CONTROL TRANSFORMER

The function of a control transformer is to obtain the desired low voltage control supply from the power supply system. The stepped down control supply voltage may be 220 V or less. There are two advantages of using a low control voltage.
(i) Reduced risk to the operator, and
(ii) Reduced risk of insulation break down and grounding in the control wiring and pilot devices.

In India the most commonly used control voltage is 220 V, however in situation where the operator has to continuously hold the control console in hand such as in small hoist, a lower voltage of 110 V, 48 V or 24 V is used. Electro-magnetic devices such as contactors, relays, solenoids etc., are manufactured in these standard control voltage ratings. The disadvantages of using low control voltage is that the cross-section of control wire increases and hence the cost also increases.

The electromagnetic coils of contactors, relays and solenoids pick up above 85% of their rated coil voltage. This provides a higher safe value of operation as burn out of coils may occur above 110% of the rated voltage. These devices have inrush currents of nearly 6 to 9 times the continuous holding current. Both the inrush and holding currents have low power factor. The control transformer design should be such as to provide good regulation under poor power factor loads. It should maintain 95% of the rated secondary voltage at all load to avoid
(i) chattering of relays and contactors; (ii) dropping of already energized devices; (iii) failure of solenoids being closed.

Control transformers are generally of two types. First type is the simple one having one primary winding and one secondary winding with no tappings. To use the transformer at various supply voltage the primary winding is to have tappings. For example, a typical control transformer will have tappings at the primary winding at 440, 415, 380, 230, 200 V, while the secondary may have only one winding giving constant voltage. The number of turns up to 440 V  tapping will be double the number than at 220 V tapping, thereby keeping the ratio of voltage per turn the same whether the tappings are connected to 440 V or 220 V. As the volt/turn remains same we get constant voltage on secondary side. The second type of control transformer known as Dual primary type is the most widely used control transformer. It has got two identical primary winding's and one secondary winding's.

In India we have three phase supply system of 415 V line to line and 240 V line of neutral (declared voltage by the supply authorities at Consumers’ end) with ± 6% variation.
The primary of a dual transformer can be connected across line to line (415 V) if the two primary winding's are connected in series (see Fig. 2.43 (b)). In series connection the number of turns on each winding are added together. The transformer primary can be connected across line and neutral (240 V), if the two primary winding's are connected in parallel. The effect of connecting two winding's of equal number of turns in parallel is the same as connecting only one winding as in Fig. 2.43 (c) because the effective number of turns for determining the turns ratio remains the same.

Determination of Rating of Control Transformer

The primary voltage of the transformer depends upon the supply voltage available while the secondary voltage depends upon the control circuit application. The voltage ampere rating of the transformer depends upon the continuous holding volt ampere (VA) rating and inrush current rating of various relays, contactors, solenoids and indication lamps to be connected to the transformer secondary. The volt ampere rating of the components are available from the manufacturers literature. The method of finding out the approximate VA rating of the control transformer is to:
(i) Calculate the maximum continuous holding current by adding the holding current of all the coils that will get energized at the same time and then multiplying this amount by a factor of 5/4.
(ii) Calculate the total maximum inrush current by adding the inrush current of all the coils that will be energized together at any one time and then multiplying this figure by 1/4.

Now taking the larger of the two figures calculated above, multiply this current by the control voltage. This product is the volt ampere (VA) required by the transformer.
When it is difficult to find out how many contactors are switched on simultaneously, take 80% of the holding capacity of all contactors and relays. Another simple method for finding out approximate VA rating of the transformer is to make calculations using the following formula:

PTR   = ΣPh  + P1C  + ΣPL
 

where, PTR
= The nominal power (VA) of the control transformer.

 
ΣPh
= The holding apparent power (VA) of all the contactors energized simultaneously.

 
P1C
= Inrush apparent power (VA) of the contactor of largest size.
 

ΣPL
= Active power of all signal lamps.

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LIMIT SWITCHES

Limit switch is an important control element. Limit switch contacts change over their position when its actuating lever or knob is actuated by the mechanical part of a machine. The mechanical  part attached to the machine which actuates the limit switch lever or knob is known as actuator or dog. Limit switches are used to stop a mechanical movement of a machine and may also be used to stop a particular movement, and initiate another movement. The simple application of a limit switch is in producing automatic to and fro movement of a planar machine bed shown in Fig. 2.32. It must be understood here that a limit switch is not used as a mechanical stop. A limit switch controls the electrical signal which is responsible for mechanical stop/movement.

Actual control circuit for achieving this to and fro movement with help of limit switches shown in Fig. 2.32, will be discussed at a later stage.

         

•   Simple Limit Switch

The simplest construction of a limit switch is shown in Fig. 2.33.
When the knob or pin is pushed the plunger attached to the knob move down against
spring pressure. The moving contacts mounted on the plunger also moves down. Thus the terminals 1 and 2, which are normally closed, become open and terminals 3 and 4, which are normally open, get closed. When the pressure on the knob is released the contacts return to their normal position. Another commonly used limit switch is the lever type. This may be single side actuation type or double side actuation type, as shown in Figs. 2.33 (b) and (c).
In a single side actuation type limit switch the contacts operate when the limit switch arm moves in one direction, either the right or to the left. Movement on the other side does not actuate the contacts. The arm returns to the normal position when the actuating force is removed. The limit switch can also be of maintained contact type in which arm has to be brought

 back manually to normal. However this type of limit switches are rarely used. In double side actuation type limit switch, when the arm is moved in one direction, say towards right, contact block A operates; and when the arm is moved towards left, contact block Boperates. This type of a limit switch can serve the purpose of two single side actuation limit switches needed in controlling to and fro movement of a machine. If the contacts of the limit switch change-over independent of the speed of operation, the limit switch is known as snap-action limit switch. At a particular position of the arm, the contacts change-over. If the contacts also move at the same speed as that of the arm of the limit switch, it is called a slow-action limit switch. Most limit switches used are the snapaction type. There are only few applications where slow acting limit switches are preferred.

• Rotary Cam Type Limit Switches

In this type of limit switches, the contacts are mounted on the stationary frame. The cams, which have to actuate the contacts, are mounted on the rotating shaft. The position of the cams is adjustable. The rotating shaft is coupled to the driving motor of the machinery either through chain and sprockets or by gear arrangement. The rotation of the driving motor is thus transmitted to the shaft of the limit switch. The cams mounted on the shaft are shaped and their position adjusted to actuate the contacts at the desired instant. In the machinery, the rotary motion of the driving motor is converted into longitudinal motion through mechanical arrangement. Thus, the cam shaft of the limit switch rotates proportional to the longitudinal motion of the machine. The limits of travel of the machinery, therefore, can be adjusted by changing the cam position. The advantage of a rotary cam limit switch over an ordinary limit switch is that the instant of actuation and hence limits of travel can be varied easily over a wide range by changing the cam positions. In an ordinary limit switch, however, once the limit switch is fixed, only very little adjustment is possible by adjusting the lever position. The reliability of rotary limit switch using sprocket and chain is however low and, therefore, wherever these are used a back-up-protection of ordinary limit switches is generally provided to avoid damage due to over travel, in case the rotary limit switch fails due to slipping or breaking of chain. Rotary limit switches are preferred where frequent adjustment of limits of travel is required. One of its important application is in over-head cranes for hoisting and lowering motion. In over-head crane, for hoisting operation, the rotary motion of the motor is converted into longitudinal motion with the help of a rope wound on the drum attached to the motor shaft. The rope winds on the drum in one direction giving hoisting (raising) operation and unwinds in other direction giving lowering motion. The limits of hoisting and lowering can be easily adjusted by varying the cam positions which have to actuate the hoisting and lowering limit switch contacts.

• Heavy Duty Limit Switches

As the name signifies, these limit switches are used in heavy duty machines such as cranes, conveyors and heavy material handling equipment. The basic difference between an ordinary limit switch and a heavy duty limit switch is that the former is used in the control circuit while the latter is used in the power circuit. Actuation of a heavy duty limit switch cuts off power supply to the motor. Heavy duty limit switch requires a minimum of two normally closed contacts because to stop a motor two phases have to be cut off. A heavy duty limit switch may be either lever type or rotary cam type. They are rugged, and the current rating of their contacts matches with the motor rating. If the rating is quite high, arc chutes are also provided over the contacts. These heavy duty limit switches are used in addition to the control limit switches. They are generally used as back-up-protection for control limit switches. If the control limit switch fails or contactor gets stuck or welded, then the heavy duty limit switch operates to disconnect power supply to the motor. The limit switch contacts are connected in series with the main contactor contacts. These limit switches therefore avoid damage and accidents due to over-travel of machinary like overhead crane and other heavy material handling equipment.

•  Speed Actuating Sensing Switches

These are also commonly known as plugging switches or zero speed switches. These switches have two sets of contacts, one each for either direction or rotation of the motor. The contacts for a particular direction of rotation open or close when a predetermined rotational speed in that direction is achieved. These switches are used in circuits where the motor is to be stopped quickly (i.e.,to be brought to standstill quickly by plugging method of braking). In plugging, the motor is stopped by reversing its two supply leads till motor comes to standstill. The torque developed during reversing causes the motor to come to standstill in a short time. The most commonly used design of this device is the one which use the effect of induced magnetic forces. In this type, a shaft having a permanent magnet on it rotates inside a copper cup. The copper cup can rotate in either direction against spring pressures. The shaft of the switch is coupled with the motor shaft. As the motor shaft rotates, the permanent magnet on the switch shaft rotates inside the cup. Magnetic induction due to its rotation causes the cup to follow the rotation of the shaft. An actuator is connected on the copper cup which can actuate the contacts in either direction. The speed at which the contacts would operate can be adjusted by varying the spring tension which restrain the movement of the cup. The contacts get reset to their original position when the motor speed falls below the value set by the spring tension. If the contact of the speed switch is so adjusted that contacts operate immediately on rotation in either direction and re-close only when zero speed is reached the switch is known as zero speed switch. The plugging switch or zero speed switch is represented in a control circuit as shown in Fig. 2.35.

 

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SELECTOR SWITCHES

Many machines and process operations are designed so as to function in any one of the several ways. For example, the three different modes of operation may be Manual, Semi-automatic and Automatic. A selector switch will enable the operator to predetermine the manner in which his machine is to operate. As with push-button switches, selector switches also have two main parts, the mechanical actuator and the contact block. Selector switches are usually of the maintained position type, although momentary spring return selectors are also available. Selector switches can have single break contacts or double break contacts. The symbols for both these type of contacts have been shown in Figs. 2.28 and 2.29.

The selector switches are generally made for four positions. As the number of positions increase to four, manufacturer provide charts of tables which display positions of the selector switch actuator. Fig. 2.30 shows a four position selector switch. Note the use of × under the position number. This indicates that the contact in line with × is closed in that position. Below position 1 (Fig. 2.30) × is against contact B and C i.e.,in position 1 contact Band Care closed.

In position 2 contacts B and D will close, further in position 3 contact A and D will close and in position 4 contacts A and C will close.

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Power System Voltage Control

 The control of voltage levels is accomplished by controlling the production, absorption, and flow of reactive power at all levels in the system. The generating units provide the basic means of voltage control; the automatic voltage regulators control field excitation to maintain a scheduled voltage level at the terminals of the generators. Additional means are usually required to control voltage throughout the ystem 

The devices used for this purpose may be classified as follows:

1. Sources or sinks of reactive power, such as shunt capacitors, shunt reactors,synchronous condensers, and static var compensators (SVCs).                                                                                                                                                                          
2. Line reactance  compensators, such as series capacitors.
3. Regulating transformers, such as tap-changing transformers and boosters.

Shunt capacitors and reactors, and series capacitors provide passive compensation. They are either permanently connected to the transmission and distribution system, or switched. They contribute to voltage control by modifying the network characteristics.

Synchronous condensers and SVCs provide active compensation; the reactive power absorbed/supplied by them is automatically adjusted so as to maintain voltages  at other locations in the system are determined by active and reactive power flows through various circuit elements, including the passive compensating devices.

The following is a description of the basic characteristics and forms of application of devices commonly used for voltage and reactive power control.

• Shunt reactors

Shunt reactors are used to compensate for the effects of line capacitance, particularly to limit voltage rise on open circuit or light load.

They are usually required for EHV overhead lines longer than 200 km. A shorter overhead line may also require shunt reactors if the line is supplied from a weak system (low short-circuit capacity). When the far end of the line is opened, the capacitive line-charging current flowing through the large source inductive reactance (Xs) will cause a rise in voltage Es at the sending end of the line. 

A shunt reactor of sufficient size must be permanently connected to the line to limit fundamental-frequency temporary over voltages to about 1.5 pu for a duration of less than 1 second. Such line-connected reactors also serve to limit energization over voltages (switching transients). Additional shunt reactors required to maintain normal voltage under light-load conditions may be connected to the EHV bus .

During heavy loading conditions some of the reactors may have to be disconnected. This is achieved by switching reactors using circuit-breakers.

For shorter lines supplied from strong systems, there may not be a need for reactors connected to the line permanently. In such cases, all the reactors used may be switchable, connected either to the tertiary windings of transformers or to the EHV bus. in some applications, tapped reactors with on-voltage tap-change control facilities have been used, to allow variation of the reactance value. 

Shunt reactors are similar in construction to transformers, but have a single winding (per phase) on an iron core with air-gaps and immersed in oil. They may be of either single-phase or three-phase construction.

• Shunt Capacitors 

Shunt capacitors supply reactive power and boost local voltages. They are used throughout the system and are applied in a wide range of sizes.

Shunt capacitors were first used in the mid-1910s for power factor correction. The early capacitors employed oil as the dielectric. Because of their large size and weight, and high cost, their use at the time was limited. In the 1930s, the introduction of cheaper dielectric materials and other improvements in capacitor construction brought about significant reductions in price and size. The use of shunt capacitors has increased phenomenally since the late 1930s. Today, they are a very economical means of supplying reactive power. The principal advantages of shunt capacitors are their low cost and their flexibility of installation and operation. They are readily applied at various points in the system, thereby contributing to efficiency of power transmission and distribution. 

The principal disadvantage of shunt capacitors is that their reactive power output proportional to the square of the voltage. Consequently, the reactive power output reduced at low voltages when it is likely to be needed most.

• Series Capacitors 

Series capacitors are connected in series with the line conductors to compensate for the inductive reactance of the line. This reduces the transfer reactance between the buses to which the line is connected, increases maximum power that can be transmitted, and reduces the effective reactive power loss. Although series capacitors are not usually installed for voltage control as such, they do contribute to improved voltage control and reactive power balance. The reactive power produced by a series capacitor increases with increasing power transfer; a series capacitor is self-regulating in this regard.

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Bouchloz Relay

• INTRODUCTION

Power Transformers are considered to  be  a  highly  reliable  type  of equipment, yet, in order to ensure the continuity  of  service  that  modern conditions  demand,  protective devices are required. The purpose of such devices is to disconnect faulty apparatus before large-scale damage is caused by a fault to the apparatus or  to  other  connected  apparatus. Such devices generally respond to a change  in  the  current  or  pressure arising from the faults and are used for  either  signaling  or  tripping  the circuits. 

Protective devices in the ideal case must be sensitive to all faults, simple in operation, robust for service and economically  feasible.  Considering liquid  immersed  transformers,  a near-ideal  'protective  device'  is available in the form of Gas and Oil relay  described  here.  The  relay operates on the well-known fact that almost every type of electric fault in a 'liquid  immersed  transformer'  gives rise to gas. This gas is collected in the body of the relay and is used in some way or other to cause the alarm or the tripping circuit to operate. 

The principle of the Gas and Oil relay was first successfully demonstrated and utilized by 'Buchholz' many years back.  In  a  series  of  experiments carried out extensively in Germany it was proved that the Relay is capable of  bringing  to  light  incipient  fault thereby preventing further spreading of the fault and extensive damage and thus saving expensive and protracted repairs. So successful is the principle of  this  Relay  that  despite  the continued search for better protective devices in other electrical fields the Gas-and-Oil Relay is still on its own in providing protection against a variety of faults.

• TYPES OF BOUCHLOZ RELAY

1.  Buchholz Relay with Mercury Switches
2.  Buchholz Relay with Magnetic Switches

• Bouchloz Relay with Mercury Switches

1. Uses mercury in switches which is toxic and also a carcinogen
   substance. Mercury is now prohibited in most parts of the world.
2. Relays with mercury switches are not accepted internationally by
   utilities and OEMs in most countries of the world.
3. Huge variation in gas volume and surge velocity readings from
   one relay to another.
4. Mercury  susceptible  to  oxidation's  resulting  in  no/false  signal
   upon prolonged use.
5. Switch activated by flow of mercury.
6. UN-branded locally made mercury switches prone to rejections in
   incoming, process and final inspection as well as transit.
7. May maloperate in one or more of the following conditions :a. External shocks to a transformer resulting in vibration.b.  Turbulence of oil due to starting of pump in forced cooled
   transformer.c.  Variation in angle of mounting of the Relay.d.  Earthquake of minor intensity.

• Buchholz Relay with Magnetic Switches

1. No use of mercury.
2. Consistent readings of gas volume and surge velocity.
3.  No affect of ageing
4. Switch activated by a magnet.
5. Branded  switches  imported  from  USA/Japan  are  free  from
   rejection in all stages.
6. Worldwide acceptance:
   a. Immune to such vibrations.
   b.  Highly stable and resistant and will not operate due to oil
   pump operation.
   c.  Immune  to  variations  of  angle  as  experienced  in
   transformer mounting.
   d.  Vibration proof to 6g accelerations.

• OPERATION

The  function  of  a  double  element relay will be described here. During normal operation of a transformer the Buchholz  relay  is  completely  filled with oil. Buoyancy and t he moment due t o counterweights keep the floats in their original top positions. In the event of some fault in the interior of the transformer tank, gas bubbles are produced  which  accumulate  in  the Buchholz  relay  on  the  way  to  the conservator. In consequence, the oil level  in  the  relay  enclosure  drops which  in  turn  lowers  the  upper bucket. 

This causes the magnetic switch to operate an alarm signal. The lower bucket does not change its position,  because  when  the  gas reaches the upper inside wall of the pipe  it  can  escape  into  the conservator. Hence, minor fault in the transformer tank will not trigger the lower  switching  assembly  and  will not trip the transformer. 

In case the liquid continues to drop due to loss of oil, the lower bucket also goes down. In consequence, the lower  switching  system  operates  if the level of oil goes below the bottom level  of  the  pipe  connected  to  the relay. 

Alternately in the event the liquid flow exceeds  a  specific  value  (which  is continuously adjustable, by means of a  flap)  the  lower  bucket  is  forced down,  thus  triggering  the  lower switching system to operate.

As the liquid flow rate decreases, or the level of the liquid rises, the bucket returns  to  its  original  position.  The single  element  relay  has  only  Trip element  and  it  responds  to  only  oil surges.  The  method  of  operation  is similar  to  that  described  for  double element relay. Single element relays are suitable for potential transformers and on load lap changers. 

The single element oil Surge relay has been  specifically  designed  for  use with on load tap change equipment and it will by-pass normal amounts of gas  which  are  generated  by  tap change  operations  and  will  only respond to oil surges and loss of oil. 

• APPLICATIONS

Double element relays can be used in detecting minor or major faults in a transformer.  The  alarm  element  will operate, after a specified volume of gas  has  collected  to  give  an  alarm indication.  Examples  of  incipient faults are :

1.  Broken-down core bolt insulation
2. Shorted lamination
3. Bad contacts
4. Overheating of part of winding's

The alarm element will also operate in the event of oil leakage, or if air gets into the oil system. 

The trip element will be operated by an  oil  surge  in  the  event  of  more serious faults such as :

1. Earth faults
2. Winding short circuits
3.  Puncture of bushings
4. Short circuit between phases

The trip element will also be operated if  a  rapid  loss  of  oil  occurs.  Single element relays can be used to detect either incipient or major faults in oil filled  potential  transformers, reactors,  capacitors  etc.  A  special single  element  relay  is  available  for the protection of on load tap-change equipment. 

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Thyristor

Thyristors, or silicon-controlled rectifiers (SCRs) have been the traditional workhorses for bulk power conversion and control in industry. The modern era of solid-state power electronics started due to the introduction of this device in the late 1950s.

Volt-Ampere Characteristics

The figure shows the thyristor symbol and its volt-ampere characteristics. Basically, it is a three-junction P-N-P-N device, where P-N-P-N and N-P-N component transistors are connected in regenerative feedback mode. The device blocks voltage in both the forward and reverse directions. When the anode is positive, the device can be triggered into conduction by a short positive gate current pulse; but once the device is conducting , the gate losses its control to turn off the device. A thyristor can also turn on by excessive anode voltage. Its rate of rise (dv/dt), by a rise in junction temperature ( TJ ), or by light shining on the junctions. 

The volt-ampere characteristics of the device indicate that at gate current IG = 0 , if forward voltage is applied on the device, there will be a leakage current due to blocking of the middle junction. If the voltage exceeds a critical limit (break over voltage), the device switches into conduction. With increasing magnitude of IG, the forward break over voltage is reduced. And eventually at IG3 , the device behaves like a diode with the entire forward blocking region removed. The device will turn on successfully if a minimum current, called a latching current, is maintained. During conduction, if the gate current is zero and the anode current falls below a critical limit, called the holding current, the device reverts to the forward blocking state. With reverse voltage, the end P-N junctions of the device become reverse-biased and the V-I curve becomes essentially similar to that of a diode rectifier. Modern thyristors are available with very large voltage (Several KV) and current (Several KA) ratings.

Switching Characteristics

Initially, when forward voltage is applied across a device, the off-state, or static (dv/dt), must  be limited so that it does not switch on spuriously. The (dv/dt) creates displacement current in the depletion layer capacitance of the middle junction, which induces emitter current in the component transistors and causes Switching action. When the device turns on, the anode current (di/dt) can be excessive, which can destroy the device by heavy current concentration. During conduction, the inner P-N regions remain heavily saturated with minority carries and the phenomena are similar to that of a diode. However, when the recovery current goes to zero, the middle junction still remains forward-biased. This junction eventually blocks with an additional delay when the minority carries die by the recommendation process. The forward voltage can then be applied successfully, but the reapplied (dv/dt) will be somewhat less than the static (dv/dt) because of the presence of minority carriers. For example, POWEREX SCR/diode module CM4208A2 (800 V , 25 A) has limiting (di/dt)=100 A/m

and off-state dv/dt =500 V/ parameters. A suitably-designed snubber circuit (discussed later) can limit di/dt and dv/dt within acceptable limits. In a converter circuit, a thyristor can be turned off (or commutated) by a segment of reverse AC line or load voltage (defined as line or load commutation, respectively), or by an inductance capacitance circuit-induced transient reverse voltage (defined as forced commutation).

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How to choose , use and calculations of the Flywheels !

 

 

 Load Equalization :

If the load fluctuates between wide limits in space of few seconds, then large peak demands of current will be taken from supply and produce

heavy voltage drops in the system. Large size of conductor is also required for this Process of smoothing out these fluctuating
loads is commonly referred to as load equalization and involves storage of energy during light load periods which can be given out during the peak load period, so that demand from supply is approximately constant. Tariff is also affected as it is based on M.D. (Maximum Demand) For example, in steel rolling mill, when the billet is in between the rolls it is a peak load period and when it comes out it is a light load period, when the motor has to supply only the friction and internal losses, as shown in figure

 

 Use Of Flywheels :

 

The method of Load Equalization most commonly employed is by means of a flywheel. During
peak load period, the flywheel decelerates and gives up its stored kinetic energy, thus reducing the
load demanded from the supply. During light load periods, energy is taken from supply to accelerate
flywheel, and replenish its stored energy ready for the next peak. Flywheel is mounted on the motor
shaft near the motor. The motor must have drooping speed characteristics, that is, there should be a
drop in speed as the load comes to enable flywheel to give up its stored energy. When the Ward -Leonard system is used with a flywheel, then it is called as Ward - Leonard Aligner control.

There are two choices left for selecting a flywheel to give up its maximum stored energy:

 

1.  Large drop in speed and small flywheel (But with this the quality of production will suffer, since a speed drop of 10 to 15% for maximum load is usually employed).
2.  Small drop in speed and large flywheel. (This is expensive and creates additional friction losses. Also design of shaft and bearing of motor is to be modified.) So compromise is made between the two and a proper flywheel is chosen.

 

Flywheel Calculations :

The behavior of flywheel may be determined as follows 

 Fly wheel Decelerating :- (or Load increasing)

Let :

TL : Load torque assumed constant during the time for which load is applied in (Kg.m)

Tf  : Torque supplied by freewheel in (Kg.m)

To : Torque required on no load to overcome friction internal losses in (Kg.m)

Tm : Torque supplied by the motor at any instant in (Kg.m)

ωo  : No load speed of the motor in (rad/sec)

ω : Speed of motor at any instant in (rad/sec)

s : Motor slip speed ( ωo-ω) in (rad/sec)

I : Moment of inertia of flywheel in (Kg.m2)

g : Acceleration due to gravity in (m/sec2)

t : Time in (sec)

When the flywheel decelerates, it gives up its stored energy. 

Tm = TL - Tf  ......... Equation (1)

Energy stored by flywheel when running at speed ‘ω’ is 1/2 Iω2/g.

If speed is reduced from ω0 to ω. 

The energy given up by flywheel is

= (I/2g)*[(ωo)² - (ω)²]   ......... Equation (2)

(ωo +ω)/2 : Mean speed . Asumming speed drop of not than 10% , this may be assumed equal to ω .
(ωo +ω)/2 = ω 
Also   (ωo - ω)= s 
From equation (2),energy given up = (I/g).ω.s
Power given up = (I/g).ω.(ds/dt)
But the torque = ( Power/ω )
Torque supplied by freewheel
 Tf = (I/g)*(ds/dt)
From equation (1), Tm = TL - (I/g)*(ds/dt)
For values of slip speed up to 10% of No-load speed,slip is proportional to torque or , 
S = K * Tm
Tm = TL - (I/g) * K *(dTm/dt)

This equation is similar to the equation for heating of the motor 

(TL - Tm) = (I/g) * K * (dTm/dt) 

g*(dt/IK) = dTm / (TL - Tm )

By integrating both sides 

ln (TL - Tm) = [(t*g)/(I*K)] + C1     ............ Equation (3)

At t = 0 , When load starts increasing from no load

 i.e. Tm = To

Hence, at t = 0       Tm = To 

C1 = - ln ( TL - To )

By substituting the value of C1 above, in equation (3) in ( TL - Tm ) = ( t*g)/(I*K) - ln ( TL - To )

ln [(TL - Tm)/(TL - To)] = - (t*g)/(I*K)

Tm = TL - (TL - To)* e^[( - t*g)/(I*K)]

If the load torque falls to zero between each rolling period, then Tm = TL - [1 - e^(- t*g)/(I*K)] , so  ( To = 0 )

Load Removed (Flywheel Accelerating)

Slip speed is decreasing and therefore ( ds/dt) is negative .

Tm = To + Tf  = To - [(I/g)*(ds/dt)]

To - Tm =(I/g)*K*(dTm/dt) 

(g*dt)/(I*K) = dTm / (To - Tm) 

After integrating both sides, 

ln (To - Tm) = [(t*g)/(I*K)] + C

At t = 0 , Tm = Tm'  motor torque at the instant, when load is removed .

C = - ln ( To - Tm' ) putting this value of C in the above equation 

ln (To - Tm) = [(t*g)/(I*K)] - ln(To - Tm')

ln [(To - Tm)/(To - Tm') = (- t*g)/(I*K)

To - Tm = (To - Tm')* e^[(-t*g)/(I*K)

Tm = To + (Tm' - To) * e^[(-t*g)/(I*K)]

Where Tm' = the motor torqur, at the instant the load is removed .

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