Phasor Diagram

The Phasor Diagram

In the last tutorial, we saw that sinusoidal waveforms of the same frequency can have a Phase Difference  between themselves which represents the angular difference of the two sinusoidal waveforms. Also the terms "lead" and "lag" as well as "in-phase" and "out-of-phase" were used to indicate the relationship of one waveform to the other with the generalized sinusoidal expression given as: A_(t) = A_m sin(ωt ± Φ) representing the sinusoid in the time-domain form. But when presented mathematically in this way it is sometimes difficult to visualise this angular or phase difference between two or more sinusoidal waveforms so sinusoids can also be represented graphically in the spacial or phasor-domain form by aPhasor Diagram, and this is achieved by using the rotating vector method.

Basically a rotating vector, simply called a "Phasor" is a scaled line whose length represents an AC quantity that has both magnitude ("peak amplitude") and direction ("phase") which is "frozen" at some point in time. A phasor is a vector that has an arrow head at one end which signifies partly the maximum value of the vector quantity ( V or I ) and partly the end of the vector that rotates.

Generally, vectors are assumed to pivot at one end around a fixed zero point known as the "point of origin" while the arrowed end representing the quantity, freely rotates in an anti-clockwisedirection at an angular velocity, ( ω ) of one full revolution for every cycle. This anti-clockwise rotation of the vector is considered to be a positive rotation. Likewise, a clockwise rotation is considered to be a negative rotation.

Although the both the terms vectors and phasors are used to describe a rotating line that itself has both magnitude and direction, the main difference between the two is that a vectors magnitude is the "peak value" of the sinusoid while a phasors magnitude is the "rms value" of the sinusoid. In both cases the phase angle and direction remains the same.

The phase of an alternating quantity at any instant in time can be represented by a phasor diagram, so phasor diagrams can be thought of as "functions of time". A complete sine wave can be constructed by a single vector rotating at an angular velocity ofω = 2πƒ, where ƒ is the frequency of the waveform. Then a Phasor is a quantity that has both "Magnitude" and "Direction". Generally, when constructing a phasor diagram, angular velocity of a sine wave is always assumed to be: ω in rad/s. Consider the phasor diagram below.

Phasor Diagram of a Sinusoidal Waveform

As the single vector rotates in an anti-clockwise direction, its tip at point A will rotate one complete revolution of 360^o or 2πrepresenting one complete cycle. If the length of its moving tip is transferred at different angular intervals in time to a graph as shown above, a sinusoidal waveform would be drawn starting at the left with zero time. Each position along the horizontal axis indicates the time that has elapsed since zero time, t = 0. When the vector is horizontal the tip of the vector represents the angles at 0^o, 180^o and at 360^o.

Likewise, when the tip of the vector is vertical it represents the positive peak value, ( +Am ) at 90^o or π/2 and the negative peak value, ( -Am ) at 270^o or 3π/2. Then the time axis of the waveform represents the angle either in degrees or radians through which the phasor has moved. So we can say that a phasor represent a scaled voltage or current value of a rotating vector which is "frozen" at some point in time, ( t ) and in our example above, this is at an angle of 30^o.

Sometimes when we are analysing alternating waveforms we may need to know the position of the phasor, representing the alternating quantity at some particular instant in time especially when we want to compare two different waveforms on the same axis. For example, voltage and current. We have assumed in the waveform above that the waveform starts at time t = 0with a corresponding phase angle in either degrees or radians. But if if a second waveform starts to the left or to the right of this zero point or we want to represent in phasor notation the relationship between the two waveforms then we will need to take into account this phase difference, Φ of the waveform. Consider the diagram below from the previous Phase Difference tutorial.

Phase Difference of a Sinusoidal Waveform

The generalised mathematical expression to define these two sinusoidal quantities will be written as:

The current, i is lagging the voltage, v by angle Φ and in our example above this is 30^o. So the difference between the two phasors representing the two sinusoidal quantities is angle Φ and the resulting phasor diagram will be.

Phasor Diagram of a Sinusoidal Waveform

The phasor diagram is drawn corresponding to time zero ( t = 0 ) on the horizontal axis. The lengths of the phasors are proportional to the values of the voltage, ( V ) and the current, ( I ) at the instant in time that the phasor diagram is drawn. The current phasor lags the voltage phasor by the angle, Φ, as the two phasors rotate in an anticlockwise direction as stated earlier, therefore the angle, Φ is also measured in the same anticlockwise direction.

If however, the waveforms are frozen at time t = 30^o, the corresponding phasor diagram would look like the one shown on the right. Once again the current phasor lags behind the voltage phasor as the two waveforms are of the same frequency.

However, as the current waveform is now crossing the horizontal zero axis line at this instant in time we can use the current phasor as our new reference and correctly say that the voltage phasor is "leading" the current phasor by angle, Φ. Either way, one phasor is designated as thereference phasor and all the other phasors will be either leading or lagging with respect to this reference.

Phasor Addition

Sometimes it is necessary when studying sinusoids to add together two alternating waveforms, for example in an AC series circuit, that are not in-phase with each other. If they are in-phase that is, there is no phase shift then they can be added together in the same way as DC values to find the algebraic sum of the two vectors. For example, two voltages in phase of say 50 volts and 25 volts respectively, will sum together as one 75 volts voltage. If however, they are not in-phase that is, they do not have identical directions or starting point then the phase angle between them needs to be taken into account so they are added together using phasor diagrams to determine their Resultant Phasor or Vector Sum by using the parallelogram law.

Consider two AC voltages, V₁ having a peak voltage of 20 volts, and V₂ having a peak voltage of 30 volts where V₁ leads V₂by 60^o. The total voltage, V_T of the two voltages can be found by firstly drawing a phasor diagram representing the two vectors and then constructing a parallelogram in which two of the sides are the voltages, V₁ and V₂ as shown below.

Phasor Addition of two Phasors

By drawing out the two phasors to scale onto graph paper, their phasor sum V₁ + V₂ can be easily found by measuring the length of the diagonal line, known as the "resultant r-vector", from the zero point to the intersection of the construction lines 0-A. The downside of this graphical method is that it is time consuming when drawing the phasors to scale. Also, while this graphical method gives an answer which is accurate enough for most purposes, it may produce an error if not drawn accurately or correctly to scale. Then one way to ensure that the correct answer is always obtained is by an analytical method.

Mathematically we can add the two voltages together by firstly finding their "vertical" and "horizontal" directions, and from this we can then calculate both the "vertical" and "horizontal" components for the resultant "r vector", V_T. This analytical method which uses the cosine and sine rule to find this resultant value is commonly called the Rectangular Form.

In the rectangular form, the phasor is divided up into a real part, x and an imaginary part, y forming the generalised expression   Z = x ± jy. ( we will discuss this in more detail in the next tutorial ). This then gives us a mathematical expression that represents both the magnitude and the phase of the sinusoidal voltage as:

So the addition of two vectors, A and B using the previous generalised expression is as follows:

Phasor Addition using Rectangular Form

Voltage, V₂ of 30 volts points in the reference direction along the horizontal zero axis, then it has a horizontal component but no vertical component as follows.

·         Horizontal component = 30 cos 0^o = 30 volts

·         Vertical component = 30 sin 0^o = 0 volts

·         This then gives us the rectangular expression for voltage V₂ of:  30 + j0

Voltage, V₁ of 20 volts leads voltage, V₂ by 60^o, then it has both horizontal and vertical components as follows.

·         Horizontal component = 20 cos 60^o = 20 x 0.5 = 10 volts

·         Vertical component = 20 sin 60^o = 20 x 0.866 = 17.32 volts

·         This then gives us the rectangular expression for voltage V₁ of:  10 + j17.32

The resultant voltage, V_T is found by adding together the horizontal and vertical components as follows.

·         V_Horizontal = sum of real parts of V₁ and V₂ = 30 + 10 = 40 volts

·         V_Vertical = sum of imaginary parts of V₁ and V₂ = 0 + 17.32 = 17.32 volts

Now that both the real and imaginary values have been found the magnitude of voltage, V_T is determined by simply usingPythagoras's Theorem for a 90^o triangle as follows.

Then the resulting phasor diagram will be:

Resultant Value of V_T

Phasor Subtraction

Phasor subtraction is very similar to the above rectangular method of addition, except this time the vector difference is the other diagonal of the parallelogram between the two voltages of V₁ and V₂ as shown.

Vector Subtraction of two Phasors

This time instead of "adding" together both the horizontal and vertical components we take them away, subtraction.

The 3-Phase Phasor Diagram

Previously we have only looked at single-phase AC waveforms where a single multi turn coil rotates within a magnetic field. But if three identical coils each with the same number of coil turns are placed at an electrical angle of 120^o to each other on the same rotor shaft, a three-phase voltage supply would be generated. A balanced three-phase voltage supply consists of three individual sinusoidal voltages that are all equal in magnitude and frequency but are out-of-phase with each other by exactly 120^o electrical degrees.

Standard practice is to colour code the three phases as Red, Yellow and Blue to identify each individual phase with the red phase as the reference phase. The normal sequence of rotation for a three phase supply is Red followed by Yellow followed by Blue, ( R, Y, B ).

As with the single-phase phasors above, the phasors representing a three-phase system also rotate in an anti-clockwise direction around a central point as indicated by the arrow marked ω in rad/s. The phasors for a three-phase balanced star or delta connected system are shown below.

Three-phase Phasor Diagram

The phase voltages are all equal in magnitude but only differ in their phase angle. The three windings of the coils are connected together at points, a₁, b₁ and c₁ to produce a common neutral connection for the three individual phases. Then if the red phase is taken as the reference phase each individual phase voltage can be defined with respect to the common neutral as.

Three-phase Voltage Equations

If the red phase voltage, V_RN is taken as the reference voltage as stated earlier then the phase sequence will be R – Y – B so the voltage in the yellow phase lags V_RN by 120^o, and the voltage in the blue phase lags V_YN also by 120^o. But we can also say the blue phase voltage, V_BN leads the red phase voltage, V_RN by 120^o.

One final point about a three-phase system. As the three individual sinusoidal voltages have a fixed relationship between each other of 120^o they are said to be "balanced" therefore, in a set of balanced three phase voltages their phasor sum will always be zero as:   V_a + V_b + V_c = 0

credits & thanks electronics-tutorials.ws

Direct Online Starter - DOL

OL Starter | Direct On Line Motor Starter

A DOL starter is mainly consists of some push switches, metallic contacts, connectors, overload sensing relay coil and a under voltage protection relay coil. Hope you are already aware about the functionality of a relay coil; it generally designed with a predetermined current  value. If somehow the limit is reached then it energies the coil and due to the electromagnetic action, it attracts a mechanical plunger within it. The plunger is fitted with the contacts of the circuit, so pulling the plunger will break the circuit and the current flow will terminate. In this way it helps to clear the faults from the circuit. In DOL starter two relay coils are used for two different purposes, one is used to sense the over current and trip the circuit immediately in order to safeguard the motor from the effect of huge current, and another is used to trip the circuit in under voltage condition. The main functionalities of a DOL starter are depended up on the relay coils.

Relay Coil and the Mechanical Plungers

The animation will show the basic relay operation. Relay coils are designed with proper current value such that it will able to attract the plunger after a certain predetermined value of current. After that the plungers are coupled with the relay coil and the contacts. In this way the entire control that is connecting or interrupting the circuit depends up on the operation of relay coil. Generally three types of relays are used in DOL starter. They are Magnetic Relay, Thermal Relay and Electronic Relay. The operation of magnetic relay solely depends up on the current value, and senses only when the current tends to approach the predetermined limiting value. Thermal relay is able to sense the heat generated due to the high current flow, and also very reliable and fast. The electronic relays are modern engineering marvel, which use the solid state devices for sensing the over current or faults. The operation is quite easy and fast, additionally they are highly reliable.

Modern direct on line (DOL) starters are using this type of relays.

Push Switches and Contacts

DOL starter is generally used in three phase 415 Volt motor starting, but sometime it also used to start 230 Volt single phase induction motors. So operation with 415 Volt requires some safety arrangement for the operator. That is why the entire starter is packed into a casing and the push switches are connected with the circuit for operating. These switches are common button type switches and able to connect the two ports or terminals with lesser risk of sparking. Fuses are installed in series with each phase to increase the safety factor and to protect the starter and motor both. In the wiring diagram the entire connection of the circuit is shown and the positions of push switches are also shown.

Operation of Direct ON Line (DOL) Starter

 The operation of DOL starter is very simple. The full line voltage is applied across the motor, by means of the starter circuit. The coil C is called as the under voltage relay coil as well as it also acts like the starting coil. This coil is controlled via the push switch “S1”. This switch is called as START switch and installed at a convenient place from where the remote operation is easily possible; generally the switch can be pushed from outside of casing. The START switch kept at open position by a spring and if the switch is pressed then the coil C is energized (or charged) from the two line conductors (Here R phase and Y Phase).

A mechanical plunger is connected with the coil C. When it energizes from the line, it attracts the plunger by relay operation. There are one auxiliary contact and three Main contacts which all are synchronized with the movement of the plunger. Three main contacts connect the three phases with the motor terminals and the auxiliary terminal is used to control the starter circuit. When the Coil C is energized, the plunger is attracted by it and automatically all the contacts become short circuited. In this way the motor is energized from the full line voltage. If the START switch S1 is released then also the motor will continue running, because the contact A will hold on the circuit at ON mode. That is why this contact is also termed as hold on contact. The “Off” and “Remote Off” switches are generally closed with the help of spring arrangement. Both of them are in series connection so pressing any of them will break the circuit and terminate the current flow. In this way the coil C will be discharged and automatically it release the plunger, which h will break the main motor contacts and in this way the motor will stop after some time. The coil C should be charged by pressing START switch to start the motor from this condition.

Under Voltage protection of DOL Starter

This protection is done with the help of relay of relay coil C (under voltage protection coil) and the plunger. Two phases are directly connected with the coil and the coil is designed such a way that it is able to attract the plunger for a certain level of current. If the level of current falls due to the low voltage supply then the coil will release the plunger and all the contacts become open circuited. In this way the circuit will open and the current flow breaks. Typically electromagnetic relays are used for this purpose, but in modern days electronic solid state relays are used also.

Overload protection of DOL Starter

The DOL starters are widely used for starting of 3 phase cage motors with 415 Volt rating. Fuses are connected in series with each line conductor, which provides the primary safety from high line current. The 3 phase induction motors are costly so additionally a relay mechanism is installed with the DOL starter with overload coils (OLC) in each phase conductors. If the line current is sufficiently high then the overload release coil are energized with such a value it can push the plunger D. The plunger is generally kept at closed position, so pushing it will break the circuit of relay of C. Which will further breaks the main contacts. The entire operation is very fast, and it depends up on the sensing capability of both of the relays.

credits & Thanks electrical4us.com