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3.3 Induced Voltage on a Conductor Moving in a Magnetic Field
1. If a conductor moves or ‘cuts’ through a magnetic field, voltage will be induced between the terminals of the conductor at which the magnitude of the induced voltage is dependent upon the velocity of the wire assuming that the magnetic field is constant. This can be summarised in terms of formulation as shown:
eind = (v x B) l
where:
v – velocity of the wire
B – magnetic field density
l– length of the wire in the magnetic field
2. Note: The value of l (length) is dependent upon the angle at which the wire cuts through the magnetic field. Hence a more complete formula will be as follows:
eind= (v x B)l cosθ
where:
q - angle between the conductor and the direction of (v x B)
3. The induction of voltages in a wire moving in a magnetic field is fundamental to the operation of all types of generators.
4. The Linear DC Machine
Linear DC machine is the simplest form of DC machine which is easy to understand and it operates according to the same principles and exhibits the same behaviour as motors and generators. Consider the following:
Equations needed to understand linear DC machines are as follows:
Production of Force on a current carrying conductor
Voltage induced on a current carrying conductor moving in a magnetic field
eind = (v x B) l
Kirchoff’svoltage law
Newton’s Law for motion
Fnet = ma
Starting the Linear DC Machine
1. To start the machine, the switch is closed.
2. Current will flow in the circuit and the equation can be derived from Kirchoff’s law:
At this moment, the induced voltage is 0 due to no movement of the wire (the bar is at rest).
3. As the current flows down through the bar, a force will be induced on the bar. (Section 1.6 a current flowing through a wire in the presence of a magnetic field induces a force in the wire).
Direction of movement: Right
4. When the bar starts to move, its velocity will increase, and a voltage appears across the bar.
Direction of induced potential: positive upwards
5. Due to the presence of motion and induced potential (eind), the current flowing in the bar will reduce (according to Kirchhoff’s voltage law). The result of this action is that eventually the bar will reach a constant steady-state speed where the net force on the bar is zero. This occurs when eindhas risen all the way up to equal VB. This is given by:
6. The above equation is true assuming that R is very small. The bar will continue to move along at this no-load speed forever unless some external force disturbs it. Summarization of the starting of linear DC machine is sketched in the figure below:
The Linear DC Machine as a Motor
1. Assume the linear machine is initially running at the no-load steady state condition (as before).
2. What happen when an external load is applied? See figure below:
3. A force Fload is applied to the bar opposing the direction of motion. Since the bar was initially at steady state, application of the force Floadwill result in a net force on the bar in the direction opposite the direction of motion.
4. Thus, the bar will slow down (the resulting acceleration a = Fnet/mis negative). As soon as that happen, the induced voltage on the bar drops (eind = v↓ Bl).
5. When the induced voltage drops, the current flow in the bar will rise:
6. Thus, the induced force will rise too. (Find ↑ = i↑ lB)
7. Final result à the induced force will riseuntil it is equal and opposite to the load force, and the bar again travels in steady state condition, but at a lower speed. See graphs below:
8. Now, there is an induced force in the direction of motion and power is being converted from electrical to mechanical form to keep the bar moving.
9. The power converted is Pconv = eindI = Find v à An amount of electric power equal to eind i is consumed and is replaced by the mechanical power Find v à MOTOR
10. The power converted in a real rotating motor is: Pconv = τind ω
The Linear DC Machine as a Generator
1. Assume the linear machine is operating under no-load steady-state condition. A force in the direction of motion is applied.
2. The applied force will cause the bar to accelerate in the direction of motion, and the velocity vwill increase.
3. When the velocity increase, eind= V ↑ Bl will increase and will be larger than VB.
4. When eind > VB the current reverses direction.
5. Since the current now flows up through the bar, it induces a force in the bar (Find= ilB to the left). This induced force opposes the applied force on the bar.
6. End result à the induced force will be equal and opposite to the applied force, and the bar will move at a higher speed than before. The linear machine no is converting mechanical power Find v to electrical power eind i à GENERATOR
7. The amount of power converted : Pconv= τindω
NOTE:
· The same machine acts as both motor and generator. The only difference is whether the externally applied force is in the direction of motion (generator) or opposite to the direction of motion (motor).
· Electrically, eind> VB à generator
· eind < VB à motor
· whether the machine is a motor or a generator, both induced force (motor action) or induced voltage (generator action) is present at all times.
· Both actions are present, and it is only the relative directions of the external forces with respect to the direction of motion that determine whether the overall machine behaves as a motor or as a generator.
· The machine was a generator when it moved rapidly and a motor when it moved more slowly. But, whether it was a motor or a generator, it always moved in the same direction.
· There is a merely a small change in operating speed and a reversal of current flow.
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