now, focus on that sum. We know that any coverging geometric sum has a simple sum:

\(\ \sum_{i=0}^{\infty} x^{i}=\frac{1}{1-x}\)

and that a truncated sum is:

\(\ \sum_{i=0}^{m-1}=\sum_{i=0}^{\infty}-\sum_{i=m}^{\infty}\)

Then the useful sum can be written as:

\(\ \sum_{i=0}^{m-1} e^{-j n i \gamma}=\left(1-e^{j n m \gamma}\right) \sum_{i=0}^{\infty} e^{-j n i \gamma}=\frac{1-e^{j n m \gamma}}{1-e^{-j n \gamma}}\)

Now, the breadth factor is found:

\(\ k_{b n}=\frac{\sin \frac{n m \gamma}{2}}{m \sin \frac{n \gamma}{2}}\)