## Basic Machines - Stationary Fields

### Simple System Description

The figure shows one rectangular loop of wire (a single turn) which is free to rotate about the z-axis (out of the plane), perpendicular to a constant uniform magnetic field of magnitude $$B$$ that is aligned with the x-axis. The conductors have length $$l$$ and trace out a cylinder of radius $$r$$. The coil is rotating at constant mecahnical angular speed $$\omega_m$$, therefore the conductors have a linear speed $$|v|=r\omega_m$$. Some important points to note at the position shown are:

1. The conductor 1-2 is moving upwards
2. The conductor 3-4 is moving downwards
3. Angle $$\theta$$ is defined between the x-axis and a line normal to the plane of the coil

### Calculating the Voltage Induced in a Coil

There are two approaches that can be used to calculate the voltage induceed in a coil: Generator Action or Transformer Action

#### Generator Action

$e=( \vec{v} \times \vec{B} ) \cdot \ell$

We can consider the application of the generator equation to each branch of the loop (1-2, 2-3, 3-4, 4-1).

Wire 1-2. Using the right hand rule for cross product, the wire is moving upwards with the component of velocity that is perpenicular to flux density given by $$vsin\theta$$; the flux density vector is left to right, therefore the cross product is $$B v \sin\theta$$ into the page. Due to the arrangement of the coil, the conductor is parallel with the result of the cross product, therefore a voltage will be induced such that the voltage at 2 is positive with respect to 1. Therefore:

$e_{21}=vBl\sin\theta$

Wires 2-3 and 4-1 . In these segments, at any point in the conductor, the cross product $$\vec{v} \times \vec{B}$$ is always perpendicular to the conducto. Therefore there is no induced voltage in either of these segments.

Wire 3-4. Using the right hand rule for cross product, the wire is moving downwards with velocity perpendicular to flux density given by $$v \sin\theta$$; the flux density vector is left to right, so the cross product is out of the plane. Due to the arrangement of the coil, the conductor is parallel with the result of the cross product, so there will be a voltage induced such that the voltage at 4 is positive with respect to 3. Therefore:

\begin{aligned} e_{21} & = vBl\sin\theta \\ e_{41} & = 2vBl \sin\theta \\ e_{loop} & = 2rl\omega_m B \sin (\omega_m t) \end{aligned}

The equation above gives the voltage induced in a rectangular loop of wire as it rotates about an axis perpendicular to a uniform magnetic field. Note that in this simple machine the induced voltage is at an electrcal frequency $$\omega_e$$ equal to the mechanical rotational speed $$\omega_m$$. The magnitude of the induced voltage is proportional to speed, the flux density and the dimensions of the loop. We can also define this voltage in terms of the flux passing through the loop. Note that the area enclosed by the circumference of the rectangular coil is given by $$A=2rl$$. Using the relationship between flux, flux density and area:

\begin{aligned} e_{loop} & = 2rlB \omega_m \sin (\omega_m t) \\ e_{loop} & = AB\omega_m \sin (\omega_m t) \\ e_{loop} & = \hat{\phi}\omega_m \sin (\omega_m t) \end{aligned}

Here, $$\hat{\phi}$$ denotes the maximum possible flux linking, or passing through the turn. The maximum flux passes through the when it is perpendicular to the flux density $$\theta = 0$$. If we had more than one loop or turn then the voltage induced in a coil of $$N$$ turns may be written as

$e_{coil} = \hat{\lambda}\omega_m \sin (\omega_m t)$

If the flux in linking each turn is the same (the turns are closely packed and effectively are all in the same position) then

$\hat{\lambda} = N\hat{\phi}$

#### Transformer Action

We can also calculat the voltage induced in a single turn using the theory for transformer action. Consider the flux passing through of linking the coil, when $$\theta=0$$ then $$\lambda= \hat{\lambda}$$ and the variation of flux linkage with time can be written as

$\lambda (t) = \hat{\lambda}\cos(\omega_m t)$

\begin{aligned} e_{loop} & = -\frac{d\lambda}{dt} \\ e_{loop} & = \hat{\lambda}\omega_m \sin (\omega_m t) \end{aligned}

#### Summary

Transformer action and generator action give the same result. Induced voltage is a function of Flux Density, coil size, rotational speed and turns.

### Torque Production

We now consider the the loop with current flowing. If a load, e.g. a resistor, is connected to the terminals when the voltage is positive 4-1 then current will flow out of 4, through the load and return to 1. In the current loop, positive current will flow from 1 to 4. This may seem counter intuitive at first, but can be thought of as similar to the current in a battery which flows into the negative terminal and out of the positive terminal.

The torque can be calculated by applying the motor action equation to each segment of the loop in turn.

$\vec{F} = i ( \vec {\ell} \times \vec{B} )$

Wire 1-2 current flows into the plane. Using the right hand rule, the force will be down and constant independent of position.

$F_{21}=iBl \quad \downarrow$

Wires 2-3 and 4-1. These conductors are in a plane perpendicular to the flux density and are perpendicular to the axis of rotation, giving a cross product along the axis of rotation. Since the direction of 2-3 is opposite 4-1, the resulting forces will cancel. No net axial force is exerted.

Wire 4-1Current flows out of the plane. Again, using the right hand rule, the force will be upwards and constant independent of position.

$F_{43}=iBl \quad \uparrow$

Torque is defined as force multiplied by the distance perpendicular to the force from the pivot. In this case, the forces are constant in the (positive or negative) y direction. Distance from the pivot is a function of angle so we find:

\begin{aligned} \tau_{21} & = Bilr \sin \theta \\ \tau_{43} & = Bilr \sin \theta \\ \tau_{loop} & = 2rlBi \sin \theta \\ \tau_{loop} & = \hat{\phi} i \sin \theta \end{aligned}

Again, we can extend this for a coil with made up of $$N$$ turns.

$\tau_{coil} = \hat{\lambda} i \sin \theta$

This torque will act in a clockwise direction. If current is constant, then the torque will be a sinusoidal function of area, flux density, current and the number of turns. Note that if the current is produced using a resistor connected to the generator terminals then the current will be sinusoidal and torque will be unidirectional, acting against the direction of motion.

### Summary

This page covers the basic fundamentals of torque production and induced voltages. Both are fundamental to the theory of electric machines. In addition, the idea of flux linkage is recapped. Almost all modern drive systems have control schemes that are based on the fact that that torque can be written as the cross product of current and flux linkage, as seen in the very simple system on this page.