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Armature Controlled DC Motor

A motor is an actuator, converting electrical energy in to rotational mechanical energy. Motor requiring DC supply for operation is termed as DC motor. DC motors are widely used in control applications like robotics, tape drives, machines and many more. Separately excited DC motors are suitable for control applications because of separate field and armature circuit. Two ways to control DC separately excited motors are:

1. Armature Control.

2. Field Control.

Modelling of armature control DC motor is discussed in the article.

It represents basic structure of separately excited DC motor.

Basic operating mechanism of a DC motor^[1]

DC motor consists of two parts i.e. rotor and stator. Stator consists of field winding while rotor (also called armature) consists of armature winding. When both armature and field are excited by DC supply, current flows through windings and magnetic flux proportional to the current is produced. When the flux of field interacts with the flux of armature, it results in motion of the rotor. Armature control is the most common control technique for DC motors. In order to implement this control, the flux is required to be kept constant. To this aim, either the stator voltage is constant or the stator coils are replaced by a permanent magnet. In the latter case, the motor is said to be a permanent magnet DC motor and is driven by means of the only armature coils.

As in armature control, field flux is constant; equations governing operation of motor become linear that is

$\displaystyle T=K_{\phi }\Phi \mathrm {I}$ (1)

where $T$ is motor torque, $\Phi$ is flux and $I$ is armature current. When field flux is constant, equation (1) becomes

$\displaystyle T=K'\mathrm {I}$ (2)

where $\displaystyle K'=K_{\phi }\Phi$ as $\Phi$ is constant.

In addition, the motor has an intrinsic negative feedback structure, hence at the steady state, the speed $\omega$ is proportional to the reference input $V_a$ .

These two facts, in addition to the cheaper price of a permanent magnet motor with respect to a standard DC motor (as a matter of fact only the rotor coils need to be winded), are the main reasons why armature controlled motors are widely used. However, several disadvantages arise from this control technique, of which major is the flow of large currents during transients. For example, when started speed ω is zero initially hence back EMF (Electromotive Force) governed by the following relation, would be zero.

$\displaystyle E_{b}=K_{\phi }\Phi \omega$ (3)

And also armature current is given by $\displaystyle I=\left({\frac {V-E_{b}}{R_{a}}}\right)$ (4)

will be very high causing increase in heating of machine and it may damage the insulation.

Transfer Function

Essential Equations [1]:

Electrical Equations:

$\displaystyle E_{b}(s)=K_{\phi }\Phi \omega (s)$ in Laplace domain $\displaystyle E_{b}(s)=K_{\phi }\Phi \omega (s)$ (5)

$\displaystyle I=\left({\frac {V-E_{b}}{R_{a}}}\right)$ in Laplace domain $\displaystyle I(s)=\left({\frac {V(s)-E_{b}(s)}{R_{a}}}\right)$ (6)

$\displaystyle T=K_{\phi }\Phi \mathrm {I}$ (7)

$\displaystyle T-T_{L}=J{d\omega \over dt}+F\omega$ in Laplace domain $\displaystyle T(s)-T_{L}(s)=J{d\omega (s) \over dt}+F\omega (s)$ (8)

Block Diagram of Separately Excited DC Motor is given in Fig.2

It represents block diagram of separately excited DC motor with armature control.

Various parameters in figure are described as

$\displaystyle K_{a}={1 \over R_{a}}$ is the rotor gain.
$\displaystyle \tau _{a}={L \over R_{a}}$ is the electrical time constant.
$\displaystyle \mathrm {T} _{m}$ is the motor torque.
$\displaystyle K$ is a constant depending on field flux.
$\displaystyle K_{m}={1 \over F}$ is mechanical gain.
F is viscous friction coefficient.
$\displaystyle \tau _{m}={J \over F}$ is mechanical time constant, where J is moment of inertia of the load.
$\displaystyle \omega (s)$ is the resulting angular velocity.

The transfer matrix of the system may be written as

$\displaystyle \omega (s)={\begin{bmatrix}W_{1}(s)&W_{2}(s)\end{bmatrix}}{\begin{bmatrix}V_{a}\\T_{L}(s)\end{bmatrix}}$ (9)

where $\displaystyle W_{1}(s)={\frac {K_{a}K_{\phi }K_{m}\phi }{(1+\tau _{a}s)(1+\tau _{m}s)+K_{a}K_{m}(K_{\phi }\phi )^{2}}}$ (10)

$\displaystyle W_{2}(s)={\frac {K_{m}(1+\tau _{a}s)}{(1+\tau _{a}s)(1+\tau _{m}s)+K_{a}K_{m}(K_{\phi }\phi )^{2}}}$ (11)

能不感覺望文生義耶？

此時或不如鉅細靡遺咀嚼大部頭文章也！

DC motors: dynamic model and control techniques

然由『學問』貴在『善用』來講，恐又略嫌不足呦！？

比方說，『轉矩』是馬達『轉子』轉動原因，簡單的例釋︰

Torque in DC Motor

This is an active graphic. Click on bold type for further illustration.

可知它與銜鐵『幾何結構』 $L$ 以及『定子』磁場強度 $B$ 密切相關，即使將之視為『固定』， $T_{orque} = f_{unction} (I(t)) = Constant \cdot I(t)$ 怕尚未考慮『渦電流』勒！？

Eddy current

Eddy currents (also called Foucault currents) are loops of electrical current induced within conductors by a changing magnetic field in the conductor due to Faraday’s law of induction. Eddy currents flow in closed loops within conductors, in planes perpendicular to the magnetic field. They can be induced within nearby stationary conductors by a time-varying magnetic field created by an AC electromagnet or transformer, for example, or by relative motion between a magnet and a nearby conductor. The magnitude of the current in a given loop is proportional to the strength of the magnetic field, the area of the loop, and the rate of change of flux, and inversely proportional to the resistivity of the material.

By Lenz’s law, an eddy current creates a magnetic field that opposes the change in the magnetic field that created it, and thus eddy currents react back on the source of the magnetic field. For example, a nearby conductive surface will exert a drag force on a moving magnet that opposes its motion, due to eddy currents induced in the surface by the moving magnetic field. This effect is employed in eddy current brakes which are used to stop rotating power tools quickly when they are turned off. The current flowing through the resistance of the conductor also dissipates energy as heat in the material. Thus eddy currents are a cause of energy loss in alternating current (AC) inductors, transformers, electric motors and generators, and other AC machinery, requiring special construction such as laminated magnetic cores or ferrite cores to minimize them. Eddy currents are also used to heat objects in induction heating furnaces and equipment, and to detect cracks and flaws in metal parts using eddy-current testing instruments.

Origin of term

The term eddy current comes from analogous currents seen in water in fluid dynamics, causing localised areas of turbulence known as eddies giving rise to persistent vortices. Somewhat analogously, eddy currents can take time to build up and can persist for very short times in conductors due to their inductance.

History

The first person to observe eddy currents was François Arago (1786–1853), the 25th Prime Minister of France, who was also a mathematician, physicist and astronomer. In 1824 he observed what has been called rotatory magnetism, and that most conductive bodies could be magnetized; these discoveries were completed and explained by Michael Faraday (1791–1867).

In 1834, Heinrich Lenz stated Lenz’s law, which says that the direction of induced current flow in an object will be such that its magnetic field will oppose the change of magnetic flux that caused the current flow. Eddy currents produce a secondary field that cancels a part of the external field and causes some of the external flux to avoid the conductor.

French physicist Léon Foucault (1819–1868) is credited with having discovered eddy currents. In September, 1855, he discovered that the force required for the rotation of a copper disc becomes greater when it is made to rotate with its rim between the poles of a magnet, the disc at the same time becoming heated by the eddy current induced in the metal. The first use of eddy current for non-destructive testing occurred in 1879 when David E. Hughes used the principles to conduct metallurgical sorting tests.

Explanation

Eddy currents (I, red) induced in a conductive metal plate (C) as it moves to right under a magnet (N). The magnetic field (B, green) is directed down through the plate. The increasing field at the leading edge of the magnet (left)induces a counterclockwise current, which by Lenz’s lawcreates its own magnetic field (left blue arrow) directed up, which opposes the magnet’s field, producing a retarding force. Similarly, at the trailing edge of the magnet (right), a clockwise current and downward counterfield is created(right blue arrow) also producing a retarding force.

A magnet induces circular electric currents in a metal sheet moving past it. See the diagram at right. It shows a metal sheet (C) moving to the right under a stationary magnet. The magnetic field (B, green arrows) of the magnet’s north pole N passes down through the sheet. Since the metal is moving, the magnetic flux through the sheet is changing. At the part of the sheet under the leading edge of the magnet(left side) the magnetic field through the sheet is increasing as it gets nearer the magnet, $\displaystyle {dB \over dt}\;>\;0$ . From Faraday’s law of induction, this creates a circular electric field in the sheet in a counterclockwise direction around the magnetic field lines. This field induces a counterclockwise flow of electric current (I, red), in the sheet. This is the eddy current. At the trailing edge of the magnet (right side) the magnetic field through the sheet is decreasing, $\displaystyle {dB \over dt}\;<\;0$ , inducing a second eddy current in a clockwise direction in the sheet.

Another way to understand the current is to see that the free charge carriers (electrons) in the metal sheet are moving with the sheet to the right, so the magnetic field exerts a sideways force on them due to theLorentz force. Since the velocity v of the charges is to the right and the magnetic field B is directed down, from the right hand rule the Lorentz force on positive charges F = q(v × B) is toward the rear of the diagram (to the left when facing in the direction of motion v). This causes a current I toward the rear under the magnet, which circles around through parts of the sheet outside the magnetic field, clockwise to the right and counterclockwise to the left, to the front of the magnet again. The mobile charge carriers in the metal, the electrons, actually have a negative charge (q < 0) so their motion is opposite in direction to theconventional current shown.

Due to Ampere’s circuital law each of these circular currents creates a counter magnetic field (blue arrows), which due to Lenz’s law opposes the change in magnetic field which caused it, exerting a drag force on the sheet. At the leading edge of the magnet (left side) by the right hand rule the counterclockwise current creates a magnetic field pointed up, opposing the magnet’s field, causing a repulsive force between the sheet and the leading edge of the magnet. In contrast, at the trailing edge (right side), the clockwise current causes a magnetic field pointed down, in the same direction as the magnet’s field, creating an attractive force between the sheet and the trailing edge of the magnet. Both of these forces oppose the motion of the sheet. The kinetic energy which is consumed overcoming this drag force is dissipated as heat by the currents flowing through the resistance of the metal, so the metal gets warm under the magnet.

顯然『轉子』若不動，實沒有『應生 $v_{emf}$ 』乎？！

Motional EMF and Faraday’s Law

The motional emf expression is an application of Faraday’s Law, as can be seen from:

那麼假設『暫態』已過、『穩定』轉動時，『轉子』之角速度為 $\Omega, \ \frac{d}{dt} \Omega =0$ ，我們能夠得出什麼結論嘛☆★

一八二一年，法國數學家『柯西』曾經考慮了一個現今稱為『柯西函數方程』 Cauchy’s functional equation 的『加性函數』additive functions $f(x + y) = f(x) + f(y)$ 。假使 $x, y$ 都是『有理數』，可以證明 $f(x) = c \cdot x$ ，這個『函數族』是它的『唯一解』。同時『柯西』也證明了︰如果 $f(x)$ 是一個『實數』的『連續函數』，那麼 $f(x) = c \cdot x$ 這個『函數族』也是它的『唯一解』。要是對於『實數函數』 $f(x)$ 不加上任何『限制條件』，一九零五年，德國數學家 Georg Hamel 證明了它可以有『無窮解』。在此我們僅再次的『演示』如何用『無窮小分析』來『求取』這個『加性函數』的『平滑解』︰

$f(x + \delta x) = f(x) + f(\delta x)$

$f(x + \delta x) = f(x) + f^{\prime}(x) \cdot \delta x + \epsilon \cdot \delta x$ ，於是

$f(x) + f(\delta x) = f(x) + f^{\prime}(x) \cdot \delta x + \epsilon \cdot \delta x$ ，所以

$f^{\prime}(x) = \frac{f(\delta x)}{\delta x} - \epsilon$ ，因此

$f^{\prime}(x) = c = constant$

，再從 $f(0) = f(0 + 0) = f(0) + f(0)$ 可得 $f(0) = 0$ ，如是就得到了 $f(x) = c \cdot x$ 這個『函數族』。

如果從『恆等式』identity 的『觀點』來看，『 泛函數方程式』可以看成是『泛函數恆等式』 functional identities，就像 ${[\sin{x}]}^2 + {[\cos{x}]}^2 = 1$ 這個『三角恆等式』一樣，假使我們藉由上式將 $\sin{(x + y)} = \sin{(x)} \cos{(y)} + \cos{(x)} \sin{(y)}$ 恆等式改寫成 $\sin{(x + y)} = \sin{(x)} \sqrt{1 - {[\sin{y}]}^2} + \sqrt{1 - {[\sin{x}]}^2} \sin{(y)}$ ，儼然是一個『 泛函數方程式』的了！

因此我們也可以用『相同』的『觀點』將『微分方程式』看成是一種『泛函數恆等式』，進一步『明白』即使『不求解』那個方程式，我們依然能夠藉之得到有關『解函數』的許多重要有用的『資訊』的啊！！

─── 摘自《【SONIC Π】電聲學之電路學《四》之《一》》

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每日彙整: 2018-08-12

STEM 隨筆︰古典力學︰轉子【五】《電路學》五【電感】 IV‧馬達‧二‧D

Armature Controlled DC Motor

Basic operating mechanism of a DC motor^[1]

Transfer Function

Torque in DC Motor

Eddy current

Origin of term

History

Explanation

Motional EMF and Faraday’s Law

輕。鬆。學。部落客

Armature Controlled DC Motor

Basic operating mechanism of a DC motor[1]

Transfer Function

Torque in DC Motor

Eddy current

Origin of term

History

Explanation

Motional EMF and Faraday’s Law

輕。鬆。學。部落客

Basic operating mechanism of a DC motor^[1]