The space phasor (or space vector) provides a shorthand graphical way of representing sinusoidally distributed spatial quantities such as the m.m.f. and flux waves that we explored in Chapter 5. It avoids us having to consider individual currents by focusing on their combined effects, and thus makes things easier to understand.

We begin by taking a fresh look at the rotating stator m.m.f., making the reasonable assumption that each of the three phase windings produces a sinusoidally distributed m.m.f. with respect to distance around the air-gap, which in turn implies that the winding itself is sinusoidally distributed (rather than sitting in clearly defined groups of coils as in the real machine discussed previously). For convenience, we will consider a 2-pole winding.

We can represent the relative position of the windings in space as shown in Fig. 8.1. We will use the standard notation (UVW) from here onwards, although the notation ABC is still often used.

In Fig. 8.1A phases V and W are on open-circuit so that we can focus on how the m.m.f. of phase U is represented. When the current in phase U is positive (i.e. current flows into the dotted end), we have chosen to represent its sinusoidal m.m.f. pattern by a vector along the axis of the winding and pointing away from it (Fig. 8.1A), and so when the current is negative the vector points towards the coil (Fig. 8.1B). The length of the vector is directly proportional to the instantaneous value of the current, as indicated by the relative sizes of the two vectors.

In Fig. 8.1C, phase U has its maximum positive current, while phases V and W both have negative currents of half the maximum value. Because each m.m.f. is distributed sinusoidally in space, we can find their resultant (R) using the approach that is probably more familiar in the context of a.c. circuits, i.e. by adding the three components vectorially. In this particular example, the resultant m.m.f., R (Fig. 8.1D) is co-phasal with the m.m.f. of phase U, but one and half times larger.

We can now use the approach outlined above to find the resultant m.m.f. when the windings are supplied with balanced 3-phase currents of equal amplitude but displaced in time by one third of a cycle (i.e. 120°). The axes of the phases are displaced in space as shown in Fig. 8.1, and the three currents are shown as functions of time in the upper part of Fig. 8.2. Four consecutive times are identified, separated by one twelfth of a complete cycle, or 30° in angle terms.

The lower part of the diagram represents the m.m.f.’s in a space phasor diagram. At each instant the three individual phase m.m.f.’s are shown in magnitude and position, together with the resultant m.m.f. (R). At time t₀ for example, the situation is the same as in Fig. 8.1, with phase U at maximum positive current and phases V and W having equal currents of half the magnitude of that in phase U; at t₁ phase V is zero while phases U and W have equal but opposite currents; and so on.

The four sketches suggest that the resultant m.m.f. describes an arc of constant radius, and it can easily be shown analytically that this is true. So although each phase produces a pulsating m.m.f. along its axis, the overall m.m.f. is constant in amplitude (with a value equal to 1.5 times the phase peak), and it rotates at a uniform rate, completing one mechanical revolution per cycle if the field is 2-pole (as here), half a mechanical revolution if 4-pole, etc. This is in line with our findings in Chapter 5.

We should note that although we have developed the idea of space phasors by focusing on steady-state sinusoidal operation, the approach is equally valid for any set of instantaneous currents, and is therefore applicable under transient conditions, for example during acceleration when the instantaneous frequency of the currents may change continuously.

An alternative way of representing the resultant m.m.f. pattern produced by a set of balanced 3-phase windings follows naturally from the discussion above. We imagine a hypothetical single m.m.f. vector that has a constant magnitude but rotates relative to the stator at the synchronous speed. This turns out to be an exceptionally useful mental picture when we come to study the behaviour of the inverter-fed induction motor, because the currents will be under our control and we are able to specify precisely the magnitude, speed and angular position of the stator m.m.f. vector in order to achieve precise control of torque.

8.2.2 Transformation of reference frames

In the previous section we saw that the resultant m.m.f. was of constant amplitude and rotated at a constant angular velocity with respect to a reference frame fixed to the stator. As far as an observer in the stationary reference frame is concerned, the same m.m.f. could equally well be produced by a sinusoidally distributed winding fed with constant (d.c.) current and mounted on a structure that rotated at the same angular velocity as the actual m.m.f. wave. On the other hand, if we were attached to a reference frame rotating with the m.m.f., the space phasor would clearly appear to us to be constant.

Transformations between reference frames have long been used to simplify the analysis of electrical machines, especially under dynamic conditions, but until fast signal-processing became available it was seldom used for live control purposes. We will see later in this chapter that the method is used in field-oriented control schemes to transform the stator currents into a rotating reference frame locked to the rotating rotor flux space phasor, thereby making them amenable for control purposes.

Transformation is usually accomplished in two stages, as shown in Fig. 8.3.

The first stage involves replacing the three windings by two in quadrature, with balanced sinusoidal currents of the same frequency but having a 90° phase shift. In this case the ‘α β’ stationary reference frame has phase α aligned with phase U. To produce the same m.m.f., either the two windings need more turns, or more current, or a combination of both. This is known as the Clarke transformation. The resultant space phasor (at the bottom of the diagram) is of course identical with the three-phase one on the left.

The second stage (the Park transformation) is more radical as the new variables I_d and I_q are in a rotating reference frame, and they remain constant under steady state conditions, as shown in Fig. 8.3. Again we need to specify the turns ratio and/or the current scaling. (Strictly speaking there is no need for the intermediate (2-phase) transformation, because we can transform directly from 3-phase to two-axis, but we have included it because it is often mentioned in the literature.) The resultant space phasor is again identical to the three-phase one.

It should be clear that the magnitude of the currents I_d and I_q will depend on the angle λ, which is the angle between the two reference frames at a specified instant, typically at t = 0. As far as we are concerned, it is sufficient to note that there are well-established formulae relating the input and output variables, both for the forward transformation (U, V, W to d, q) and for the inverse transformation, so it is straightforward to construct algorithms to perform the transformations. We should also note that whilst we have considered the transformation of sinusoidal currents, the technique is equally valid for instantaneous values.

8.2.3 Transient and steady-states in electric circuits

Field-oriented control allows us to obtain (almost) instantaneous (step) changes in torque on demand, and in essence it does this by jumping directly from one steady-state condition to another, without any unwelcome transient period of adjustment.

Given the very poor inherent transient response of the induction motor to sudden changes in load or utility supply (see for example Fig. 6.7), it will come as no surprise when we learn later that this sudden transition between steady states can only be achieved by precise control of the magnitude, frequency and instantaneous position of the stator current space vector. As will emerge, the key requirement for a successful sudden transition is that it must not involve a step change in the stored energy of the system.

As an introduction to the underlying principle of changing from one steady state to another without any transient, we can look at the behaviour of a series resistor and inductor circuit fed by an ideal voltage source (Fig. 8.4). This is much simpler than the induction motor, (it only has one energy storage element—the inductor) but it demonstrates the key requirement to be satisfied for transient-free switching.

First, we look at the current when the voltage is a step at t = 0 (Fig. 8.4A). The steady-state current is simply V/R, but the current cannot rise instantaneously because that would require the energy stored in the inductor ($\frac{1}{2}L{i}^2$) to be supplied in zero time, which corresponds to an impulse of infinite power. So in addition to the steady state term $i_{\mathit{ss}}=\frac{V}{R}$, there is a transient term given by $i_{\mathit{tr}}=-\frac{V}{R}{e}^{-\frac{t}{\tau }}$, where the time-constant, $\tau =\raisebox{1ex}{$L$}\!\left/ \!\raisebox{-1ex}{$R$}\right.$. The total current is the sum of the steady-state and transient components, as shown by the dotted blue line in Fig. 8.4A.

Now consider a more relevant situation, where we wish the current to jump suddenly from a steady state at one frequency (in this case zero amplitude at zero frequency (or ‘d.c.’) for t < 0 to a sinusoidal steady state for t > 0.

Fig. 8.4B shows what happens if we make the sudden transition in the applied voltage (from zero d.c.) at a point where the new voltage waveform is zero but rising, i.e. at t = 0. We note that the current does not immediately assume its steady state, but displays the characteristic decaying transient, lasting for several cycles before the steady-state is reached, with the current finally lagging the voltage by an angle ϕ. Examination of the steady state current waveform shows that the current is negative as the voltage rises through zero, so this particular attempt to jump straight into the steady state is clearly doomed from the outset because it would have required the circuit to anticipate the arrival of the voltage by having a negative current already in existence!

The fundamental reason for the transient adjustment in Fig. 8.4B is that we are seeking an instantaneous increase in the energy stored in the inductor from its initial value of zero, which is clearly impossible. It turns out that if we want to avoid the transient, we must make the jump without requiring a change in the stored energy, which in this example means at the point when the current passes through zero, as shown in Fig. 8.4C. The voltage that has to be applied therefore starts abruptly at a value Vsinϕ, as shown, and the current immediately enters its steady state, with no transient term.

We will see later that the principle of not disturbing the stored energy is essential to obtain sudden step changes in torque from an induction motor.

8.3.1 Coupled circuits, induced EMF, and flux linkage

By ‘coupled circuits’ we mean two or more circuits, often in the form of multi-turn coils sharing a magnetic circuit, where the magnetic flux produced by the current in one coil not only links with its own winding, but also with those of the other coils. The coupling medium is the magnetic field, and as we will see the electrical effect of the coupling is manifested when the flux changes.

We know from Faraday's law that when the magnetic flux (ϕ) linking a coil of N turns changes, an e.m.f. (e) is induced in the coil, given by

$e\mathit{=}\mathit{-}N\frac{d\varphi }{dt}\text{,}$

i.e. the e.m.f. is proportional to the rate of change of the flux. (The minus sign indicates that if the induced e.m.f. is allowed to drive a current, the m.m.f. produced by the current will be in opposition to that which produced the original changing flux.) This equation only applies if all the flux links all N turns of the coil, the situation most commonly approached in transformer windings that share a common magnetic circuit, and are thus very tightly coupled.

We have seen that windings for induction motors are distributed, and the flux wave produced by the current in the winding is also distributed around the air-gap. As a result not all of the flux produced by one winding links with all of its turns, and we have to perform a summation (integration) of all the ‘turns times flux that links them’ contributions to find the ‘total effective self flux linkage’ which we denote by the symbol ψ (psi). The e.m.f. induced when the self-produced flux linkage changes in, say, a stator winding (subscript s) is then given by

$e_S\mathit{=}\frac{d{\psi }_S}{dt}\text{.}$

In an induction motor there are three distributed windings on the stator, and either a cage or three more distributed windings on the rotor, and some of the flux produced by current in any one of the windings will link all of the others. We term this ‘mutual flux linkage’, and often denote it by a double suffix: for example the symbol ψ_SR is the mutual flux linkage between a stator winding and a rotor winding.

In the same way that an e.m.f. is induced in a winding when its self-produced flux changes, so also are e.m.f.’s induced in all other windings that are mutually coupled to it. For example if the flux produced by the stator winding changes, the e.m.f. in the rotor (subscript R) is given by

$e_R\mathit{=}\frac{d{\psi }_{SR}}{dt}\text{.}$

8.3.2 Self and mutual inductance

The self and mutual flux linkages produced by a winding are proportional to the current in the winding: the ratio of flux linkage to the current that produces it is therefore a constant, and is defined as the inductance of the winding. The self inductance (L), is given by

$\mathit{L}=\frac{\mathit{Self\; flux\; linkage}}{\mathit{Current}}\mathit{=}\frac{{\mathit{\psi }}_{\mathit{S}}}{{\mathit{i}}_{\mathit{S}}},$

while the mutual inductance (M) is defined as

$M_{SR}\mathit{=}\frac{\mathit{Mutual\; flux\; linkage}}{\mathit{Current}}\mathit{=}\frac{{\psi }_{SR}}{i_S}\text{.}$

The self and mutual inductances therefore depend on the design of the magnetic circuit and the layout of the windings. In an induction motor the self inductances are constant, but the mutual inductance between a stator and a rotor winding varies with the angular position of the rotor.

We can now re-cast the expressions for e.m.f. derived above so that they involve the rates of change of the currents and the inductances, rather than the fluxes. This is a very important simplification because it allows us to represent the distributed effects of the magnetic coupling in single lumped-parameter electric circuit terms. The self-induced and mutually-induced e.m.f.’s are now given by

$e_S\mathit{=}L\frac{d{i}_S}{dt}\text{,}$

and

$e_R\mathit{=}{M}_{SR}\frac{d{i}_S}{dt}\text{.}$

As mentioned above, the mutual inductance between stator and rotor windings varies with the rotor position, so M_SR is a function of θ, and the rotor e.m.f. has to be expressed as

$e_R\mathit{=}{M}_{SR}\frac{d{i}_S}{dt}\mathit{+}{i}_S\frac{d{M}_{SR}}{dt}\mathit{=}{M}_{SR}\frac{d{i}_S}{dt}\mathit{+}{i}_s\frac{d{M}_{SR}}{d\theta }\left(\frac{d\mathit{\theta }}{dt}\right)$

The first term of the equation is the ‘transformer e.m.f.’ that results from changes in the stator current, while the second is present even when the stator current is constant, and it is proportional to the speed of the rotor. We have already seen the vital role that this ‘motional’ e.m.f. plays in the energy conversion process. When the term occurs in a circuit model, it is often referred to as a ‘speed’ voltage.

8.3.3 Obtaining torque from a circuit model

We represent the two sets of 3-phase distributed windings of the induction motor by means of the six fictitious ‘equivalent’ coils shown in Fig. 8.5. (We are using the well-proven fact that a cage rotor behaves in essentially the same way as one with a wound rotor, as explained in Chapter 5.) The three stator coils remain stationary, while those on the rotor obviously move when the angle θ changes.

Because the air-gap is smooth, and the rotor is assumed to be magnetically homogeneous, all the self inductances are independent of the rotor position, as are the mutual inductances between pairs of stator coils and between pairs of rotor coils. Symmetry also means that the mutuals between any two stator or rotor phases are the same.

However, it is obvious that the mutual inductance between a stator and a rotor winding will vary with the position of the rotor: when stator and rotor windings are aligned, the flux linkage will be maximum, and when they are positioned at right angles, the flux linkage will be zero. With windings that are distributed so as to produce sinusoidal flux waves, the mutual inductances vary sinusoidally with the angle θ.

To a circuit theory practitioner, it is this variation of mutual inductance with position that immediately signals that torque production is possible. In fact it is straightforward (if somewhat intellectually demanding) to show that the torque produced when the sets of windings in Fig. 8.5 carry currents is given by the rather fearsome-looking expression

$T=\sum i_S{i}_R\frac{{\mathit{dM}}_{\mathit{SR}}}{\mathit{d\theta }}$

What this means is that to find the total torque we have to find the summation of nine terms, each of which represents a contribution to the total torque from one of the nine stator-rotor pairs. So we need the instantaneous value of each of the six currents, and the rate of change of inductance with rotor position for each stator-rotor pair. For example the term representing the contribution to torque made by stator coil U interacting with rotor coil V is given by

$T_{\mathit{SURV}}=i_{\mathit{SU}}{i}_{\mathit{RV}}\frac{{\mathit{dM}}_{\mathit{SURV}}}{\mathit{d\theta }}$

In practice we can use various expedients to simplify the torque expression, for example we know that mutual inductance is a reciprocal property, i.e. M_UV = M_VU, and we can exploit symmetry, but the important thing to note here is that it is a straightforward business to find the torque from the circuit model, provided that we know the currents and the angle-dependancy of the inductances.

8.3.4 Finding the rotor currents

In the induction motor, the rotor currents are induced, and to find them we have to solve the set of six equations relating them to the applied stator voltages, using Kirchoff's voltage law.

So for example the voltage equation below relating to rotor phase U includes a term representing the resistive volt drop, another representing the self-induced e.m.f. and five others representing the mutual coupling with the other windings. There are two more rotor equations and three similar ones for the stator windings.

$v_{RU}\mathit{=}{i}_{RU}{R}_R\mathit{+}{L}_{RU}\frac{d{i}_{RU}}{dt}\mathit{+}{M}_{\mathit{RURV}}\frac{d{i}_{RV}}{dt}\mathit{+}{M}_{\mathit{RURW}}\frac{d{i}_{RW}}{dt}\mathit{+}{M}_{\mathit{RUSU}}\frac{d{i}_{SU}}{dt}\mathit{+}{M}_{\mathit{RUSV}}\frac{d{i}_{SV}}{dt}\mathit{+}{M}_{\mathit{RUSW}}\frac{d{i}_{SW}}{dt}$

In the induction motor the rotor windings are usually short-circuited, so there is no applied voltage and the left-hand side of each rotor equation is zero.

If we have to solve these six simultaneous differential equations when the stator terminal voltages are specified (typical of utility-fed constant-frequency conditions), we have a very challenging task that demands computer assistance, even under steady-state conditions. However, when the stator currents are specified (as we will see is the norm in an inverter-fed motor under vector control), the equations can be solved much more readily. Indeed under steady-state locked rotor conditions we can employ an armoury of techniques such as j notation and phasor diagrams to solve the equations by hand.

We have now seen in principle how to predict the torque, and how to solve for the rotor currents, when the stator currents are specified. So we are now in a position to see what can be learned from a study of the known outcomes under two specific conditions.

In the next section, we look at how the torque varies when the stator windings are fed with a balanced set of 3-phase a.c. currents of constant amplitude but variable frequency, and the rotor is stationary. Although this is not of practical importance, it is very illuminating, and it points the way to the second and much more significant mode of operation, in which the net rotor flux is kept constant at all frequencies: this forms the basis for field-oriented control.

8.4.1 Torque vs slip frequency—Constant stator current

An alert reader might question why the title of this section includes reference to slip frequency, when we have specified so far that the rotor is stationary, in which case the effective slip is 1 and the frequency induced in the rotor will always be the same as the stator frequency. The reason for referring to slip frequency is that, as far as the reaction of the rotor is concerned, the only thing that matters is the relative speed of the travelling stator field with respect to the rotor.

So if we study the static model with an induced rotor frequency of 2 Hz, the torque that we predict can represent locked rotor conditions with 2 Hz on the stator; or the rotor running with a slip of 0.1 with 20 Hz on the stator; or a slip of 0.04 with 50 Hz on the stator, and so on. In short, under current-fed conditions, our model correctly predicts the rotor behaviour, including the torque, when we supply the stator windings at the slip frequency. (Note that all other aspects of behaviour on the stator side are not represented in this model, notably the fact that the stator voltage has to vary appropriately in order to keep the current constant.)

The variation in the flux linkage triangle with slip frequency, assuming that the amplitude of the stator current is constant, is shown in Fig. 8.7. The locus of the resultant rotor flux linkage as the slip is varied is shown by the semi-circles; the rotor current increases progressively with slip, but at the same time it moves out of space phase with the stator phasor.

The right hand side relates to low values of slip frequency, where the rotor self flux linkage is much less than the stator mutual flux linkage, so the resultant rotor flux linkage (ψ_R) is not much less than when the slip is zero. In other words, at low slips the presence of the rotor currents has little effect on the magnitude of the resultant flux, as we saw in Chapter 5. Low-slip operation is the norm in controlled drives.

The left hand figure relates to high values of slip, where the large induced currents in the rotor lead to a rotor m.m.f. that is almost able to wipe out the stator m.m.f., leaving a very small resultant flux in the rotor. We will not be concerned with this end of the diagram in an inverter drive.

There is a simple formula for the angle ϕ, which is given by

$tan\varphi ={\omega }_s\tau$

where $\tau =\frac{L_R}{R_R}$, the rotor time-constant.

We noted earlier that the torque is proportional to the area of the triangle, so it should be clear that the peak torque is reached when the slip increases from the point T and moves to T_max. At this point, ϕ = π/4 and the slip frequency is given by ${\omega }_S=\frac{1}{\tau }=\frac{R_R}{L_R}.$ Under these constant-current conditions, the slip at which maximum torque occurs is much less than under constant-voltage conditions, because the rotor self inductance is much larger than the rotor leakage inductance.

8.4.2 Torque vs slip frequency—Constant rotor flux linkage

As already mentioned, it is clear that to make full use of the flux-carrying capacity of the rotor iron, we will want to keep the amplitude of the rotor flux ψ_R constant. Given that the majority of the rotor flux links the stator (see Fig. 8.6C), keeping the rotor flux constant also means that for most operating conditions, the stator flux is also more or less constant, as we assumed in Chapter 5.

In this section we explore how steady-state torque varies with slip when the rotor flux is maintained constant: this is illuminating, but much more importantly it prepares us for the final section, which deals with how we are able to achieve precise control of torque even under dynamic conditions.

We can see from Fig. 8.7 that to keep the rotor flux constant we will have to increase the stator current with slip. This is illustrated graphically in Fig. 8.8, in which the rotor flux linkage ψ_R is shown vertically to make it easier to see that it remains constant. In the left hand sketch, the slip is very small, so the induced rotor current and the torque (which is proportional to the area of the triangle) are both small. The rotor flux is more or less in phase with the applied stator flux linkage because the ‘opposing’ influence of the rotor m.m.f. is small.

In the middle and right-hand diagrams the slip is progressively higher, so the induced rotor current is larger and the stator current has to increase in order to keep the rotor flux constant.

There is a simple analytical relationship that gives the value of stator current required to keep ψ_R constant as slip varies, but of more interest is the relationship between the induced rotor current and the slip. From Fig. 8.8, we can see that the tangent of the angle ϕ is given by

$tan\varphi =\frac{L_R{I}_R}{{\psi }_R}$

Combining this with Eq. (8.1) we find that the rotor current is given by

${\mathit{I}}_R\mathit{=}\left(\frac{{\psi }_R}{R_R}\right){\omega }_{\mathit{slip}}$

The bracketed term is constant, therefore the rotor current is directly proportional to the slip. Hence the horizontal sides of the triangles in Fig. 8.8 are proportional to slip, and since the vertical side is constant, the area of each triangle (and torque) is also proportional to slip. To emphasise this simple relationship, the right-hand diagram in Fig. 8.8 has been drawn to correspond to a slip three times higher than that of the middle one, so the horizontal side of the right-hand sketch is three times as long, and the area of the triangle (and torque) is trebled.

We note that when the rotor flux is maintained constant, the torque-speed curve becomes identical to that of the d.c. motor. In this respect the behaviour differs markedly from that under both constant-voltage and constant-current conditions, where a peak or pull-out torque is reached at some value of slip. With constant rotor flux there is no theoretical limit to the torque, but in practice the maximum will be governed by thermal limits on the rotor and stator currents.

For those who prefer the physical viewpoint it is worth noting that the results discussed in this section could have been deduced directly from Fig. 8.6C, which indicates that the resultant rotor flux and rotor current waves are always aligned (i.e. the peak flux density coincides with the peak current density) so that if the flux is held constant, the torque is proportional to the rotor current. The rotor current is proportional to the motionally induced e.m.f., which in turn depends on the velocity of the flux wave relative to the rotor, i.e. the slip speed.

8.4.3 Flux and torque components of stator current

If we resolve the stator flux-linkage phasor MI_s into its components, parallel and perpendicular to the rotor flux, the significance of the terms ‘flux component’ and ‘torque component’ of the stator current becomes obvious (Fig. 8.9).

We can view the ‘flux’ component as being responsible for setting up the rotor flux, and this is the component that we must keep constant in order to maintain the working flux of the machine at a constant value for all slips. It is clearly analogous to the field current that sets up the flux in a d.c. motor.

The other (‘torque’) component (which is proportional to the rotor current) can be thought of as being responsible for nullifying the opposing effect of the rotor current that results when the rotor conductors are ‘cut’ by the travelling flux wave. This current component is therefore seen as the counterpart of the armature or work current in the d.c. motor.

Looking back to the left-hand diagram in Fig. 8.8, we see that at small slips (light load) the stator current is small and practically in phase with the flux; this is what we referred to as the magnetising current in previous chapters. At higher slips, the stator current is larger, reflecting that it now has a torque or ‘work’ component in addition to its magnetising component, which again accords with our findings in previous chapters.

8.5.1 Special property of closely-coupled circuits

In Chapter 5 we saw that, when looked at from its terminals, the induction motor under steady-state conditions always looks—to a greater or lesser extent—inductive. At no-load, for example, the (magnetising) current was relatively small, but it lagged the voltage by almost 90°, so that the motor looked more or less like a high inductance. But as the load torque and slip increased, a load component of stator current came into play, so that the total current became much larger and moved closer in phase with the voltage, making the motor overall look more like a resistor, but still with a significant inductive element.

On the face of it therefore, the presence of stator inductance makes the prospect of making rapid changes to the rotor current look daunting. However, the important difference is that we are no longer talking about the steady-state (i.e. when all the initial transients have settled down) but instead we must focus on how the stator-rotor combination reacts immediately after sudden changes in the voltage applied to the stator. The full treatment is complex and beyond our scope, but fortunately we can illustrate the essence of the matter by looking at the behaviour of a pair of coupled coils, one representing the stator winding and the other notionally representing the short-circuited cage rotor winding.

We will begin with a recap of the relationship between voltage and current in a pure inductor, the aim being to emphasise the difficulty of making a rapid change in the current. The differential equation linking voltage and current for a pure inductance is

$\mathit{v}\mathit{=}\mathit{L}\frac{\mathit{di}}{\mathit{dt}}\mathit{,}\mathit{or}\mathit{di}\mathit{=}\frac{1}{\mathit{L}}\left(\mathit{v}\mathit{dt}\right)$

It follows that unlike in a resistor, where the current follows the voltage immediately, in an inductor the current is determined by the time-integral of the applied voltage, so that to increase the current by di we have to apply a fixed volt-second product vdt.

To increase the current (and hence increase the stored energy in the inductor) we can choose to apply a high voltage (and high power) for a short time or a lower voltage for longer, but, whatever the inductance, we can never obtain a perfect step change in current because that would require an impulse of infinite voltage and zero duration, i.e. a pulse of infinite power.

For example, Fig. 8.10 shows the result of applying successive voltage-time pulses of the same area (shown shaded) but of voltages V, 2 V and 4 V, respectively, to a pure inductance, L. Each pulse raises the current by the same amount (VT/L), but the rate of rise of the current increases in direct proportion to the voltage. (Finally, the current is brought back down to zero by applying a negative pulse of area 3VT.) It should be clear that to effect a rapid change in current, the higher the voltage and the lower the inductance, the better.

Some readers may feel uneasy when observing Fig. 8.10 to note that the current remains constant when the voltage is zero, but they will be reassured when reminded that this is only true when the inductance is ideal, i.e. it has zero resistance. In practice there will always be resistance, and then, as soon as the voltage is reduced to zero, the current will begin to fall with a time-constant of L/R. The reason that we neglect resistance is not only that the analysis is very much simpler, but also we are interested in what happens immediately after a step change in voltage, in which case the effect of resistance is negligible, all the voltage then appearing across the inductive element.

Having looked at how a single inductor behaves, we return to the matter of what the stator ‘looks like’ to the supply under transient conditions, by considering a pair of magnetically coupled coils, one representing the stator and the other the rotor. We will assume that the coils are stationary (i.e. the rotor is at rest), but the arguments are equally valid when the rotor is moving because at every instant the cage rotor looks the same when viewed from the stator.

Let the self inductances of the coils representing stator and rotor be L_s and L_r, respectively, let their mutual inductance be M, and ignore their (very low) resistances. Because of the good magnetic circuit, all three inductances will be large. If we were able to open-circuit the rotor cage, the stator would therefore look like a high inductance (and its reactance at the utility supply frequency would correspond to what we earlier called the magnetising reactance). However, the rotor circuits are short-circuited, so we model this by setting the rotor voltage term to zero in the Kirchoff's voltage equations:-

${\mathit{v}}_{\mathit{s}}\mathit{=}{\mathit{L}}_{\mathit{s}}\frac{{\mathit{di}}_{\mathit{s}}}{\mathit{dt}}\mathit{+}\mathit{M}\frac{{\mathit{di}}_{\mathit{r}}}{\mathit{dt}}$

${\mathit{v}}_{\mathit{r}}\mathit{=}0\mathit{=}{\mathit{L}}_{\mathit{r}}\frac{{\mathit{di}}_{\mathit{r}}}{\mathit{dt}}\mathit{+}\mathit{M}\frac{{\mathit{di}}_{\mathit{s}}}{\mathit{dt}}$

Eliminating the rotor current we obtain

${\mathit{v}}_{\mathit{s}}\mathit{=}{\mathit{L}}_{\mathit{s}}\left(1\mathit{-}\frac{{\mathit{M}}^2}{{\mathit{L}}_{\mathit{s}}{\mathit{L}}_{\mathit{r}}}\right)\frac{\mathit{d}{\mathit{i}}_{\mathit{s}}}{\mathit{dt}}$

The effective inductance at the stator is therefore given by

$L_s^{\prime }=L_s\left(1-k^2\right),$

where k is the coupling coefficient, defined as

$\mathit{k}=\frac{\mathit{M}}{\sqrt{{\mathit{L}}_{\mathit{s}}{\mathit{L}}_{\mathit{r}}}}$

The coupling coefficient always lies between 0 (no mutual flux) and 1 (no leakage flux), so we see the very welcome news that the inductance looking in at the stator is reduced from its open-circuit value by the factor 1 – k², and that if the coils were perfectly coupled (i.e. k = 1), the effective inductance becomes zero.

At first sight this is a very unexpected (and welcome) result given that both windings taken separately have high self inductances. (But for those familiar with the idea of referred impedance in the context of a transformer, it is perhaps not surprising!)

In an induction motor the good magnetic circuit means that the coupling coefficient is high, perhaps 0.95, in which case the effective transient inductance looking in at the stator is barely 10% of its self-inductance (and it corresponds loosely to the so-called leakage inductance). It is this remarkably fortunate outcome that allows us to achieve rapid changes in the stator currents without requiring extremely high voltages from the inverter.

It remains for us to see how the rotor currents react when the stator current is changed so, knowing that we can come close to effecting a sudden change in practice, we will assume that the current in the stator suddenly increases from zero to I_s. Under this condition, it emerges that the rotor current also suddenly increases, its magnitude being given by

${\mathit{I}}_{\mathit{r}}=\mathit{-}\left(\frac{\mathit{M}}{{\mathit{L}}_{\mathit{r}}}\right){\mathit{I}}_{\mathit{s}}$

Rearranging this expression to highlight the flux linkage produced by the rotor yields

$L_r{I}_r=-M{I}_s.$

Hence the rotor reacts by producing self flux linkages that exactly cancel those reaching the rotor from the stator, leaving the rotor flux linkage at its initial value of zero. As mentioned earlier, in practice the rotor current will decay because of its finite resistance, but this will not be significant over the short timescale that we will be interested in for torque control.

For those who prefer to argue from a physical basis, we can see why the effective inductance is reduced by the closely-coupled and short-circuited secondary coil by invoking Faraday's and Lenz's laws. If the primary current changes, the flux linking the secondary also changes, inducing an e.m.f. in the secondary. The secondary is short-circuited, so a secondary current is induced in a direction that makes its associated flux oppose the changing flux that was responsible for producing the e.m.f.. If the circuits are perfectly coupled via zero-reluctance paths, the opposing flux completely neutralises the primary flux, and there is never any resultant flux, and the effective primary inductance is therefore zero. In the non-ideal case, cancellation is incomplete, but the effective primary inductance is always reduced.

To conclude this section, it is appropriate to say that in order to cover a potentially tricky matter we have deliberately made use of a very simple model, and we have been somewhat loose in suggesting that the stator and rotor can be represented by single coils. Our experience suggests that non specialist readers will accept such gross simplifications where the main message emerges without too much difficulty, and we will make use of similar arguments in the next section, that deals with establishing the rotor flux.

8.5.2 Establishing the flux

In Section 8.4 we assumed that steady-state conditions prevailed, with the rotor flux linkage remaining of constant magnitude and rotating relative to the rotor at the slip speed. We now look at how the rotor flux wave was first established.

We start with the rotor at rest, no current in any of the windings, and hence no flux. With reference to Fig. 8.1, we suppose that we supply a step (d.c.) current into phase U, which will split with half exiting from each of phases V and W, and producing a stationary sinusoidally distributed m.m.f. that, ultimately, will produce the flux pattern labelled ‘final state’ in Fig. 8.11.

But of course the rotor windings are short-circuited, with no flux through them, and as we showed in the previous section, closed electrical circuits behave like many things in the physical world in that they react to change by opposing it. In this context, if the flux linking a winding changes, there will be an induced e.m.f. that produces a current, the m.m.f. of which will be in opposition to the ‘incoming’ m.m.f..

So, when the stator m.m.f. phasor suddenly comes into existence, the immediate reaction of the rotor is the production of a negative stationary rotor m.m.f. pattern, i.e. in direct opposition to the stator m.m.f.: this is labelled ‘Initial instant’ in Fig. 8.11. Instantaneously, the magnitude of the rotor m.m.f. is such as to keep the rotor flux linkage at zero, as it was previously. This outcome agrees with what we found from a study of two coupled coils.

However, because of the rotor resistance, the rotor current needs a voltage to sustain it, and the voltage can only be induced if the flux changes. So the rotor flux begins to increase, rising rapidly at first (high e.m.f.) then with ever-decreasing gradient leading to lower current and lower rotor m.m.f. The response is a first-order one, governed by the rotor time-constant, so after one time constant (middle sketch) the flux linking the rotor reaches about 63% of its final value, while the rotor current has fallen to 37% of its initial value. Finally the rotor's struggle to prevent the flux changing comes to an end and the rotor flux linkage reaches a steady value determined by the stator current. If the resultant rotor flux linkage is the target value for steady-state running (ψ_R), the corresponding stator current is what we previously referred to as the ‘flux component’.

The physical reason why it takes time to build the flux is that energy is stored in a magnetic field, so we cannot suddenly produce a field because that would require an impulse of infinite power. If we want to build up the flux more rapidly, we can put in a bigger step of stator current at first, so that the flux heads for a higher final value than we really need, then reduce the stator current when we get close to the flux we are seeking.

We began this section with d.c. current in the stator, which in effect corresponds to zero slip frequency, all the field patterns being stationary in space. Because there is no relative motion involved, there is no motional e.m.f. and hence no torque. The ‘torque component’ only comes into play when there is relative motion between the rotor and the rotor flux wave, i.e. when there is slip. Obviously, to cause rotation the frequency must be increased, and as we have seen in the previous section the stator current then has to be adjusted with slip and torque to keep the rotor flux linkage constant.

Finally, it is worth revisiting Fig. 8.9 briefly to reconcile what we have discussed in this section with our picture of steady-state operation, where the rotor currents are at slip frequency. On the left we have the fictitious ‘flux component’ of stator current, which remains constant in magnitude and aligned with the rotor flux linkage, ψ_R, along the so-called ‘direct’ axis. When we first established this flux, the rotor reacted as we have discussed above, but after a few time-constants the flux settled to a constant value along the direct axis in the direction of the flux component of the stator m.m.f.. This is why the arrows on ψ_R point in the same direction as the stator flux producing component.

However, we note from Fig. 8.9 that the rotor flux linkage phasor (L_RI_R) is always equal in magnitude to the torque component of the stator mutual flux linkage phasor, but, as shown by the arrows, it is in the opposite direction. There is therefore no resultant m.m.f. or tendency for flux to develop along this, the so-called ‘quadrature’ axis. This is what we would expect in the light of the previous discussion, where we saw that the reaction of mutually coupled windings to any suggestion of change is for currents to spring up so as to oppose the change. In the literature when, as here, the ‘torque’ current does not affect the flux, the axes are said to be ‘decoupled’.

8.5.3 Mechanism of torque control

In the previous section, our aim was to grow the rotor flux, which, because of its stored energy, took a while to reach the steady state. However if, subsequently, we keep the rotor flux linkage constant (by ensuring that the flux component of the stator current is constant and aligned with the flux) we can cause sudden changes to the motionally-induced rotor current by making sudden changes in the torque component of the stator current.

We achieve sudden step changes in the stator currents by means of a fast-acting closed-loop current controller. Fortunately, we have seen that under transient conditions the effective inductance looking in at the stator is quite small (it is equal to the leakage inductance), so it is possible to obtain very rapid changes in the stator currents by applying high, short-duration impulsive voltages to the stator windings. In this respect the stator current controller closely resembles the armature current controller used in the d.c. drive.

When a step change in torque is required the magnitude, frequency, and phase of the stator currents are changed (almost) instantaneously in such a way that the rotor current jumps suddenly from one steady-state to another. But in this transition it is only the torque component of stator current that is changed, leaving the flux component aligned with the rotor flux. There is therefore no change in the magnitude of the rotor flux wave and no change in the stored energy in the field, so the change can be accomplished almost instantaneously.

We can picture what happens by asking what we would see if we were able to observe the stator m.m.f. wave at the instant that a step increase in torque was demanded. For the sake of simplicity, we will assume that the rotor speed remains constant, and consider an increase in torque by a factor of three (as between the middle and right-hand sketches in Fig. 8.8), in which case we would find that:

1. (a) the stator m.m.f. wave suddenly increases its amplitude;
2. (b) the frequency of the stator m.m.f. wave suddenly increases, defining a new synchronous speed such that the slip increases by a factor of three (as the rotor speed has not changed);
3. (c) the stator m.m.f. wave jumps forward to retain its correct relative phase with respect to the rotor flux, i.e. the angle between the stator m.m.f. and the rotor flux increases from ϕ₂ to ϕ₃ as shown in Fig. 8.8.

Thereafter the stator m.m.f. retains its new amplitude, and rotates at its new speed.

The rotor current experiences a step change from a steady state at its initial slip frequency to a new steady state with three times the amplitude and frequency, and there is a step increase in torque by a factor of three, as shown in Fig. 8.12A. The new current is maintained by the new (higher) stator currents and slip frequency.

We should note particularly that it is the jump in the angular position (i.e. space phase angle) that accompanies the step changes in magnitude and frequency of the stator m.m.f. phasor that allows the very rapid and transient-free control of torque. Given that the definition of a vector quantity is one which has magnitude and direction, and that the angular position of the phasor defines the direction in which it is pointing, it is clear why this technique is sometimes known by the name ‘vector control’.¹

To underline the importance of the sudden change in phase of the stator current (i.e. the sudden jump in angular position of the stator m.m.f. in achieving a step of torque), Fig. 8.12B shows what happens typically if only the magnitude and frequency, but not the position of the stator m.m.f. phasor are suddenly changed. The steady state conditions are ultimately reached, but only after an undesirable transient governed by the (long) rotor time-constant, which may persist for several cycles at the slip frequency. The fundamental reason for the transient is that if the magnitude of the stator current is suddenly increased without a change of position, the flux and torque components both increase proportionately. The change in the flux component portends a change in the rotor flux (and associated stored energy), which in turn is resisted by induced rotor currents until they decay and the steady-state is reached.

This section has described the underlying principles by which very rapid and precise torque control can be achieved from an induction motor, but we should remember that until sophisticated power electronic control became possible the approach outlined here was only of academic interest. The fact that the modern inverter-fed drive is able to implement torque control and achieve such outstandingly impressive performance from a motor whose inherent transient behaviour is poor, represents a major milestone in the already impressive history of the induction motor. The way in which such drives achieve field oriented control is discussed next.

8.6.1 PWM controller/vector modulator

In the inverters we have looked at so far (see Section 2.4) we have supposed that the periodic switching required to approximate a sinusoidal output was provided from a master oscillator. The frequency of the oscillator determined the frequency of the a.c. voltage applied to the motor, and the amplitude was controlled separately. In terms of space phasors this allows control of the amplitude and frequency, but not the instantaneous angular position of the voltage and current phasors. As we have seen, it is the additional ability to make instantaneous changes to the angular position of the output phasor that is the key to dynamic torque control, and this is the key feature provided by the ‘vector modulator’.

We now explore what the inverter can produce in terms of its output voltage phasor. We recall that there are six devices (switches) in three legs (see Fig. 2.21), and to avoid a short-circuit across the inverter d.c. link both switches in one leg must not be turned on at the same time. If we make the further restriction that each phase winding must at all times be connected to one or other of the d.c. link terminals, there are only eight possible combinations, as shown in Fig. 8.13.

The six switching combinations labelled 1–6 each produce an output voltage phasor of equal amplitude but displaced in phase by 60° as shown in the lower part of each sketch, while the final two combinations have all three terminals joined together, so the line voltage is zero. The six unit vectors are shown with their correct relative phase, but rotated so as to bring U6 horizontal, in Fig. 8.14.

Having only six states of the voltage phasor at our disposal is clearly not satisfactory, because we need to exert precise control over the magnitude and position of the voltage phasor at any instant, so this is where the ‘time modulation’ aspect comes into play. For example, if we switch rapidly between states U1 and U6, spending the same time with each, we will effectively have synthesised a voltage phasor lying half way between them, and of magnitude U1cos 30° (or 86.7% of U1), as shown by the vector U(x) in the upper part of Fig. 8.14. If we spend a higher proportion of the time on U1 and the remainder on U2, we could produce the vector U(y). As long as we spend the whole of the sample time on either U1 or U6, we will end up somewhere along the line joining U1 to U6.

We have used the terms ‘switch rapidly’ and ‘the time’ without specifying what they mean. In practice, we would expect the switching or modulating frequency to be perhaps a few kHz up to the low tens of kHz, so ‘the time’ means one cycle at this frequency, say 100 microseconds at 10 kHz. So for as long as we wished the voltage phasor to remain at U(x), we would spend 50 μs of each sample period alternately connected to U1 and U6.

Recalling that ideally we want to be able to choose both magnitude and position it is clearly not satisfactory to be constrained to the outer edges of the hexagon. So now we bring the zero vector into play. For example, suppose we wish the voltage phasor to be U(z), as in the lower sketch. This is composed of (0.5)U5 plus (0.3)U6. Hence in each modulating cycle of 100 μs, we will spend times of 50 μs on U5, 30 μs on U6 and 20 μs with one of the zero states.

The precise way in which these periods are divided within one cycle of the modulating frequency is a matter of important detail in relation to the distribution and minimisation of losses between the six switching devices, but need not concern us here. Suffice it to say that it is a straightforward matter to arrange for digital software/hardware that has input signals representing the magnitude and instantaneous position of the output voltage phasor, and which selects and modulates the six switches appropriately to create the desired output until told to move to a new location.

When we introduced the idea of space phasors earlier in this chapter, we saw that if we begin with balanced three-phase sinusoidal voltages, the voltage phasor is of constant length and rotates at a uniform rate. Looking at it the other way round, it should be clear that if we arrange for the output of the inverter to be a voltage phasor of constant length, rotating at a constant rate, then the corresponding phase voltages must form a balanced sinusoidal set, which is what we want for steady state running.

We conclude that in the steady state, the magnitude of the input signal to the vector modulator would have a constant amplitude and its angle would increase at a linear rate corresponding to the desired angular velocity of the output. Clearly in order to avoid having to deal with ever-increasing angles it will reset each time a full cycle of 360° is reached, as shown in Fig. 8.15.

Away from the steady-state condition, for example during acceleration, we should recall that to preserve the linear relation between torque and the stator current component (I_t), the flux component of the stator current phasor (I_F) must remain aligned with the rotor flux. As we will see in the next section, this is achieved by deriving the angle input to the vector modulator directly from the absolute angular position of the rotor flux.

8.6.2 Torque control scheme

A simplified block diagram of a typical field-oriented torque control system is shown in Fig. 8.16.

The first and most important fact to bear in mind in the discussion that follows is that Fig. 8.16 represents a torque control scheme, and that for applications that require speed control, it will form the ‘inner loop’ of a closed-loop speed control scheme. The torque and flux inputs will therefore be outputs from the speed controller, as indicated in Fig. 8.17. Note that the flux demand will be constant up to the base speed of the motor, and then falls. This is field-weakening operation (Section 7.3), as a consequence of the drive not being able to produce any more volts, and the V/f ratio therefore reducing. In Field oriented control, we still have the same practical limitations of maximum available voltage and so the flux must be reduced accordingly. As in any good control scheme, in order to facilitate good dynamic operation, it is necessary to keep a margin of volts in order to change the current quickly. Nonetheless, the transient performance in the field-weakening region remains impressive (though obviously not as good as at full flux).

Returning to Fig. 8.16 it has to be acknowledged that it looks rather daunting, and getting to grips with it is not for the faint-hearted. However, if we examine it a bit at a time, it should be possible to grasp the essential features of its operation. To simplify matters, we will focus on steady-state conditions, despite the fact that the real merit of the system lies in its ability to provide precise torque control even under transient conditions.

Taking the broad overview first, we can see that there are similarities with the d.c. drive with its inner current (torque) control loop (see Chapter 4), notably the stator current feedback and the use of proportional and integral (PI) controllers to control the torque and flux components of the stator current. It would be good if we could measure the flux and torque components directly, but of course the current components do not have separate existences: they are merely components of the stator current, which is what we can measure. The motor has three phases, but because we are assuming that there is no neutral connection, it suffices to measure only two of the line currents (because the sum of the three is zero). The information from these two currents allows us to keep track of the angular position of the stator current phasor with respect to the stationary reference frame (θ_S) as shown in Fig. 8.18.

However, the stator current feedback signals are alternating at the frequency supplied by the inverter, and the corresponding stator space phasor is rotating at the supply frequency with respect to a stationary reference frame. Before the flux and torque components of these signals (I_F and I_T) can be identified (and subsequently fed back to the PI controllers) they must first be transformed (see Section 8.2.2) into a reference frame that rotates with the rotor flux. As explained previously, the rotor flux angle θ_Ref is therefore an essential input to the transformation algorithm, as shown in Fig. 8.16.

The vertical dotted line in the middle of Fig. 8.16 separates quantities defined in the stationary reference frame (on the right) from those in the rotating reference frame (on the left). In the steady state, all those on the left are d.c., while all those on the right are time-varying.

The reader might wonder why, when we follow the signal path of the current control loops, beginning on the right with the phase current transducers, there is no matching ‘inverse transform’ to get us back from the rotating reference frame on the left to the stationary reference frame on the right. The answer lies in the nature of the input signal to the PWM/vector modulator and inverter, which we discussed above. Let us suppose that the motor is running in the steady state, so that the output voltage phasor rotates at a constant rate with angular frequency ω. Under these conditions the rotor flux phasor also rotates with constant angular velocity ω, so the angle of the flux vector with respect to the stationary reference frame (θ_Ref) increases linearly with time. Also, in the steady state, the output from the PI controllers is constant, so the angle θ_V (Fig. 8.16) is constant. Hence the input angle to the modulator (θ_m in Fig. 8.16) which is the sum of θ_Ref and θ_V is also a ramp in time, and this is what provides the rotation of the output voltage phasor. In effect, the system is self-sustaining: the primary time-varying input angle to the modulator comes from the flux position signal (which is already in the stationary reference frame), and the PI controller provides the required magnitude signal (| Vs |) and the additional angle (θ_V).

Turning now to the action of the PI controllers, we see from Fig. 8.16 that the outputs are voltage commands in response to the differences between the feedback (actual) values of the transformed currents and their demanded values. The flux demand will usually be constant up to base speed, while the torque demand will usually be the output from the speed or position controller, as shown in Fig. 8.17. The proportional term gives an immediate response to an error, while the integral term ensures that the steady-state error is zero. The outputs from the two PI controllers (which are in the form of quadrature voltage demands, V_F and V_T) are then converted from rectangular to polar form, to produce amplitude and phase signals, | V_s | and θ_V, where

$\left|V_S\right|=\sqrt{V_F^2+V_T^2},\mathrm{and}{\theta }_V={\mathit{tan}}^{-1}\frac{V_T}{V_F},$

as shown in Fig. 8.19.

The amplitude term specifies the magnitude of the output voltage phasor (and thus the three phase voltages applied to the motor), with any variation of the dc link voltage (V_dc) being compensated in the PWM controller. The phase angle (θ_V) represents the desired angle between the stator voltage phasor and the rotor flux phasor, both of which are measured in the stationary reference frame. The angle of the rotor flux phasor is θ_Ref, so θ_V is added at the input to the vector modulator to yield the stator voltage phasor angle, θ_m, as shown in Fig. 8.16.

We can usefully conclude our look at the steady state by adding the stator voltage phasor to Fig. 8.18 to produce Fig. 8.20, to provide reassurance that, in the steady state, the rather different approach we have taken in this section is consistent with the classical approach taken earlier.

8.6.3 Transient operation

We concluded earlier that for the motor torque to be directly proportional to the torque component of stator current, it is necessary to keep the magnitude of the rotor flux constant and to ensure that the flux component of stator current is aligned with the rotor flux. This is achieved automatically because the principal angle input to the vector modulator comes directly from the rotor flux angle (θ_Ref), as shown in Fig. 8.16. So during acceleration, for example, the instantaneous angular velocity of the rotor flux wave will remain in step with that of the stator current phasor, so that there is no possibility of the two waves falling out of synchronism with one another.

In Section 8.5.3 we discussed a specific example of how to obtain a step change in torque by making near-instantaneous changes to the magnitude, speed and position of the stator m.m.f. wave, and we are now in a position to see how this particular strategy is effected using the control scheme shown in Fig. 8.16.

A step demand for torque causes a step increase in | V_s | and θ_V at the output of the rectangular to polar converter in order to effect a very rapid increase in the magnitude and instantaneous position of the stator current phasor. At the same time, the algorithm that calculates the slip velocity of the flux wave (see later, Eq. 8.3) yields a step increase because of the sudden increase in the torque component of stator current. The principal angular input to the vector modulator (the flux angle (θ_Ref)) therefore changes gradient abruptly, as shown in Fig. 8.21.

Recalling that the steady-state stator frequency is governed by the angular velocity of the flux (i.e. $\frac{{\mathit{d\theta }}_{\mathit{Ref}}}{\mathit{dt}}$), this lines up with our expectation that (assuming the rotor velocity is constant) the stator frequency will increase in order to increase the slip and provide the new higher torque.

8.6.4 Acceleration from rest

The practical results in Fig. 8.22 ably demonstrate the impressive performance of a field-oriented torque control system. These relate to a motor whose rotor time-constant is approximately 0.1 s, and covers an overall time period of 0.5 s. The motor is initially unexcited, rotor flux is then established and it is then accelerated up to a steady speed.

The upper diagram shows the demanded values for the transformed flux and torque components of stator current; the middle diagram shows the measured (actual) flux and torque components; and the lower shows the three phase currents.

This particular motor requires a stator current flux component (I_F) of 10 A to maintain full rotor flux linkage, but by applying an initial demand of 30 A, the initial rate of rise of the flux is trebled. After about one rotor time-constant the demand is reduced to 10 A to maintain the flux. Without this short-term ‘forcing’ it would have taken about five time-constants to establish the flux.

The measured value of the transformed flux component of stator current is shown in the middle plot, and it is seen to follow the demanded signal closely, with only slight overshoot. This signal is the transformed version of the actual stator winding currents, so the fact that it is on target demonstrates that the phase currents are established rapidly, and held while the flux builds up, as we can see in the lower figure. After 0.2 s phase U carries a positive d.c. current of 10 A while phases V and W each carry a negative d.c. current of 5 A. The rotor remains at rest, with full rotor flux now established, and at this time there is no demand for torque, so the motor remains stationary, the slip being zero.

At 0.2 s, a step demand signal (I_T) equivalent to a torque component of stator current of 20 A is applied, in order to accelerate the motor. The flux demand remains at 10 A during the acceleration, in order to keep the rotor flux linkage constant, thereby ensuring that torque is proportional to slip. The torque demand is maintained until 0.4 s, when the torque producing reference is reduced to zero, and the motor stops accelerating.

We note the almost immediate and transient-free transition of the three-phase currents from their initial steady (d.c.) values immediately prior to 0.2, into constant amplitude, ‘smoothly increasing frequency’ a.c. currents over the next 0.2 s. And then there is a similarly near-perfect transition to reduced amplitude steady-state conditions (at about 40 Hz) after 0.4 s. In the steady state, the torque component is negligible because the motor is unloaded, and the stator current consists only of the flux component, which traditionally would be referred to as the magnetising current.

During the acceleration the controller keeps the stator phase currents at the amplitude corresponding to the vector sum of the demanded flux and torque components (Fig. 8.19), and it continuously estimates the rotor flux position in order to keep the stator flux component aligned with the rotor flux. Hence while the rotor is accelerating, the instantaneous angular velocity of the rotor flux is greater than that of the rotor by an amount equal to the slip, as shown in the Fig. 8.23.

Younger readers will doubtless not require convincing of the validity of these remarkable results, but they might find it salutary to know that until the 1970’s it was widely believed that such performance would never be possible.

To conclude this section we can draw a further parallel between the field-oriented induction motor and the d.c. motor. We see from Fig. 8.22 that as the motor accelerates, the frequency of the stator currents increases with the speed. If we stationed ourselves on the rotor of a d.c. motor as it accelerated, the rate at which the current in each rotor coil reversed as it was commutated would also increase in proportion to the speed, though of course we are not aware of it when we are in the stationary reference frame.

8.6.5 Deriving the rotor flux angle

By now, the key role played by the rotor flux angle should have become clear, so finally we look at how it is obtained. It is not practical nor economic to fit a flux sensor to the motor, so industrial control schemes invariably estimate the position of the flux.

We will first establish an expression for absolute rotor flux angle (θ_Ref) in the stationary reference frame in terms of quantities that can either be measured or estimated. Readers who find the derivation indigestible need not worry as it is the conclusions that are important, not the analytical detail.

If we let the angle of the rotor body with respect to the stationary reference frame be θ, then the instantaneous angular velocities of the rotor flux wave and the rotor itself are given by

${\omega }_{\mathit{flux}}\mathit{=}\frac{d{\theta }_{Ref}}{dt}$

${\omega }_{\mathit{rotor}}=\frac{d\mathit{\theta }}{dt}$

The rotor motional e.m.f. is directly proportional to the rotor flux linkage and the slip velocity, i.e.

$V_R={\psi }_R\left({\omega }_{\mathit{flux}}-{\omega }_{\mathit{rotor}}\right),$

and the rotor current is therefore given by

$I_R=\frac{{\psi }_R\left({\omega }_{\mathit{flux}}-{\omega }_{\mathit{rotor}}\right)}{R_R}.$

The corresponding component of stator current is given (see Fig. 8.12) by

$I_{\mathit{ST}}=\frac{L_R}{M}{I}_R$

Combining these equations and rearranging gives

$\frac{d{\theta }_{Ref}}{dt}\mathit{=}\frac{{MR}_R}{{\psi }_R{L}_R}{I}_{ST}\mathit{+}{\omega }_{\mathit{rotor}}=\left(\frac{M}{\tau {\psi }_R}\right)I_{ST}+{\omega }_{\mathit{rotor}}$

where τ is the rotor time-constant. Hence to find the rotor flux angle at time t we must integrate the expression above.

The mutual inductance M is a constant, and although the time-constant will vary because the rotor resistance varies with temperature, it will change relatively slowly, so we can treat it as constant, in which case the rotor flux angle is given by

${\theta }_{Ref}\mathit{=}{\int }_0^{\mathit{t}}{\omega }_{\mathit{rotor}}dt\mathit{+}\frac{\mathit{M}}{\mathit{\tau }}{\int }_0^{\mathit{t}}\frac{{\mathit{I}}_{ST}}{{\psi }_{\mathit{R}}}dt\mathit{=}\theta \mathit{+}\frac{\mathit{M}}{\mathit{\tau }}{\int }_0^{\mathit{t}}\frac{{\mathit{I}}_{ST}}{{\psi }_{\mathit{R}}}dt$

Note that because of the symmetry of the rotor, we only need the time-varying element of the rotor body angle (θ), not the absolute position, so the constant of integration is not required. (In contrast, for vector control of permanent-magnet motors, the absolute position is important, because the rotor has saliency.)

The various methods that are used to keep track of the flux angle are what differentiate the various practical and commercial implementations of field-oriented control, as we will now see.

If we have a shaft encoder we can measure the rotor position (θ), or if we have a measured speed signal, we can derive θ by direct integration. This approach involves the fewest estimations, and therefore will normally offer superior performance, especially at low speeds, but is more costly because it requires extra transducers. We will refer to systems that use shaft feedback as ‘closed-loop’, but in the literature they may be also referred to as ‘direct vector control.’ In common with all schemes, the second term has to be estimated.

Many different methods of estimating the instantaneous parameter values are employed, but all employ a digital simulation or mathematical model of the motor/inverter system. The model runs in real time and is subjected to the same inputs as the actual motor, the model then being continuously fine-tuned so that the predicted and actual outputs match. Modern drives measure the circuit parameters automatically at the commissioning stage, and refine them on a near-continual basis to capture parameter variations.

The majority of vector control schemes eliminate the need for measurement of rotor position, and instead the rotor position term in Eq. (8.3) is also estimated from a motor model, based on the known motor voltage and currents. Rather confusingly, in order to differentiate them from schemes that do have shaft transducers, these systems are known as ‘open-loop’ or ‘indirect’ vector control. The term ‘open loop’ is a misleading one because at its heart is the closed-loop torque control shown in Fig. 8.16, but it is widely used: what it really means is ‘no shaft position or speed feedback’.

The main problems of the open-loop approach occur at low speeds where motor voltages become very small and measurement noise can render the algorithms unreliable. Techniques such as the injection of high frequency “diagnostic” voltage signals exist, but are yet to find widespread acceptance in the market. Open loop inverter-fed induction motors are usually unsuitable for continuous operation at frequencies below 0.75 Hz, and struggle to produce full torque in this region.

An additional difficulty is that the significant variation of rotor resistance with temperature is reflected in the value of the all-important rotor time-constant, τ. Any difference between the real rotor time constant and the value used by the model causes an error in the calculation of the flux position and so the reference frame becomes misaligned. If this happens, the flux and torque control are no longer completely decoupled which results in sub-optimum performance and possible instability. To avoid this, routines are included in the drive to provide on-going estimates of the rotor time constant.

8.7.1 Outline of operation

The block diagram of a typical direct torque control scheme is shown in Fig. 8.24. There are several similarities with the scheme shown in Fig. 8.16, notably the inverter, the phase current feedback, and the separate flux and torque demands, which may be generated by the speed controller, as in Fig. 8.17.

However, there are substantial differences. Earlier we discovered that the inverter output voltage space phasor has only six active positions, and two zero states (see Fig. 8.14), corresponding to the eight possible combinations of the six switching devices. This means that at every instant there are only eight options in regard to the voltage that we can apply to the motor terminals. In the field-oriented approach, PWM techniques are employed to alternate between adjacent unit vectors to produce an effective voltage phasor of any desired magnitude and instantaneous position. However, with direct torque control, only one of the eight intrinsic vectors is used for the duration of each sample, during which the estimated stator flux and torque are monitored.

The motor model is exposed to the same inputs as the real motor, and from it the software continuously provides updated estimates of the stator flux and torque. These are compared with the demanded values and as soon as either strays outside its target hysteresis band, a logical decision is taken as to which of the six voltage phasors is best placed to drive the flux and/or torque back onto target. At that instant the switching is changed to bring the desired voltage phasor into play. The duration of each sample therefore varies according to the rate of change of the two parameters being monitored: if they vary slowly it will take a long time before they hit the upper or lower hysteresis limit and the sample will be relatively long, whereas if they change rapidly, the sample time will be shortened and the sample frequency will increase. Occasionally, the best bet will be to apply zero voltage, so one of the two zero states then takes over.

8.7.2 Control of stator flux and torque

We will restrict ourselves to operation below base speed, so we should always bear in mind that although we will talk about controlling the stator flux, what we really mean is keeping its magnitude close to its normal (rated) value, at which the magnetic circuit is fully utilised. We should also recall that when the stator flux is at its rated value and in the steady state, so is the rotor flux.

It is probably easiest to grasp the essence of the direct torque method by focusing on the stator flux linkage, and in particular on how (a) the magnitude of the stator flux is kept within its target limits and (b) how its phase angle with respect to the current is used to control the torque.

The reason for using stator flux linkage as a reference quantity is primarily the ease with which it can be controlled. When we discussed the basic operation of the induction motor in Chapter 5, we concluded that the stator voltage and frequency determined the flux, and we can remind ourselves why this is by writing the voltage equation for the stator as

$V_S=I_S{R}_S+\frac{{\mathit{d\psi }}_S}{\mathit{dt}}$

(We are being rather loose here, by treating space phasor quantities as real variables, but there is nothing to be gained by being pedantic when the message we take away will be valid.) In the interests of clarity we will make a further simplification by ignoring the resistance voltage term, which will usually be small compared with V_S. This yields

$V_S=\frac{{d\psi }_S}{dt}\mathrm{or},\mathrm{in}\mathrm{integral}\mathrm{form},{\psi }_S=\int V_{S}dt$

The differential form shows us that the rate of change of stator flux is determined by the stator voltage, while the integral form reminds us that to build the flux (e.g. from zero) we have to apply a fixed volt-second product, with either a high voltage for a short time, or a low voltage for a long time. We will limit ourselves to the fine-tuning of the flux after it has been established, so we will only be talking about very short sample intervals of time (Δt) during which the change in flux linkage that results (Δψ_S) is given by

$\mathrm{\Delta }{\psi }_S=V_S\mathrm{\Delta }t$

As far as we are concerned, ψ_S represents the stator flux linkage space phasor, which has magnitude and direction relative to the stator reference frame, and V_S represents one of the six possible stator voltage space phasors that the inverter can deliver. So if we consider an initial flux linkage vector ψ_S as shown in Fig. 8.25A, and assume that we apply, over time Δt, each of the six possible options, we will produce six new flux-linkage vectors. The tips of the new vectors are labelled ψ₁ to ψ₆ in the figure, but only one (ψ₄) is fully drawn (dotted) to avoid congestion. There is also the option of applying zero voltage, which would of course leave the initial flux-linkage unchanged.

In (a) option 4 results in a reduction in amplitude and (assuming anticlockwise rotation) a retardation in phase of the original flux, but if the original flux linkage had a different phase, as shown in diagram (b), option 4 results in an increase in magnitude and an advance in phase. It should be clear that outcomes vary according to initial conditions, and therefore an extensive look-up table will be needed to store all this information.

Having seen how we can alter the magnitude and phase of the stator flux, we now consider the flux linkage phasor during steady-state operation with constant speed and torque, in which case we know that ideally all the space phasors will be rotating at a constant angular velocity.

The locus of the stator flux linkage space phasor (ψ_S) is shown in Fig. 8.26. In this diagram the spacing of hysteresis bands indicated by the innermost and outermost dotted lines have been greatly exaggerated in order to show the trajectory of the flux linkage phasor more clearly. Ideally, the phasor should rotate smoothly along the centre dotted line.

In this example, the initial position shown has the flux linkage at the lower bound, so the first switching brings voltage vector 1 into play to drive the amplitude up and the phase forward. When the upper bound is reached, vector 3 is used, followed by vector 1 again and then vector 3. Recalling that the change in the flux linkage depends on the time for which the voltage is applied, we can see from the diagram that the second application of vector 3 lasts longer than the first. (We should also reiterate that in this example only a few switchings take place while the flux rotates through sixty degrees: in practice the hysteresis band is very much narrower, and there may be many hundreds of transitions.)

We are considering steady-state operation, and so we would wish to keep the torque constant. Given that the flux is practically constant, this means we need to keep the angle between the flux and the stator current constant. This is where the torque hysteresis controller shown in Fig. 8.24 comes in. It has to decide what switching will best keep the phase on target, so it runs in parallel with the magnitude controller we have looked at here. Each controller will output a signal for either an increase or decrease in its respective variable (i.e. magnitude or phase) and these are then passed to the optimal switching table to determine the best switching strategy in the prevailing circumstances (see Fig. 8.24).

As we saw when discussing field-oriented control, it is not possible to make very rapid changes to the rotor flux because of the associated stored energy. Because the rotor and stator are tightly coupled it follows that the magnitude of the stator flux linkage cannot change very rapidly either. However, just as with field-oriented control, sudden changes in torque can be achieved by making sudden changes to the phase of the flux linkage, i.e. to the tangential component of the phasor shown in Fig. 8.26.