Induction motorWhen I was a child, I got very disappointed when I first opened a squirrel cage induction motor. There was nothing inside. Only some wires that do not move and a heavy iron cylinder fixed on an axle. The most interesting parts were bearings and a cooling fan but they didn’t explain how an induction motor work. To me, the induction motor remained a mystery for a long time. However, slowly, it revealed some of its secrets. Now I must say that an induction, squirrel cage electric motor is absolutely an ingenious device. It is rugged, long lasting, quiet, clean and most of all, it is cheap. Most of its good properties come from its apparent “nothing inside” simplicity.
A side crosssection and a frontal crosssection of a squirrel cage induction motor. Stator parts are in grayish colors, rotor parts are in bluish colors and electrical windings (coils) are in reddish colors.
The statorOutside is the enclosure designed to protect the motor from any damage and dust and also to help in cooling. Often the enclosure has cooling fins on its outer surface – as the motor rotates it drives a small centrifugal fan mounted on its rear axle stub that blows air over cooling fins to promote cooling effect. Inside, most of the stator consists of thin highsilicalowcarbon steel ring sheets that are tightly packed together (laminated stator). The purpose of these steel sheets is to conduct magnetic field while the motor is engaged. The reason why is it made form many thin steel sheets, and not form a solid block of steel is the same as is for transformer cores – to lower eddy current loses and to lower hysteresis loses. There are deep grooves on the stator’s laminated package on its inner side. The wires that make stator windings (coils) are placed into these grooves and are fixed there with some resin. A stator winding can have many loops of wire – it depends on the voltage the motor is intended to run on.
A detail of stator and rotor grooves filled with conductors.
The rotorAt the very center of the rotor there is a steel axle (shaft) that can rotate on ball bearings. The rest of the rotor also consists mostly of thin steel sheets – much like the stator. These steel sheets have grooves on the outer side. The rotor windings (coils) are placed into these grooves. Rotor windings are quite different than stator windings. A rotor winding only consist of a single wire loop. This wire must be quite thick to allow for high amperages that are induced on it. In practice, rotor windings are made as straight aluminum or copper rods that are all shortcircuited at both ends. Made this way, rotor’s winding assembly looks similar to a squirrel cage (a rodent running cage).
The squirrel cage of an induction motor represents rotor’s windings. Thick conductive rods are shortcircuited by conductive rings on each side. Usually, the squirrel cage is made by casting the molten aluminum directly into the rotor grooves. This way, the whole cage is made from one single piece of material and so there are no highresistance junction points on the current path. Also usually, cage wires are intentionally skewed a bit to make rotor rotation smoother.
Rotating magnetic fieldThe foundation of a 3phase squirrel cage induction motor is the rotating magnetic field. The rotating field is generated by stator windings. All the stator windings can be divided into three main groups; each group belongs to one phase – that is, each of these three winding groups must have its electric current in offset by 120 degrees, and to obtain this it is the easiest to connect the motor to the threephase power grid system. Stator windings must be arranged around the stator in a specific order to create the rotating magnetic field effect. There should be even number of windings per each phase (therefore there can be 6, 12, 18, 24…. windings) where every first of them should be winded clockwise, and every second counterclockwise (looking to the center of the motor). Let’s take, for example, the simplest case – a 3phase motor with 6 stator windings (the, socalled, 2pole motor). There are two windings per phase, each of these two is winded in opposite direction.
The left picture shows how you arrange stator windings on a 2pole induction motor. Each color represents a winding pair that belongs to one particular phase. On the right you see in detail how the red phase windings are wound  note that windings that belong to the same phase are actually made from one single piece of wire that is looped many times in each of them. What happens when you connect the motor to the threephase power grid? The current will start to flow through stator windings. But as this is an AC current it changes in time. In addition, the current that flows through each of three main groups of windings have the offset of 120 degrees. It means that in one moment the current will have its maximal value in first group of windings, then after 1/3 cycle in the second, and then in the third group. The overall magnetic field is a vector sum of magnetic fields produced by all three sets of windings. As the result we will have a rotating magnetic field.
The upper portion of this picture shows how currents change in each main group of windings during the time (sinusoids at 120 degrees offset). The bottom portion shows the magnetic field lines inside an induction motor at given moments. Note that the magnetic field rotates once per AC current period (50Hz in Europe, 60Hz in USA). By the way, a motor with 6 stator windings is called a 2pole motor (because there is one pole where magnetic field enters, and one pole where it exits the stator). If it has 12winding it is called a 4pole motor, 18winding is called a 6pole motor and so on. What if we want our motor to rotate more slowly? Easy, we only have to make a 4pole, or a 6pole stator (or any 2Npole). The picture below shows how you arrange windings in a 6pole motor.
The picture shows schematics of a 6pole motor stator – arrangement of windings and an example of magnetic field lines configuration. The rotating field will turn three times slower than in a 2pole motor. In an induction motor the rotating magnetic field is performed, as you can see, without any moving part. This was a brilliant idea – the usage of the threephase power grid system makes our electric motors very simple and rugged.
A squirrel cage rotor inside a rotating magnetic fieldWhat happens when you place a squirrel cage rotor inside a rotating magnetic field? As you expected, it starts turning. (Note that you should not even try to power an induction motor without having the rotor in place – without the rotor the inductance of stator windings is very low so a high current will burn stator windings immediately). The rotating magnetic field induces a current inside a squirrel cage (the rotor windings) and that, in turn, creates torque on rotor. The torque will accelerate the rotor in the direction of the rotating magnetic field. If there is no load applied to the motor shaft, the rotation speed of the rotor will get very close to the rotation speed of the magnetic field (called the synchronous speed) but will never reach it – that is why this type of motor is also called asynchronous motor. The torque that motor delivers depends on its rotating speed. This curve is very important and is shown on the picture below
You see four interesting points on the above diagram. The nominal torque is the torque that can be given by the motor at full time. However, a typical induction motor is capable to give much greater maximal torque than its nominal but only for a short period of time (the motor soon overheats if you keep it at higherthannominal torque). It can be seen that as you increase the load (torque) the speed of the motor rotation slows down. However, after some maximal torque the motor stops and will restart again only if the torque drops down below the starting torque level. By the way, most motors have their nominal torque lower than their starting torque so they can start at their full nominal load. But what will happen if we try to force the rotor faster than the synchronous speed. On the picture below you have the full version of the torquespeed diagram.
When the motor rotation rate is lower than synchronous speed then the motor is in motor regime, and when it is greater than the synchronous speed then the motor is in generator regime. At both sides, the torque curve asymptotically tends to zero as the rotation rate tends toward infinity. If we try to rotate the rotor faster than synchronous speed, the motor will try to resist by providing countertorque. This way it enters into the generator regime – it starts producing electric power.
The mathUsing a very simplified induction motor model we will try to explain why the torque curve has its shape. Our approach will be more qualitative than quantitative but at least it will use some straightforward math. (You can find some very nice and short induction motor calculations on the web but they all tend to use math that is beyond my knowledge level – that is why I decided to calculate it myself). Our induction motor model is shown on the picture. It basically consists of one single closed loop of wire (that represents the motor cage) and of rotating homogeneous magnetic field. I will imagine that the loop is fixed and the field is rotating, but you can think the other way if it is easier to you.
A very simple model of an induction motor – a homogenous magnetic field is rotating around the stationary wire loop. A current is induced in the loop and as a result the force is generated on to produce a torque. Here, we don’t take in consideration how the rotating magnetic field is actually produced. We want to calculate what is the torque excreted on the wire loop depending on magnetic field rotation rate (omega). We expect to get something similar to the torque/rotationrate curve (only the origin point will be translated to the synchronous speed and the xaxis will be inverted because N=Nsomega/[2*pi]). We start with four simple equations. The equation [1] is the famous Faraday’s law – the electromotive force generated in the loop is equal to the change in magnetic flux. This electromotive force generates a current I(t) depending on the loop resistance (Ohm’s law).
The equation [2] describes that the magnetic flux through the loop, in our case, consist of two parts – the flux that comes from the rotating magnetic field and the flux that is generated by the loop itself (actually by the generated current in the loop). Equations [3] and [4] describe these two flux components. The flux component of the rotating magnetic field has the sine wave function shape, while the loop autoinduced flux is proportional to the current in the loop. Combining equations [4] and [3] into equation [2] and then substituting the result into the equation [1] we get the following equation.
This is a simple differential equation that, I am sure, you know how to handle. Good for you. Unfortunately, I am, not very good in solving differential equations so I used the Laplace transform (I am more familiar with).
Equation [6] is the Laplace transform of the equation [5]. The current, I(s) can then be expressed as in equation [7] or in turn, can be as in [8]. Where A, B and C are some factors that we have to calculate yet.
After we calculated A, B and C we substituted them into [8] and got the equation [10]. This one we can easily transform back to timedomain – finally, the equation [11] is the solution of the differential equation [5].
However, we only want to consider a stationary result, therefore we are rejecting the redoutlined part of the equation [11] (as you can see, after some time this part of equation will drop down to a very small value and we can simply reject it).
Now we have the equation [12] that is very simple and tells us what is the current in the loop. However, I am not very satisfied with the equation [12] because I was hoping to get the result in form of I(t)=Io sin (omega*t + phi). I will convert the equation [12] using the following general equations:
And finally I get the equation [13] from where I can directly read the amplitude of the current and the phase of the current in the loop.
Let’s take a closer look at the equation [13]. Both, its phase and its amplitude depend on the omega (rotation speed of the magnetic field).
We will now use the equation [13] to calculate the torque on the loop. We need two more simple equations. The equation [14] gives the general force on the straight wire (of lambda length) placed the magnetic field B(t). The equation [14] describes our particular case magnetic field and it deserves a bit of explanation.
The equation [14] is closely related to the equation [3]. First, the magnetic flux in the equation [3] can be simply calculated, for a homogenous magnetic field, by multiplying the surface are of the loop (2r x lambda) and the strength of the field (Bo). Second, the equation [3] incorporates the sine function because only sine component of the rotating magnetic field generates the magnetic flux that is perpendicular to the surface area of the loop, while the equation [14] incorporates the cosine function because only this component generates the torque on the loop. By combining equations [13], [14] and [15] we get the equation [16]. It doesn’t look very nice to me so I will transform it to the equation [17] (by using: sinA cosB = [sin(A+B)+sin(AB)]/2 ).
Again I will reject the redoutlined part of the equation [17] because this one gives no contribution to the average torque (its integral during the rotation cycle is zero). Finally we are calculating the torque by multiplying the force and the distance from the axis of rotation (r). We are also multiplying the result by two because there are two straight wire sections in our loop that contribute to the torque.
The result is the equation [20] that tells us what is the torque on the loop. If we draw this function we get something like the picture below.
This is exactly what I was hoping for – the torque/speed curve of an induction motor (only we have to invert the xaxis and translate the origin point). Okay, what can we do to make our motor have more torque? We could enlarge its dimensions (r and/or lambda), of course. We cannot enlarge the magnetic field strength much because the iron allows only around 1T. Also we are limited in decreasing the ‘L’ because we can only decrease it further if we use smaller amount of iron in our motor – and this would decrease the magnetic field strength even more dramatically than the ‘L’ generating the opposite effect. The interesting stuff is that the maximal torque doesn’t depend on the resistance of the loop. Odd. However the resistance greatly changes the shape of the torque curve and this fact is sometimes used to regulate the speed of an induction motor (wound rotor motor type) by changing the resistance of the rotor windings. Anyway, when the resistance of the cage conductor is higher then the motor will hold its rotating speed more loosely as the load changes.
The singlephase induction motorOkay, we know that a threephase power grid system can create a rotating magnetic field that we can efficiently employ in a threephase induction motor. But most of induction motors are actually singlephase type. How do they make a rotating magnetic field? Well, the answer is that they don’t. They only create a pulsating magnetic field, but it shows up that this is good enough! Cool. Even more cool is the way it can be proven. This proof you can never forget – it is so amazing. A pulsating magnetic field can be expressed as a superposition of two rotating magnetic fields that are rotating in the opposite direction. Each of these two rotating fields would generate its own torque on the rotor – we can simply sum up these torques. We can conclude that the torque curve of a singlephase motor would be equal to the summation of two standard, opposite direction 3phase motor torque curves.
Torque curves of two opposite direction 3phase induction motors are summed up – the result (the red curve) is the torque curve of a singlephase induction motor We can see that the starting torque of a singlephase induction motor would be zero, but once it gain some speed (in any direction) it will produce torque on their own – very similar to the 3phase motor. In the real word, a singlephase induction motor always have additional, second phase winding set that is used to produce starting torque so the motor can start on its own. The second phase is usually generated by a quite large capacitor – the capacitor offsets the phase to about 90? and delivers it to the second winding set. Once the motor starts, the second winding set has no much effect on the motor torque. Danijel Gorupec, 2006
