Inductance in Electrical Machines

Main Topics:

1. Definition of inductance and its determining factors
2. Common types of inductance in motors and their significance
3. Influencing factors of inductance in motors
4. Effects of inductance on motor design and performance

Main Content:

An electric motor is an electromagnetic device containing coils, and wherever there are coils, inductance inevitably exists. However, people often ask whether a motor’s inductance is large or small, and why. Because a motor contains many coils, and under different operating conditions, the coils exhibit different inductances. This gives rise to many inductance terms—for example, line inductance, phase inductance, direct‑axis (d‑axis) inductance, quadrature‑axis (q‑axis) inductance, leakage inductance, mutual inductance, self‑inductance, as well as transient inductance, sub‑transient inductance, and so on.

I have heard many industry professionals say that when they switch to motor engineering later in their careers, understanding basic principles like flux cutting is not too difficult. But when it comes to inductance, most of them get confused. So today we will sort out these inductances.

Definition of Inductance
Any coil has the issue of inductance. Usually, the term “inductance” has two meanings. One refers to a component—an electromagnetic device, i.e., a coil with or without an iron core, made into a component called an inductor (or choke), which is a circuit element. The other meaning refers to a parameter of a reactor or inductor—an inherent parameter denoted by L. Today we are talking about the second meaning: the inductance parameter of a coil or a reactor.

Since inductance is a parameter, it has a definition: L = Ψ / I is the defining formula. That is, inductance is the flux linkage produced per unit current. When a coil is energised, it generates a magnetic field; the parameter that measures the coil’s ability to generate that field is called inductance. In plain words, it means: if you pass 1 ampere of current through the coil, how many magnetic field lines (flux) are produced? The more lines produced per ampere, the larger the inductance; the fewer, the smaller. So inductance is essentially the “hollow” inductance. For example, the same coil wound on air produces weaker field‑generating capability than wound on an iron core. Hence, air‑core inductors usually have smaller inductance values.

With this definition, the unit is henry (H): if 1 ampere produces 1 weber of flux linkage, we call it 1 henry. Henry is a relatively large unit; typically, small coils in circuit components are less than 1 henry, so we often use millihenries (mH) or microhenries (µH).

Many circuit parameters or components have their inherent parameters; inductance is an inherent parameter of a coil. What does “inherent” mean? It means that the parameter depends only on the structure of the device. Once the coil is made, its inductance is fixed. Although the definition uses Ψ/I, I often ask students: if L = Ψ/I, and 1 A produces, say, 5 webers of flux linkage, then the inductance is 5 H. Then, if the coil is not energised, what is its inductance? Some say zero because no current means no flux, so Ψ = 0 and L = 0—this is wrong. That definition is only for measurement. In fact, inductance is generally independent of the operating state. Once the coil is wound, whether energised or not, the inductance is the same. However, for coils wound on iron cores, saturation can occur; when the core is heavily saturated, the inductance decreases; when unsaturated, it increases. So the definition only tells us what the concept is, but we must remember that inductance is an inherent parameter of the coil, determined solely by its own structure.

Types of Inductance: Self‑inductance and Mutual Inductance
There are two types: self‑inductance and mutual inductance. Consider two coils (see left figure). If there is only one coil, passing 1 A through it produces flux linkage linking its own turns—that is self‑inductance. If there are two or more coils close to each other, or they are magnetically coupled, then when current flows in the first coil, the flux produced also links the second coil. In that case, we call it mutual inductance. The definition: pass 1 A in the first coil, and the flux linkage in the second coil gives the mutual inductance from coil 1 to coil 2. Once the coils are made and their relative positions fixed, the mutual inductance is determined. Theory and practice show that the mutual inductance from coil 1 to coil 2 equals that from coil 2 to coil 1—i.e., mutual inductance is reversible.

Determining Factors
Now, what determines self‑inductance and mutual inductance? Ignoring saturation, for an air‑core coil, once manufactured, the inductance is fixed, independent of current or operating state. So what does it depend on? For self‑inductance, it is proportional to the square of the number of turns and proportional to the permeance (or inversely proportional to the reluctance) of the magnetic circuit. More turns give larger self‑inductance—for example, 1 turn with 1 A gives 1 flux line, 2 turns with 1 A gives 4 flux linkages. Self‑inductance also depends on the core material—air has low permeability, so air‑core inductance is small; iron or ferromagnetic materials give higher inductance.

For mutual inductance, it depends not only on the self‑inductances of the two coils but also on the mutual coupling coefficient, which is determined by their relative positions and magnetic coupling. If they are coaxially aligned in air, coupling is strong; if their axes are perpendicular, coupling is weak. In a transformer, primary and secondary coils on the same iron core have very strong coupling, with a coupling coefficient close to 1 (ignoring leakage). Thus, mutual inductance depends on both the self‑inductances (turns and magnetic paths) and the geometric/magnetic coupling between them.

Saturation and Incremental Inductance
For air‑core coils, the relationship between flux and current is linear—the inductance is constant. For iron‑core coils, the core saturates: at low currents, flux increases nearly linearly; at high currents, the incremental flux gain diminishes. Plotting magnetising current against flux gives the magnetisation curve (as shown). The ratio Ψ/I at a given operating point is the secant inductance (DC or steady‑state inductance). If we consider a small incremental change around that point, ΔΨ/ΔI (or dΨ/dI) is the tangent or incremental inductance, because it corresponds to the slope of the tangent to the curve at that point.

Series and Parallel Connections of Inductances
Unlike resistors and capacitors, the series/parallel combination of inductances is more complex, especially when mutual inductance exists. If two coils are far apart with no magnetic coupling, they behave like independent elements—series connection adds their inductances, parallel follows the reciprocal sum. However, if they are close and coupled, series connection can be either series‑aiding (same polarity, adding mutual inductance) or series‑opposing (opposite polarity, subtracting mutual). In the aiding case, the total inductance increases; in the opposing case, it decreases. If two identical coils with coupling coefficient 1 are connected in series‑opposing, the total inductance becomes zero. For parallel connection with mutual coupling, the formula becomes more complicated (as given in the equation). Therefore, when combining inductances, we must consider both the electrical connections and the magnetic coupling.

Inductances in AC Motors
Motors consist of coils and iron cores, so they naturally exhibit all these inductance issues. Usually, we refer to inductances in AC motors (synchronous and induction). DC motors also have inductance, but it mainly affects transient behaviour and commutation (sparking, etc.), not steady‑state performance. Here we focus on AC motors.

Both synchronous and induction motors have identical stator (armature) windings—typically three‑phase, symmetrically arranged 120° apart in space. The rotor differs: induction motors have a squirrel‑cage (or wound‑rotor) that is essentially a uniform cylindrical structure; synchronous motors may have salient poles or cylindrical (non‑salient) rotors with field windings and damper windings.

Symmetric Rotor (Cylindrical)
When the rotor is perfectly cylindrical and symmetric, the stator phase windings see identical magnetic paths. Thus, the self‑inductances of the three phases are equal, and the mutual inductances between any two phases are equal (and negative, because the phase axes are 120° apart). For a three‑phase system, when all phases carry current, the phase inductance is not simply the self‑inductance of one phase; it also includes the mutual effects from the other phases. Typically, we consider the resultant flux linking a phase due to all three currents, divided by that phase’s current—this is the effective phase inductance.

Asymmetric Rotor (Salient‑pole)
If the rotor is not symmetric (e.g., salient‑pole synchronous machine with d‑axis and q‑axis), the self‑inductance and mutual inductance of each stator phase vary with rotor position. For example, when the rotor’s d‑axis aligns with a phase axis, the reluctance is low (more iron), so the inductance is high; when the q‑axis aligns, the reluctance is high (more air), so the inductance is low. As the rotor rotates, each phase inductance fluctuates twice per revolution (because of two poles). The same applies to mutual inductances. These variations are expressed as sinusoidal functions of rotor angle, with average values and second‑harmonic components.

Two‑Reaction Theory
Analysing motor performance directly with these position‑dependent inductances is very complicated. Therefore, the two‑reaction theory is introduced: the armature magnetomotive force (MMF) is resolved into two components—one aligned with the direct (d) axis and one with the quadrature (q) axis. Since the d‑axis and q‑axis have fixed reluctances, the corresponding inductances (Ld and Lq) are constants, independent of rotor position. This greatly simplifies steady‑state analysis. For a salient‑pole synchronous motor, Ld (aligned with the pole) is larger than Lq for electrically excited machines; for permanent‑magnet (PM) machines, it may be the opposite, depending on magnet placement.

The 3/2 Factor
In a three‑phase motor, when phase A carries current, phases B and C also carry currents (due to the balanced three‑phase supply). The flux produced by B and C also links phase A. Because the mutual inductances are each –1/2 of the self‑inductance, the total flux linkage in phase A becomes (1 + 1/2) times that due to phase A alone—hence the factor 3/2 appears in the armature reaction inductance. This factor arises from the mutual contributions of the other two phases.

Line Inductance
Line inductance is defined as the inductance measured between two line terminals (e.g., A and B) with the third terminal open (star connection). For a star‑connected motor, the line inductance relates to Ld and Lq. For permanent‑magnet motors with interior magnets, Lq is usually larger than Ld because the q‑axis path is mostly iron, while the d‑axis path passes through magnets (low permeability). The maximum line inductance equals 2Lq, and the minimum equals 2Ld. For electrically excited synchronous machines, the opposite holds: Ld > Lq.

Determining Factors for Motor Inductances
Inductance is proportional to the square of the number of turns. The number of turns is determined by the back‑EMF design (using the 4.44 formula). As a rule of thumb, for the same voltage rating, larger‑capacity motors tend to have smaller inductances; higher voltage at the same capacity gives larger inductance. Also, the magnetic permeability of the path affects inductance. In interior PM motors, the d‑axis path crosses magnet slots (equivalent to air gaps), so its reluctance is high and Ld is small; the q‑axis path stays in iron, so Lq is large. In electrically excited machines, the d‑axis path has less air and more iron, so Ld is large, while the q‑axis path has more air, so Lq is small.

Effects of Inductance on Motor Performance

• Induction motors: Larger main (magnetising) inductance means larger magnetising reactance (Xm = ωL). This reduces the no‑load (magnetising) current, improving power factor. To increase this, the air gap is made small (typically <1 mm). Larger leakage inductance reduces starting current (good for starting), but also reduces starting torque (bad). So there is a trade‑off.
• Synchronous motors (electrically excited): Larger synchronous reactance (Xd) increases the internal impedance, worsening the steady‑state voltage regulation (voltage drop from no‑load to full‑load). It also reduces the amplitude of the power‑angle characteristic, leading to a larger power angle for the same load, which reduces stability margin. Moreover, a larger reactance reduces the short‑circuit ratio (SCR), which is important for grid‑connected generators—a small SCR indicates a weak system. On the positive side, larger inductance means less field current is needed to produce the same flux, reducing rotor copper usage and cost. So there is a cost‑performance trade‑off: smaller air gaps give larger inductance, which lowers cost but degrades some performance aspects.
• Permanent‑magnet synchronous motors (surface‑mounted): Larger inductance increases reactance, which worsens voltage regulation for stand‑alone generators and reduces short‑circuit current and power factor for motors. For interior PM motors, the short‑circuit current is determined by the back‑EMF divided by Xd.

In summary, inductance is a fundamental parameter that affects efficiency, power factor, starting, stability, and cost, and its optimal value depends on the specific application and design priorities.