Phasor Diagrams and Phasor Algebra

Phasor Diagrams are a graphical way of representing the magnitude and directional relationship between two or more alternating quantities

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” are commonly 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 phasor difference between two or more sinusoidal waveforms. One way to overcome this problem is to represent the sinusoids graphically within the spacial or phasor-domain form by using Phasor Diagrams, and this is achieved by 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-clockwise direction 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 of ω = 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/sec. 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 2π representing 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 = 0 with a corresponding phase angle in either degrees or radians.

But 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 anticlockwisedirection 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 the reference 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, if two voltages of say 50 volts and 25 volts respectively are together “in-phase”, they will add or sum together to form one voltage of 75 volts (50 + 25).

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 V₂ by 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, yforming 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:

Definition of a Complex Sinusoid

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 using Pythagoras’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

Phasor Diagram Summary

Then to summarise this tutorial about Phasor Diagrams a little.

In their simplest terms, phasor diagrams are a projection of a rotating vector onto a horizontal axis which represents the instantaneous value. As a phasor diagram can be drawn to represent any instant of time and therefore any angle, the reference phasor of an alternating quantity is always drawn along the positive x-axis direction.

• Vectors, Phasors and Phasor Diagrams ONLY apply to sinusoidal AC alternating quantities.
• A Phasor Diagram can be used to represent two or more stationary sinusoidal quantities at any instant in time.
• Generally the reference phasor is drawn along the horizontal axis and at that instant in time the other phasors are drawn. All phasors are drawn referenced to the horizontal zero axis.
• Phasor diagrams can be drawn to represent more than two sinusoids. They can be either voltage, current or some other alternating quantity but the frequency of all of them must be the same.
• All phasors are drawn rotating in an anticlockwise direction. All the phasors ahead of the reference phasor are said to be “leading” while all the phasors behind the reference phasor are said to be “lagging”.
• Generally, the length of a phasor represents the r.m.s. value of the sinusoidal quantity rather than its maximum value.
• Sinusoids of different frequencies cannot be represented on the same phasor diagram due to the different speed of the vectors. At any instant in time the phase angle between them will be different.
• Two or more vectors can be added or subtracted together and become a single vector, called a Resultant Vector.
• The horizontal side of a vector is equal to the real or “x” vector. The vertical side of a vector is equal to the imaginary or “y” vector. The hypotenuse of the resultant right angled triangle is equivalent to the “r” vector.
• In a three-phase balanced system each individual phasor is displaced by 120^o.

Phase Difference and Phase Shift

Phase Difference is used to describe the difference in degrees or radians when two or more alternating quantities reach their maximum or zero values

Previously we saw that a Sinusoidal Waveform is an alternating quantity that can be presented graphically in the time domain along an horizontal zero axis. We also saw that as an alternating quantity, sine waves have a positive maximum value at time π/2, a negative maximum value at time 3π/2, with zero values occurring along the baseline at 0, π and 2π.

However, not all sinusoidal waveforms will pass exactly through the zero axis point at the same time, but may be “shifted” to the right or to the left of 0^o by some value when compared to another sine wave.

For example, comparing a voltage waveform to that of a current waveform. This then produces an angular shift or Phase Difference between the two sinusoidal waveforms. Any sine wave that does not pass through zero at t = 0 has a phase shift.

The phase difference or phase shift as it is also called of a Sinusoidal Waveform is the angle Φ (Greek letter Phi), in degrees or radians that the waveform has shifted from a certain reference point along the horizontal zero axis. In other words phase shift is the lateral difference between two or more waveforms along a common axis and sinusoidal waveforms of the same frequency can have a phase difference.

The phase difference, Φ of an alternating waveform can vary from between 0 to its maximum time period, T of the waveform during one complete cycle and this can be anywhere along the horizontal axis between, Φ = 0 to 2π (radians) or Φ  = 0 to 360^odepending upon the angular units used.

Phase difference can also be expressed as a time shift of τ in seconds representing a fraction of the time period, T for example, +10mS or – 50uS but generally it is more common to express phase difference as an angular measurement.

Then the equation for the instantaneous value of a sinusoidal voltage or current waveform we developed in the previous Sinusoidal Waveform will need to be modified to take account of the phase angle of the waveform and this new general expression becomes.

Phase Difference Equation

• Where:
•   A_m  –  is the amplitude of the waveform.
•   ωt  –  is the angular frequency of the waveform in radian/sec.
•   Φ (phi)  –  is the phase angle in degrees or radians that the waveform has shifted either left or right from the reference point.

If the positive slope of the sinusoidal waveform passes through the horizontal axis “before” t = 0 then the waveform has shifted to the left so Φ >0, and the phase angle will be positive in nature, +Φ giving a leading phase angle. In other words it appears earlier in time than 0^o producing an anticlockwise rotation of the vector.

Likewise, if the positive slope of the sinusoidal waveform passes through the horizontal x-axis some time “after” t = 0 then the waveform has shifted to the right so Φ <0, and the phase angle will be negative in nature -Φ producing a lagging phase angle as it appears later in time than 0^o producing a clockwise rotation of the vector. Both cases are shown below.

Phase Relationship of a Sinusoidal Waveform

Firstly, lets consider that two alternating quantities such as a voltage, v and a current, ihave the same frequency ƒ in Hertz. As the frequency of the two quantities is the same the angular velocity, ω must also be the same. So at any instant in time we can say that the phase of voltage, v will be the same as the phase of the current, i.

Then the angle of rotation within a particular time period will always be the same and the phase difference between the two quantities of v and i will therefore be zero and Φ = 0. As the frequency of the voltage, v and the current, i are the same they must both reach their maximum positive, negative and zero values during one complete cycle at the same time (although their amplitudes may be different). Then the two alternating quantities, v and iare said to be “in-phase”.

Two Sinusoidal Waveforms – “in-phase”

Now lets consider that the voltage, v and the current, i have a phase difference between themselves of  30^o, so (Φ  = 30^o or π/6 radians). As both alternating quantities rotate at the same speed, i.e. they have the same frequency, this phase difference will remain constant for all instants in time, then the phase difference of  30^o between the two quantities is represented by phi, Φ as shown below.

Phase Difference of a Sinusoidal Waveform

The voltage waveform above starts at zero along the horizontal reference axis, but at that same instant of time the current waveform is still negative in value and does not cross this reference axis until 30^o later. Then there exists a Phase difference between the two waveforms as the current cross the horizontal reference axis reaching its maximum peak and zero values after the voltage waveform.

As the two waveforms are no longer “in-phase”, they must therefore be “out-of-phase” by an amount determined by phi, Φ and in our example this is 30^o. So we can say that the two waveforms are now 30^o out-of phase. The current waveform can also be said to be “lagging” behind the voltage waveform by the phase angle, Φ. Then in our example above the two waveforms have a Lagging Phase Difference so the expression for both the voltage and current above will be given as.

  where, i lags v by angle Φ

Likewise, if the current, i has a positive value and crosses the reference axis reaching its maximum peak and zero values at some time before the voltage, v then the current waveform will be “leading” the voltage by some phase angle. Then the two waveforms are said to have a Leading Phase Difference and the expression for both the voltage and the current will be.

  where, i leads v by angle Φ

The phase angle of a sine wave can be used to describe the relationship of one sine wave to another by using the terms “Leading” and “Lagging” to indicate the relationship between two sinusoidal waveforms of the same frequency, plotted onto the same reference axis. In our example above the two waveforms are out-of-phase by 30^o. So we can correctly say that i lags v or we can say that v leads i by 30^o depending upon which one we choose as our reference.

The relationship between the two waveforms and the resulting phase angle can be measured anywhere along the horizontal zero axis through which each waveform passes with the “same slope” direction either positive or negative.

In AC power circuits this ability to describe the relationship between a voltage and a current sine wave within the same circuit is very important and forms the bases of AC circuit analysis.

The Cosine Waveform

So we now know that if a waveform is “shifted” to the right or left of 0^o when compared to another sine wave the expression for this waveform becomes A_m sin(ωt ± Φ). But if the waveform crosses the horizontal zero axis with a positive going slope 90^o or π/2 radians before the reference waveform, the waveform is called a Cosine Waveform and the expression becomes.

Cosine Expression

The Cosine Wave, simply called “cos”, is as important as the sine wave in electrical engineering. The cosine wave has the same shape as its sine wave counterpart that is it is a sinusoidal function, but is shifted by +90^o or one full quarter of a period ahead of it.

Phase Difference between a Sine wave and a Cosine wave

Alternatively, we can also say that a sine wave is a cosine wave that has been shifted in the other direction by -90^o. Either way when dealing with sine waves or cosine waves with an angle the following rules will always apply.

Sine and Cosine Wave Relationships

When comparing two sinusoidal waveforms it more common to express their relationship as either a sine or cosine with positive going amplitudes and this is achieved using the following mathematical identities.

By using these relationships above we can convert any sinusoidal waveform with or without an angular or phase difference from either a sine wave into a cosine wave or vice versa.

AC Waveform and AC Circuit Theory

AC Sinusoidal Waveforms are created by rotating a coil within a magnetic field and alternating voltages and currents form the basis of AC Theory

Direct Current or D.C. as it is more commonly called, is a form of electrical current or voltage that flows around an electrical circuit in one direction only, making it a “Uni-directional” supply.

Generally, both DC currents and voltages are produced by power supplies, batteries, dynamos and solar cells to name a few. A DC voltage or current has a fixed magnitude (amplitude) and a definite direction associated with it. For example, +12V represents 12 volts in the positive direction, or -5V represents 5 volts in the negative direction.

We also know that DC power supplies do not change their value with regards to time, they are a constant value flowing in a continuous steady state direction. In other words, DC maintains the same value for all times and a constant uni-directional DC supply never changes or becomes negative unless its connections are physically reversed. An example of a simple DC or direct current circuit is shown below.

DC Circuit and Waveform

An alternating function or AC Waveform on the other hand is defined as one that varies in both magnitude and direction in more or less an even manner with respect to time making it a “Bi-directional” waveform. An AC function can represent either a power source or a signal source with the shape of an AC waveform generally following that of a mathematical sinusoid being defined as: A(t) = A_max*sin(2πƒt).

The term AC or to give it its full description of Alternating Current, generally refers to a time-varying waveform with the most common of all being called a Sinusoid better known as a Sinusoidal Waveform. Sinusoidal waveforms are more generally called by their short description as Sine Waves. Sine waves are by far one of the most important types of AC waveform used in electrical engineering.

The shape obtained by plotting the instantaneous ordinate values of either voltage or current against time is called an AC Waveform. An AC waveform is constantly changing its polarity every half cycle alternating between a positive maximum value and a negative maximum value respectively with regards to time with a common example of this being the domestic mains voltage supply we use in our homes.

This means then that the AC Waveform is a “time-dependent signal” with the most common type of time-dependant signal being that of the Periodic Waveform. The periodic or AC waveform is the resulting product of a rotating electrical generator. Generally, the shape of any periodic waveform can be generated using a fundamental frequency and superimposing it with harmonic signals of varying frequencies and amplitudes but that’s for another tutorial.

Alternating voltages and currents can not be stored in batteries or cells like direct current (DC) can, it is much easier and cheaper to generate these quantities using alternators or waveform generators when they are needed. The type and shape of an AC waveform depends upon the generator or device producing them, but all AC waveforms consist of a zero voltage line that divides the waveform into two symmetrical halves. The main characteristics of an AC Waveform are defined as:

AC Waveform Characteristics

• • The Period, (T) is the length of time in seconds that the waveform takes to repeat itself from start to finish. This can also be called the Periodic Time of the waveform for sine waves, or the Pulse Width for square waves.
• • The Frequency, (ƒ) is the number of times the waveform repeats itself within a one second time period. Frequency is the reciprocal of the time period, ( ƒ = 1/T ) with the unit of frequency being the Hertz, (Hz).
• • The Amplitude (A) is the magnitude or intensity of the signal waveform measured in volts or amps.

In our tutorial about Waveforms ,we looked at different types of waveforms and said that “Waveforms are basically a visual representation of the variation of a voltage or current plotted to a base of time”. Generally, for AC waveforms this horizontal base line represents a zero condition of either voltage or current. Any part of an AC type waveform which lies above the horizontal zero axis represents a voltage or current flowing in one direction.

Likewise, any part of the waveform which lies below the horizontal zero axis represents a voltage or current flowing in the opposite direction to the first. Generally for sinusoidal AC waveforms the shape of the waveform above the zero axis is the same as the shape below it. However, for most non-power AC signals including audio waveforms this is not always the case.

The most common periodic signal waveforms that are used in Electrical and Electronic Engineering are the Sinusoidal Waveforms. However, an alternating AC waveform may not always take the shape of a smooth shape based around the trigonometric sine or cosine function. AC waveforms can also take the shape of either Complex Waves, Square Waves or Triangular Waves and these are shown below.

Types of Periodic Waveform

The time taken for an AC Waveform to complete one full pattern from its positive half to its negative half and back to its zero baseline again is called a Cycle and one complete cycle contains both a positive half-cycle and a negative half-cycle. The time taken by the waveform to complete one full cycle is called the Periodic Time of the waveform, and is given the symbol “T”.

The number of complete cycles that are produced within one second (cycles/second) is called the Frequency, symbol ƒ of the alternating waveform. Frequency is measured in Hertz, ( Hz ) named after the German physicist Heinrich Hertz.

Then we can see that a relationship exists between cycles (oscillations), periodic time and frequency (cycles per second), so if there are ƒ number of cycles in one second, each individual cycle must take 1/ƒ seconds to complete.

Relationship Between Frequency and Periodic Time

AC Waveform Example No1

1. What will be the periodic time of a 50Hz waveform and 2. what is the frequency of an AC waveform that has a periodic time of 10mS.

1).

2).

Frequency used to be expressed in “cycles per second” abbreviated to “cps”, but today it is more commonly specified in units called “Hertz”. For a domestic mains supply the frequency will be either 50Hz or 60Hz depending upon the country and is fixed by the speed of rotation of the generator. But one hertz is a very small unit so prefixes are used that denote the order of magnitude of the waveform at higher frequencies such as kHz, MHz and even GHz.

Definition of Frequency Prefixes

Prefix	Definition	Written as	Periodic Time
Kilo	Thousand	kHz	1ms
Mega	Million	MHz	1us
Giga	Billion	GHz	1ns
Terra	Trillion	THz	1ps

Amplitude of an AC Waveform

As well as knowing either the periodic time or the frequency of the alternating quantity, another important parameter of the AC waveform is Amplitude, better known as its Maximum or Peak value represented by the terms, Vmax for voltage or Imax for current.

The peak value is the greatest value of either voltage or current that the waveform reaches during each half cycle measured from the zero baseline. Unlike a DC voltage or current which has a steady state that can be measured or calculated using Ohm’s Law, an alternating quantity is constantly changing its value over time.

For pure sinusoidal waveforms this peak value will always be the same for both half cycles ( +Vm = -Vm ) but for non-sinusoidal or complex waveforms the maximum peak value can be very different for each half cycle. Sometimes, alternating waveforms are given a peak-to-peak, Vp-p value and this is simply the distance or the sum in voltage between the maximum peak value, +Vmax and the minimum peak value, -Vmax during one complete cycle.

The Average Value of an AC Waveform

The average or mean value of a continuous DC voltage will always be equal to its maximum peak value as a DC voltage is constant. This average value will only change if the duty cycle of the DC voltage changes. In a pure sine wave if the average value is calculated over the full cycle, the average value would be equal to zero as the positive and negative halves will cancel each other out. So the average or mean value of an AC waveform is calculated or measured over a half cycle only and this is shown below.

Average Value of a Non-sinusoidal Waveform

To find the average value of the waveform we need to calculate the area underneath the waveform using the mid-ordinate rule, trapezoidal rule or the Simpson’s rule found commonly in mathematics. The approximate area under any irregular waveform can easily be found by simply using the mid-ordinate rule.

The zero axis base line is divided up into any number of equal parts and in our simple example above this value was nine, ( V₁ to V₉ ). The more ordinate lines that are drawn the more accurate will be the final average or mean value. The average value will be the addition of all the instantaneous values added together and then divided by the total number. This is given as.

Average Value of an AC Waveform

Where: n equals the actual number of mid-ordinates used.

For a pure sinusoidal waveform this average or mean value will always be equal to 0.637*V_max and this relationship also holds true for average values of current.

The RMS Value of an AC Waveform

The average value of an AC waveform that we calculated above as being: 0.637*V_max is NOT the same value we would use for a DC supply. This is because unlike a DC supply which is constant and and of a fixed value, an AC waveform is constantly changing over time and has no fixed value. Thus the equivalent value for an alternating current system that provides the same amount of electrical power to a load as a DC equivalent circuit is called the “effective value”.

The effective value of a sine wave produces the same I²*R heating effect in a load as we would expect to see if the same load was fed by a constant DC supply. The effective value of a sine wave is more commonly known as the Root Mean Squared or simply RMS value as it is calculated as the square root of the mean (average) of the square of the voltage or current.

That is V_rms or I_rms is given as the square root of the average of the sum of all the squared mid-ordinate values of the sine wave. The RMS value for any AC waveform can be found from the following modified average value formula as shown.

RMS Value of an AC Waveform

Where: n equals the number of mid-ordinates.

For a pure sinusoidal waveform this effective or R.M.S. value will always be equal too: 1/√2*V_max which is equal to 0.707*V_max and this relationship holds true for RMS values of current. The RMS value for a sinusoidal waveform is always greater than the average value except for a rectangular waveform. In this case the heating effect remains constant so the average and the RMS values will be the same.

One final comment about R.M.S. values. Most multimeters, either digital or analogue unless otherwise stated only measure the R.M.S. values of voltage and current and not the average. Therefore when using a multimeter on a direct current system the reading will be equal to I = V/R and for an alternating current system the reading will be equal to Irms = Vrms/R.

Also, except for average power calculations, when calculating RMS or peak voltages, only use V_RMS to find I_RMS values, or peak voltage, Vp to find peak current, Ip values. Do not mix them together as Average, RMS or Peak values of a sine wave are completely different and your results will definitely be incorrect.

Form Factor and Crest Factor

Although little used these days, both Form Factor and Crest Factor can be used to give information about the actual shape of the AC waveform. Form Factor is the ratio between the average value and the RMS value and is given as.

For a pure sinusoidal waveform the Form Factor will always be equal to 1.11. Crest Factor is the ratio between the R.M.S. value and the Peak value of the waveform and is given as.

For a pure sinusoidal waveform the Crest Factor will always be equal to 1.414.

AC Waveform Example No2

A sinusoidal alternating current of 6 amps is flowing through a resistance of 40Ω. Calculate the average voltage and the peak voltage of the supply.

The R.M.S. Voltage value is calculated as:

The Average Voltage value is calculated as:

The Peak Voltage value is calculated as:

The use and calculation of Average, R.M.S, Form factor and Crest Factor can also be use with any type of periodic waveform including Triangular, Square, Sawtoothed or any other irregular or complex voltage/current waveform shape. Conversion between the various sinusoidal values can sometimes be confusing so the following table gives a convenient way of converting one sine wave value to another.

Sinusoidal Waveform Conversion Table

Convert From	Multiply By	Or By	To Get Value
Peak	2	(√2)2	Peak-to-Peak
Peak-to-Peak	0.5	1/2	Peak
Peak	0.707	1/(√2)	RMS
Peak	0.637	2/π	Average
Average	1.570	π/2	Peak
Average	1.111	π/(2√2)	RMS
RMS	1.414	√2	Peak
RMS	0.901	(2√2)/π	Average

Sinusoidal Waveforms

When an electric current flows through a wire or conductor, a circular magnetic field is created around the wire and whose strength is related to the current value.

 

If this single wire conductor is moved or rotated within a stationary magnetic field, an “EMF”, (Electro-Motive Force) is induced within the conductor due to the movement of the conductor through the magnetic flux.

From this we can see that a relationship exists between Electricity and Magnetism giving us, as Michael Faraday discovered the effect of “Electromagnetic Induction” and it is this basic principal that electrical machines and generators use to generate a Sinusoidal Waveform for our mains supply.

In the Electromagnetic Induction, tutorial we said that when a single wire conductor moves through a permanent magnetic field thereby cutting its lines of flux, an EMF is induced in it.

However, if the conductor moves in parallel with the magnetic field in the case of points A and B, no lines of flux are cut and no EMF is induced into the conductor, but if the conductor moves at right angles to the magnetic field as in the case of points C and D, the maximum amount of magnetic flux is cut producing the maximum amount of induced EMF.

Also, as the conductor cuts the magnetic field at different angles between points A and C, 0 and 90^o the amount of induced EMF will lie somewhere between this zero and maximum value. Then the amount of emf induced within a conductor depends on the angle between the conductor and the magnetic flux as well as the strength of the magnetic field.

An AC generator uses the principal of Faraday’s electromagnetic induction to convert a mechanical energy such as rotation, into electrical energy, a Sinusoidal Waveform. A simple generator consists of a pair of permanent magnets producing a fixed magnetic field between a north and a south pole. Inside this magnetic field is a single rectangular loop of wire that can be rotated around a fixed axis allowing it to cut the magnetic flux at various angles as shown below.

Basic Single Coil AC Generator

As the coil rotates anticlockwise around the central axis which is perpendicular to the magnetic field, the wire loop cuts the lines of magnetic force set up between the north and south poles at different angles as the loop rotates. The amount of induced EMF in the loop at any instant of time is proportional to the angle of rotation of the wire loop.

As this wire loop rotates, electrons in the wire flow in one direction around the loop. Now when the wire loop has rotated past the 180^o point and moves across the magnetic lines of force in the opposite direction, the electrons in the wire loop change and flow in the opposite direction. Then the direction of the electron movement determines the polarity of the induced voltage.

So we can see that when the loop or coil physically rotates one complete revolution, or 360^o, one full sinusoidal waveform is produced with one cycle of the waveform being produced for each revolution of the coil. As the coil rotates within the magnetic field, the electrical connections are made to the coil by means of carbon brushes and slip-rings which are used to transfer the electrical current induced in the coil.

The amount of EMF induced into a coil cutting the magnetic lines of force is determined by the following three factors.

•  Speed – the speed at which the coil rotates inside the magnetic field.
•  Strength – the strength of the magnetic field.
•  Length – the length of the coil or conductor passing through the magnetic field.

We know that the frequency of a supply is the number of times a cycle appears in one second and that frequency is measured in Hertz. As one cycle of induced emf is produced each full revolution of the coil through a magnetic field comprising of a north and south pole as shown above, if the coil rotates at a constant speed a constant number of cycles will be produced per second giving a constant frequency. So by increasing the speed of rotation of the coil the frequency will also be increased. Therefore, frequency is proportional to the speed of rotation, ( ƒ ∝ Ν ) where Ν = r.p.m.

Also, our simple single coil generator above only has two poles, one north and one south pole, giving just one pair of poles. If we add more magnetic poles to the generator above so that it now has four poles in total, two north and two south, then for each revolution of the coil two cycles will be produced for the same rotational speed. Therefore, frequency is proportional to the number of pairs of magnetic poles, ( ƒ ∝ P ) of the generator where P = the number of “pairs of poles”.

Then from these two facts we can say that the frequency output from an AC generator is:

Where: Ν is the speed of rotation in r.p.m. P is the number of “pairs of poles” and 60 converts it into seconds.

Instantaneous Voltage

The EMF induced in the coil at any instant of time depends upon the rate or speed at which the coil cuts the lines of magnetic flux between the poles and this is dependant upon the angle of rotation, Theta ( θ ) of the generating device. Because an AC waveform is constantly changing its value or amplitude, the waveform at any instant in time will have a different value from its next instant in time.

For example, the value at 1ms will be different to the value at 1.2ms and so on. These values are known generally as the Instantaneous Values, or V_i Then the instantaneous value of the waveform and also its direction will vary according to the position of the coil within the magnetic field as shown below.

Displacement of a Coil within a Magnetic Field

The instantaneous values of a sinusoidal waveform is given as the “Instantaneous value = Maximum value x sin θ ” and this is generalized by the formula.

Where, V_max is the maximum voltage induced in the coil and θ = ωt, is the rotational angle of the coil with respect to time.

If we know the maximum or peak value of the waveform, by using the formula above the instantaneous values at various points along the waveform can be calculated. By plotting these values out onto graph paper, a sinusoidal waveform shape can be constructed.

In order to keep things simple we will plot the instantaneous values for the sinusoidal waveform at every 45^o of rotation giving us 8 points to plot. Again, to keep it simple we will assume a maximum voltage, V_MAX value of 100V. Plotting the instantaneous values at shorter intervals, for example at every 30^o (12 points) or 10^o (36 points) for example would result in a more accurate sinusoidal waveform construction.

Sinusoidal Waveform Construction

Coil Angle ( θ )	0	45	90	135	180	225	270	315	360
e = Vmax.sinθ	0	70.71	100	70.71	0	-70.71	-100	-70.71	-0

The points on the sinusoidal waveform are obtained by projecting across from the various positions of rotation between 0^o and 360^o to the ordinate of the waveform that corresponds to the angle, θ and when the wire loop or coil rotates one complete revolution, or 360^o, one full waveform is produced.

From the plot of the sinusoidal waveform we can see that when θ is equal to 0^o, 180^o or 360^o, the generated EMF is zero as the coil cuts the minimum amount of lines of flux. But when θ is equal to 90^o and 270^o the generated EMF is at its maximum value as the maximum amount of flux is cut.

Therefore a sinusoidal waveform has a positive peak at 90^o and a negative peak at 270^o. Positions B, D, F and H generate a value of EMF corresponding to the formula: e = Vmax.sinθ.

Then the waveform shape produced by our simple single loop generator is commonly referred to as a Sine Wave as it is said to be sinusoidal in its shape. This type of waveform is called a sine wave because it is based on the trigonometric sine function used in mathematics, ( x(t) = Amax.sinθ ).

When dealing with sine waves in the time domain and especially current related sine waves the unit of measurement used along the horizontal axis of the waveform can be either time, degrees or radians. In electrical engineering it is more common to use the Radian as the angular measurement of the angle along the horizontal axis rather than degrees. For example, ω = 100 rad/s, or 500 rad/s.

Radians

The Radian, (rad) is defined mathematically as a quadrant of a circle where the distance subtended on the circumference of the circle is equal to the length of the radius (r) of the same circle. Since the circumference of a circle is equal to 2π x radius, there must be 2πradians around the 360^o of a circle.

In other words, the radian is a unit of angular measurement and the length of one radian (r) will fit 6.284 (2*π) times around the whole circumference of a circle. Thus one radian equals 360^o/2π = 57.3^o. In electrical engineering the use of radians is very common so it is important to remember the following formula.

Definition of a Radian

Using radians as the unit of measurement for a sinusoidal waveform would give 2πradians for one full cycle of 360^o. Then half a sinusoidal waveform must be equal to 1πradians or just π (pi). Then knowing that pi, (π) is equal to 3.142, the relationship between degrees and radians for a sinusoidal waveform is therefore given as:

Relationship between Degrees and Radians

Applying these two equations to various points along the waveform gives us.

The conversion between degrees and radians for the more common equivalents used in sinusoidal analysis are given in the following table.

Relationship between Degrees and Radians

Degrees	Radians	Degrees	Radians	Degrees	Radians
0o	0	135o	3π 4	270o	3π 2
30o	π 6	150o	5π 6	300o	5π 3
45o	π 4	180o	π	315o	7π 4
60o	π 3	210o	7π 6	330o	11π 6
90o	π 2	225o	5π 4	360o	2π
120o	2π 3	240o	4π 3

The velocity at which the generator rotates around its central axis determines the frequency of the sinusoidal waveform. As the frequency of the waveform is given as ƒ Hz or cycles per second, the waveform also has angular frequency, ω, (Greek letter omega), in radians per second. Then the angular velocity of a sinusoidal waveform is given as.

Angular Velocity of a Sinusoidal Waveform

and in the United Kingdom, the angular velocity or frequency of the mains supply is given as:

in the USA as their mains supply frequency is 60Hz it can be given as: 377 rad/s

So we now know that the velocity at which the generator rotates around its central axis determines the frequency of the sinusoidal waveform and which can also be called its angular velocity, ω. But we should by now also know that the time required to complete one full revolution is equal to the periodic time, (T) of the sinusoidal waveform.

As frequency is inversely proportional to its time period, ƒ = 1/T we can therefore substitute the frequency quantity in the above equation for the equivalent periodic time quantity and substituting gives us.

The above equation states that for a smaller periodic time of the sinusoidal waveform, the greater must be the angular velocity of the waveform. Likewise in the equation above for the frequency quantity, the higher the frequency the higher the angular velocity.

Sinusoidal Waveform Example

A sinusoidal waveform is defined as: V_m = 169.8 sin(377t) volts. Calculate the RMS voltage of the waveform, its frequency and the instantaneous value of the voltage, (V_i) after a time of six milliseconds (6ms).

We know from above that the general expression given for a sinusoidal waveform is:

Then comparing this to our given expression for a sinusoidal waveform above of V_m = 169.8 sin(377t) will give us the peak voltage value of 169.8 volts for the waveform.

The waveforms RMS voltage is calculated as:

The angular velocity (ω) is given as 377 rad/s. Then 2πƒ = 377. So the frequency of the waveform is calculated as:

The instantaneous voltage V_i value after a time of 6mS is given as:

Note that the angular velocity at time t = 6mS is given in radians (rads). We could, if so wished, convert this into an equivalent angle in degrees and use this value instead to calculate the instantaneous voltage value. The angle in degrees of the instantaneous voltage value is therefore given as:

Sinusoidal Waveform

Then the generalised format used for analysing and calculating the various values of a Sinusoidal Waveform is as follows:

A Sinusoidal Waveform

Parallel & Perspective Projection

Parallel Projection:-

In parallel projection, z co-ordinate is discarded and parallel lines from each vertex on the object are extended until they intersect the view plane. The point of intersection is the projection of the vertex. We connect the projected vertices by line segments which correspond to connections on the original object.

As shown in the Fig. (21), a parallel projection preserves relative proportions of objects but does produce the realistic views.

Perspective Projection:-

The perspective projection. on the other hand, produces realistic views but does not preserve relative proportions. In perspective Projection, the lines of projection are not parallel. Instead, they all converge at a single point called the center of projection or projection reference point. The object positions are transformed to the view plane along these converged projection lines and the projected view of an object is determined by calculating the intersection of the converged projection lines with the view plane as shown in the Fig. (22).

Types of parallel Projections:-

Parallel projections are basically categorized into two types, depending on the relation between the direction of projection and the normal to the view plane. When the direction of the projection is normal (perpendicular) to the view plane, we have an orthographic parallel projection. Otherwise, we have an oblique parallel projection. Fig. (23) illustrates the two types of parallel projection.

Types of Perspective Projections:-

The perspective projection of any set of parallel lines that are not parallel to the projection plane converge to a vanishing point.The vanishing point for any set of lines that are parallel to one of the three principle axes of an object is referred to as a principle vanishing point or axis vanishing point.There are at most three such points,corresponding to the number of principle axes cut by the projection plane. The perspective projection is classified according to number of principle vanishing points in a projection: one-point, two-points or three-points Projections. Fig. (24) shows the appearance of one-point, two-points and three point Perspective projections of a cube.

 

General Parallel Projection Transformations:-

In a general parallel projection On XY Plane, we may select any direction for the lines of projection. Suppose that the direction of projection is given by the vector [xp yp zp] and that the object is to be projected onto the xy plane. If the point on the object is given as (x1, y1, z1), then we can determine the projected point (x2, y2) as given below: The equations in the parametric form for a line passing through the projected point (x2, y2, z2) and in the direction of projection are given as

For projected point z2 is 0, therefore, the third equation can be written as

General Perspective-Projection Transformations:-

The general Perspective-Projection Transformations can be obtained by Performing following two operations.

• Make the center line of the frustum perpendicular to the view plane by shearing the view volume.

• Scale the view volume with a scaling factor that depends on 1/z .

These two steps are illustrated in Fig. (26).

  

How Digital Cameras Work ?

Digital Camera Basics

Let’s say you want to take a picture and e-mail it to a friend. To do this, you need the image to be represented in the language that computers recognize — bits and bytes. Essentially, a digital image is just a long string of 1s and 0s that represent all the tiny colored dots — or pixels — that collectively make up the image. (For information on sampling and digital representations of data, see this explanation of the digitization of sound waves. Digitizing light waves works in a similar way.)

• You can take a photograph using a conventional film camera, process the film chemically, print it onto photographic paper and then use a digital scanner to sample the print (record the pattern of light as a series of pixel values).
• You can directly sample the original light that bounces off your subject, immediately breaking that light pattern down into a series of pixel values — in other words, you can use a digital camera.

Instead of film, a digital camera has a sensor that converts light into electrical charges.

The image sensor employed by most digital cameras is a charge coupled device (CCD). Some cameras use complementary metal oxide semiconductor (CMOS) technology instead. Both CCD and CMOS image sensors convert light into electrons. If you’ve read How Solar Cells Work, you already understand one of the pieces of technology used to perform the conversion. A simplified way to think about these sensors is to think of a 2-D array of thousands or millions of tiny solar cells.

Once the sensor converts the light into electrons, it reads the value (accumulated charge) of each cell in the image. This is where the differences between the two main sensor types kick in:

• A CCD transports the charge across the chip and reads it at one corner of the array. An analog-to-digital converter (ADC) then turns each pixel’s value into a digital value by measuring the amount of charge at each photosite and converting that measurement to binary form.
• CMOS devices use several transistors at each pixel to amplify and move the charge using more traditional wires.

Differences between the two types of sensors lead to a number of pros and cons:

A CCD sensor

CCD sensors create high-quality, low-noise images. CMOS sensors are generally more susceptible to noise.

• Because each pixel on a CMOS sensor has several transistors located next to it, the light sensitivity of a CMOS chip is lower. Many of the photons hit the transistors instead of the photodiode.
• CMOS sensors traditionally consume little power. CCDs, on the other hand, use a process that consumes lots of power. CCDs consume as much as 100 times more power than an equivalent CMOS sensor.
• CCD sensors have been mass produced for a longer period of time, so they are more mature. They tend to have higher quality pixels, and more of them.

It takes several steps for a digital camera to take a picture. Here’s a review of what happens in a CCD camera, from beginning to end:

• You aim the camera at the subject and adjust the optical zoom to get closer or farther away.
• You press lightly on the shutter release.
• The camera automatically focuses on the subject and takes a reading of the available light.
• The camera sets the aperture and shutter speed for optimal exposure.
• You press the shutter release all the way.
• The camera resets the CCD and exposes it to the light, building up an electrical charge, until the shutter closes.
• The ADC measures the charge and creates a digital signal that represents the values of the charge at each pixel.
• A processor interpolates the data from the different pixels to create natural color. On many cameras, it is possible to see the output on the LCD at this stage.
• A processor may perform a preset level of compression on the data.
• The information is stored in some form of memory device (probably a Flash memory card).

How Scanner Works ?

Scanners have become an important part of the home office over the last few years. Scanner technology is everywhere and used in many ways:

• Flatbed scanners, also called desktop scanners, are the most versatile and commonly used scanners. In fact, this article will focus on the technology as it relates to flatbed scanners.
• Sheet-fed scanners are similar to flatbed scanners except the document is moved and the scan head is immobile. A sheet-fed scanner looks a lot like a small portable printer.
• Handheld scanners use the same basic technology as a flatbed scanner, but rely on the user to move them instead of a motorized belt. This type of scanner typically does not provide good image quality. However, it can be useful for quickly capturing text.
• Drum scanners are used by the publishing industry to capture incredibly detailed images. They use a technology called a photomultiplier tube (PMT). In PMT, the document to be scanned is mounted on a glass cylinder. At the center of the cylinder is a sensor that splits light bounced from the document into three beams. Each beam is sent through a color filter into a photomultiplier tube where the light is changed into an electrical signal.

The basic principle of a scanner is to analyze an image and process it in some way. Image and text capture (optical character recognition or OCR) allow you to save information to a file on your computer.

Here are the steps that a scanner goes through when it scans a document:

• The document is placed on the glass plate and the cover is closed. The inside of the cover in most scanners is flat white, although a few are black. The cover provides a uniform background that the scanner software can use as a reference point for determining the size of the document being scanned. Most flatbed scanners allow the cover to be removed for scanning a bulky object, such as a page in a thick book.

• A lamp is used to illuminate the document. The lamp in newer scanners is either a cold cathode fluorescent lamp (CCFL) or a xenon lamp, while older scanners may have a standard fluorescent lamp.
• The entire mechanism (mirrors, lens, filter and CCD array) make up the scan head. The scan head is moved slowly across the document by a belt that is attached to a stepper motor. The scan head is attached to a stabilizer bar to ensure that there is no wobble or deviation in the pass. Pass means that the scan head has completed a single complete scan of the document.

• The image of the document is reflected by an angled mirror to another mirror. In some scanners, there are only two mirrors while others use a three mirror approach. Each mirror is slightly curved to focus the image it reflects onto a smaller surface.
• The last mirror reflects the image onto a lens. The lens focuses the image through a filter on the CCD array.

The filter and lens arrangement vary based on the scanner. Some scanners use a three pass scanning method. Each pass uses a different color filter (red, green or blue) between the lens and CCD array. After the three passes are completed, the scanner software assembles the three filtered images into a single full-color image.

Parameters of a Scanner

• The resolution of the image is one of the main parameters of the scanner. Each scanner varies according to its resolution and hence the cost. The resolution may be expressed in pixels per inch [ppi] and also samples per inch (spi). But, instead of defining the scanner’s correct optical resolution the manufacturers mostly define the interpolated resolution of the scanner. The latest flatbed scanner has an interpolated resolution of 5400 ppi and almost 12,000 ppi for a drum scanner.
• Interpolated resolution actually refers to the increase in the resolution of the image with the help of the scanning software. This is done by adding extra pixels in between the ones actually scanned by the CCD array. These extra pixels can be added only as an average of the adjacent pixels. Suppose a scanner has a true resolution of 300×300 dpi and the interpolated resolution declared by the manufacturer is 600×300 dpi. Thus an additional pixel is added in each row of the CCD sensor by the software. As the resolution increases, the size of the file also increases. This size can be reduced through lossy compression technique like JPEG. Through this method the quality of the picture will only be reduced to a small amount. Usually this method is done to load an image faster on the internet and also to print the image on a full page.
• A scanner has a least original resolution of about 300×300 dots per inch (dpi). This increases with the increase in the CCD sensors row wise and also by the precision of the stepper motor.
• As the scanner’s lamp brightness increases along with the use of high quality optics the sharpness of the image also increases. Density range is another parameter through which the minor shadow and brightness details can also be reproduced through scanning. The higher the density range, the higher the details.
• Another parameter used is the colour depth. In colour scanning, the colour depth refers to the number of colours that can be reproduced by the scanner. Though a 24 bit/pixel scanner is sufficient enough there are scanners with 30 bits and 36 bits available now.
• On your computer, you need software, called a driver, that knows how to communicate with the scanner. Most scanners speak a common language, TWAIN. The TWAIN driver acts as an interpreter between any application that supports the TWAIN standard and the scanner. This means that the application does not need to know the specific details of the scanner in order to access it directly. For example, you can choose to acquire an image from the scanner from within Adobe Photoshop because Photoshop supports the TWAIN standard.

DID YOU KNOW?

TWAIN is not an acronym. It actually comes from the phrase “Never the twain shall meet” because the driver is the go-between for the software and the scanner. Because computer people feel a need to make an acronym out of every term, TWAIN is known as Technology Without An Interesting Name!

 

How Inkjet Printer Works ?

Inkjet printers are a familiar sight in homes and offices, used to print homework, newsletters and photos for the family or quotes, invoices, forms and colour business documents for small businesses. But have you ever stopped to think about how they work? Inkjet printers have functioned in roughly the same way since HP launched its original ThinkJet printer in 1984, but this is changing – and these changes are revolutionising printing.

How do traditional inkjets work?

In a traditional thermal inkjet printer, ink is fed in to thousands of tiny reservoirs in the printhead from the cartridge, then heated rapidly by a tiny resistor, which causes the ink to form a bubble. This bubble then propels the miniscule droplets through a nozzle on to the page, where each forms an equally miniscule dot. These dots form the lines, characters and subtle gradations of colour that we see in a finished printout, whether that’s a simple letter, a 20-page report packed with charts and graphs or a family photo.

In black-and-white prints, dots of black ink, placed precisely in microscopically controlled quantities, create crisp, black text and smooth lines. In full-colour prints, coloured dots of up to eight different coloured inks are layered accurately in patterns or directly on top of one another, creating the impression that there are millions of different colours on the page.

It’s clever stuff, but the traditional inkjet has its limitations. The printhead is very small and can only deposit ink on a tiny section of the paper at a time, which means that the printhead has to travel right to left then left to right across the sheet, printing one line of dots at a time. When the line is finished, the paper transport mechanism – the mechanism that pulls paper in to the printer, under the printhead and in to the output tray – moves the paper in to place for the next line.

It can also impact long-term reliability, as the printhead’s horizontal movement means another mechanism that’s subject to wear and tearThis limits print speeds, since the printer has to wait for the printhead to make its way across the page before the paper transport mechanism can do its stuff. It can also impact long-term reliability, as the printhead’s horizontal movement means another mechanism that’s subject to wear and tear. It can even affect quality, as the motion of the printhead makes it harder to lay down all those dots with such precision.

Inkjet printers have improved dramatically over the last 30 years. They’ve gone from speeds of 2 pages per minute (ppm) to more than 30ppm, while resolutions have leapt from 300 dots per inch (dpi) to 2,400dpi. However, the limitations of the printhead are now holding the inkjet back. They prevent it from printing faster and handling the kind of workloads you’d expect from a laser printer.

A better way

HP’s PageWide technology removes these limitations by replacing the traditional single, moving printhead with an array of printheads that span the width of the page. In a PageWide printhead, each printhead has 1,056 nozzles for each of the four main inks – cyan, magenta, yellow and black – working out at 4,224 nozzles per printhead and 42,240 nozzles in the whole array. These nozzles enable a PageWide printer to print each line in a single burst as the paper transport moves the sheet through underneath.

That’s not just fast for an inkjet – it’s fast for any kind of printerThe resulting printhead does something incredible: it puts thousands of droplets of uniform weight and size on the page at amazing speed and with equally impressive accuracy. This enables PageWide printers, such as those in the HP OfficeJet Pro X range, to print full-colour pages at speeds of up to 70ppm. That’s not just fast for an inkjet – it’s fast for any kind of printer.

What’s more, by eliminating the need for the printhead to travel across the page, HP has made the PageWide printhead more robust and reliable, not to mention better equipped to handle higher monthly workloads. By thinking outside the conventional inkjet box, HP has transformed the way that inkjet printers operate and the tasks they can take on.

Ensuring Great Prints

HP’s first PageWide printers – those in the OfficeJet Pro X line – have been designed to compete directly with laser printers and handle workloads of up to 6,000 pages per month. Yet where a conventional printer has the print head integrated into the ink cartridge, meaning both are replaced periodically, the OfficeJet Pro X has a separate printhead designed to last the entire lifespan of the product.

It achieves this using an optical tracking system that scans both the paper and the drops of ink in flight, looking for variations in alignment and ensuring that each nozzle is depositing the correct amount of ink at the correct time. If not, the printer can intelligently substitute working neighbour nozzles for nozzles that are jammed or misaligned. Meanwhile, an integrated service cassette cleans and conditions the nozzles and refreshes the ink in each reservoir to maintain great performance. Put it all together and you have a printer that keeps pushing out great prints, year after year.

Advanced inkjet technology

The PageWide printhead is a marvel of technology. The print heads are manufactured using the same photolithographic processes used to manufacture microprocessors in a plate thinner than a single human hair, and each print head in the array overlaps the others by 30 nozzles on each side to avoid any artefacts appearing where the printheads join.

It combines with a new precision paper-transport mechanism that maintains a constant speed for the paper underneath the print head and dampens any lateral movement, while using over 300 star wheels – thin metal gears that only touch the paper with their points – to move the paper through the printer without leaving tracks in the ink. Finally, a specially formulated pigment ink improves colour saturation, clarity and definition.

Together, these enhancements are paving the way for a brighter future for inkjet technology, where printers are  faster, more versatile and more capable than ever befor

How does a laser printer work?

You can find a laser printer in almost every office.From the outside, it whirs away, churning out warm, freshly printed pages.

Inside, though, is a lightshow combining laser beams and electrical currents working perfectly together to turn million of bytes worth of information into an image of anything from a diagram of a cell dividing to a fluffy cat. But how does this all happen?

You can thank static electricity.

It all starts when the user presses the print button on their computer, sending a couple million pieces (bytes) of information to a small chip in the printer.

This mini brain converts the data into a two-dimensional image ready to be printed.

Two components are then simultaneously activated that begin the process of turning a digital image to a hard copy.

A large drum, positively charged by a wire known as a corona wire, begins to spin.

This drum, known as a photoreceptor drum, will be the main interface between the ink and the paper. It spins at a speed in time with the movement of the paper.

At the same time, a laser begins to fire at a mirror that reflects the light across the drum millions of times a second.

When the laser beam makes contact with the drum, that precise spot has its positive charge removed, making it negatively charged. This attracts positively charged toner, a fine black or coloured powder.

By passing the laser over the drum again and again, a negatively charged image begins to form.

A roller then applies toner to the drum to a thickness of around 15 microns or about a thousandth of a centimetre.

Toner only “sticks” to the drum where the laser has converted the charge to negative.

Think of it as a rolling pin that has an image traced in glue on it. If the pin rolls over something like flour, it will only stick to where glue has been applied.

COSMOS MAGAZINE

Now comes the time to put that toner onto paper.

As a feeder draws paper from the paper tray, it is given a strong negative charge by another corona wire inside a small drum. This makes the paper a lot more attractive to the toner than the photoreceptor drum, causing it to jump across to the paper.

The paper moves along, attracting the toner, with the full image eventually being transferred.

At this point, you would think the process is complete, but there is still one more important step.

Toner does not dry like ink and therefore is currently only stuck to the paper by the electrostatic charge. So the paper is then passed between two rollers known as the fuser unit, rapidly heating the surface to 200 °C and melting the toner onto the page. This is why paper is hot when it’s fresh out of the printer.

When the print is done, the photoreceptor drum is reset with a new positive charge, the leftover toner is cleaned off and the whole process begins again.

Bresenham’s Line

Program for Bresenham’s Line Drawing Algorithm in C

1.   #include<stdio.h>
2.   #include<graphics.h>
3.
4.          void drawline(int x0, int y0, int x1, int y1)
5.            {
6.                  int dx, dy, p, x, y;
7.                  dx=x1-x0;
8.                  dy=y1-y0;

9.                  x=x0;

10.

11.                y=y0;
12.
13.                p=2*dy-dx;
14.
15.                   while(x<x1)
16.                          {
17.                              if(p>=0)
18.                                 {
19.                                   putpixel(x,y,7);
20.                                   y=y+1;
21.                                  p=p+2*dy-2*dx;
22.                                 }
23.                          else
24.                               {

25.                                 putpixel(x,y,7);
26.                                 p=p+2*dy;
27.                                }
28.                            x=x+1;
29.                          }
30.        }
31.
32.         int main()
33.               {
34.                     int gdriver=DETECT, gmode, error, x0, y0, x1, y1;
35.                     initgraph(&gdriver, &gmode, “c:\\turboc3\\bgi”);
36.
37.                    printf(“Enter co-ordinates of first point: “);
38.                    scanf(“%d%d”, &x0, &y0);
39.
40.                   printf(“Enter co-ordinates of second point: “);
41.                   scanf(“%d%d”, &x1, &y1);
42.                   drawline(x0, y0, x1, y1);
43.
44.                 return 0;
45.      }

1.