mod 3 rg 3.13 AC theory

AC is the most common type of current used in the mains supply, in the field of

power engineering.

In Direct Current (DC), the current flows in the same direction over a period of time (t). Direct current often maintains

the same value over a period of time, but not in every case.

AC has two main characteristics that distinguish it from DC:

• The amount (size) of both the voltage and current changes with time.

• The direction (polarity) of both the voltage and current changes with time.

For common AC, it can also be said that:

• The changes repeat continuously.

• The average voltage and current are both zero.

• The voltage and current vary sinusoidally with time.

the current (I) varies in

direction (polarity) and magnitude, at a point in the circuit, over a period of time (t).

In the positive alternation (positive half-wave), the current flows in one direction from the zero reference line to the positive peak +Ip and from +Ip down to Then the current flows in the opposite direction

falls from 0 to a negative peak value -Ip and rises from -Ip to the zero reference line, as shown in the negative alternation (negative half-wave)

he current changes then repeats periodically. An alternation is equivalent to

one half-cycle.

positive and a negative alternation make up

one AC cycle, as shown in the graphs below. These graphs apply to both voltage and current.

The phase of a periodic function is a measure of

the fraction of the cycle that has been completed at a particular point on the x-axis. This is usually given in degrees or as fractions of π, where 360° or 2π radians represent one cycle.

the phase of the first peak is 90° or π/2 radians, as this occurs at exactly

one-quarter of a cycle from the origin.

The initial angle of a sinusoidal function, at its origin, is called

“phase offset”

In the illustration above, the phase offset is zero (in both degrees and radians) as the cycle begins at the origin.

Phase difference describes

the relative displacement between two points in a cycle (for example, peaks or zero crossings) of two waveforms with the same frequency, again given in degrees or radians as a fraction of π.

The period (T) of a wave is the time taken

to perform one complete cycle

the AC sine waveform repeats with

a period (T) (also called cycle time T).

The unit of (T) is the second (s).

The frequency f of a wave is the number of

complete cycles per second.

The unit of frequency is the hertz (Hz).

frequency is also defined as the reciprocal of the

period or cycle time T

Frequency F = 1/cycle time (T)

The frequency (f) for most aircraft AC systems with a periodic time (t) of 2.5 mS =

400 Hz

The higher the frequency, the shorter the cycle time; the lower the frequency,

the longer the cycle time.

The peak value of the voltage or current in an AC waveform is called

"the amplitude", usually shown as V Vmax I Imax Ipeak

Amplitude is measured from the

zero reference line to the positive or negative peak value of voltage or current.

Amplitude is, therefore, expressed in

volts or amperes.

The instantaneous value of current can be calculated using the formula:

i(t)= Ip sin (a)

The same is valid for the instantaneous value of voltage V:

v(t)= Vp sin (a)

Where v(t) and i(t) are the instantaneous values of voltage and current respectively, and Vp and Ip are the peak values (amplitudes). It is conventional to use lower-case letters for instantaneous values and the brackets indicate that these are

time-dependent values sin(a) is the sine of the phase angle at the time of measurement.

For an AC voltage with a peak value Vp of 340 volts, the instantaneous voltage at a phase angle of 120° can be calculated by substituting these values into the formula

v(t)= Vp sin (a)

= 340sin 120

=340 × 0.866

=294

The average value of current or voltage for a sinusoidal waveform, when calculated over a full cycle, is equal to

zero. this is because a sinusoidal waveform is symmetrical about the X-axis and the values in the positive half cancel the values in the negative half when an average is taken.

work is done in each half cycle and because work has no direction it does not cancel out over a full cycle e.g, heat is transferred to the surroundings whichever way the current flows through a resistor. An approximation to this value can be found by

summing equally spaced instantaneous values of the half cycle of an alternating waveform.

he positive half cycle is equally divided into n number of portions with equal spaces between them. The equally divided portions are called

mid-ordinates and the quotient value of each portion is called the instantaneous value.

to calculate the average voltage, each mid-ordinate value of an alternating waveform is added to its next ordinate’s value; then,

the total is divided by the number of mid-ordinates. This calculation is expressed by the following formula.

Vav= sim of all the mid ordinates/ number of mid ordinates

By using integral calculus, it can be determined that the average value of one alternation is equal to

0.637 Peak.

The Root Mean Square (RMS) value of an AC current or AC voltage is the value which, when applied to a resistance R, will produce

he same power output as an equivalent DC current or DC voltage.

To calculate the RMS value of a sinusoidal AC or AC voltage:

divide by root 2(1.41) or multiply by 1/root 2 (0.707)

Unless otherwise specified, RMS values for AC are normally

quoted

The positive or negative maximum value of the AC current or voltage is called the

peak value

If the voltage of a sine wave 100V at ¼ time (900) then peak value will be

100/ sine90 = 100/1 = 100v

Because the sine wave is in phase (starts at the root of the X-Y axis) the instantaneous value is the peak value at

900

If the peak value is 100V then the Instantaneous value at 45° would be:

sin45 × 100= 70.7 volts

it can be seen how the sine wave can be plotted, as long as

one voltage is known, e.g., the RMS, Peak or 1 Instantaneous value.

Any alternating quantity which is sinusoidal can be represented by a

PHASOR

it is assumed, by convention, that the phasor is rotating in an

anti-clockwise direction at a constant angular velocity (w rad/sec)

The length of the phasor is taken to represent

the peak value of the sine wave.

In the diagram below, (a) represents a phasor LAGGING a reference by an angle and (b) represents a phasor LEADING a reference by an angle

Phasors may be used to represent

two sinusoidal quantities which may have different amplitudes and phases thus the angle between waveforms) A and B could represent 2 voltages, two currents or more usually a voltage and a current in an inductive or capacitive circuit.

The difference between a VECTOR and a PHASOR is:

• A Vector represents a quantity in magnitude and direction.

• A Phasor represents an alternating quantity in peak value and phase.

A sine wave may be represented by a mathematical expression which clearly shows the amplitude and phase of the sine wave. This is necessary so that

circuit calculations can be performed for circuits which are carrying alternating current.

Any sine wave can be described by a mathematical equation as show

V = Vmax Sin(2pie F t + - angle)

the value f will just tell us how

compressed or stretched out the sine wave will be.

Peak-to-peak value means

twice the peak value i.e., the value measured between the positive and negative maximum values of a sine curve. However, this is only true if there is no DC component to the supply i.e., the sine curve is symmetrical about the x-axis.

Periodic voltages or currents, which do not follow a sinusoidal waveform, are referred to as

non-sinusoidal AC voltages or currents.

If the area between the sine curve and the horizontal time axis is the same above the axis as it is below, the supply is still considered to be

pure AC supply.

Some types of AC are classified according to

the shape of the waveform, for example, a square wave indicates the current changing between +Vp and -Vp in a very short time, compared to the time spent at each extreme, whereas a triangular wave indicates a steady rate of change of voltage over each half cycle.

A square waveform is a waveform that

rises quickly to a particular amplitude, remains constant for a time period and drops fast at the end.

In digital systems, square waves are the norm, because they represent

a binary digit (0 or 1). They use Clock pulses to generate and convert them from a sine wave to a square wave

The Frequency of this waveform is calculated by:

f = 1/ ontime + off time

he square wave is symmetrical in shape and has a positive pulse width equal to its negative pulse width resulting in

50% duty cycle.

f the duty cycle of the waveform is any other value than 50%, (half-ON half-OFF) the resulting waveform would then be called

Rectangular Waveform or if the “ON” time is really small, a Pulse.

This is a variation of the square wave waveform, the difference being that the two pulse widths of the waveform are of an unequal time period

Rectangular waveforms are therefore classed as

“Non-symmetrical” waveforms. Often referred to as “The Mark –Space”.

riangular Waveforms are generally

bi-directional non-sinusoidal waveforms that oscillate between a positive and a negative peak value. First, the sine wave has to be squared; converted into a square wave. This is done by applying very high gain amplification.

The square wave goes through or triggers a special circuit, whose important function is applying a constant current source for charging and discharging a capacitor. This produces a perfect linear

Triangular waveform.

Note: The capacitor however has to be valued properly for the frequency being worked with.

A variation of the Triangular waveform is the Sawtooth. The difference between the Triangular and Sawtooth waveforms is that

the rise time of the triangular wave is always equal to its fall time, while in a sawtooth wave generator, the rise time may be much higher than its fall time or vice versa.

As its name suggests, the shape of the waveform resembles the teeth of a saw blade. Sawtoothed waveforms can have a mirror image of themselves, by having either a slow-rising but extremely steep decay, or

an extremely steep almost vertical rise and a slow decay.

n practical terms, the Sawtoothed Waveform is rich in

harmonics and gives music synthesizers and musicians quality of sound or tonal colour to their music, without any distortion.

Any quantity which reverses its direction at regular intervals is said to be an

alternating quantity, e.g., the output of an alternator.

Amplitude, cycle, periodic time, and frequency are terms which are used to describe

the waveform of an alternating quantity.

The instantaneous and the average values of an alternating quantity are of limited practical importance.

• he Root Mean Square (RMS) value of an alternating quantity is the equivalent value of a direct quantity which would produce the same power in a load.

• Peak value equals 1.414 x RMS value, RMS value 0.707 x peak value.

• Unless otherwise stated, all alternating quantities are given as RMS values.

• A Phasor is a line which represents an alternating quantity. The length of the line represents the amplitude and the angle between the line and a reference direction indicates the phase of the quantity.

• A sine wave may be represented mathematically by the formula:

V = Vmax Sin(2pie F t + - angle)

In AC systems both single phase and multi-phase (polyphase) systems are common. Their understanding is needed to grasp the electrical fundamentals of AC technology. However, single-phase is not the only system used, multi-phase systems have been proven to be more

efficient than single-phase systems.

Below shows how a single-phase AC may be generated. The rotors when spinning, place the coils in the stators under the influence of a changing magnetic field, and an EMF is induced into the coils (Faraday’s Law). In the two-pole example, one complete revolution of the rotor will produce a full sine wave. The Peak of the sine wave will be when the rotor is vertical (the 12 and 6 o’clock position) and zero when horizontal (the 3 and 9 o’clock position). In the 8-pole machine, it will produce again a single phase, but the frequency will increase. The number of poles partially governs

the frequency.

Only the number of coils in the stator governs

the phases.

f 2 phases. Here there is a 2 pole rotor, but the stator has 2 windings A and B, 90° apart. One complete revolution of the rotor produce

2 complete sine waves as seen on the graph BUT are 90° displaced from each other.

The Phasor diagram is shown below. Phase A is leading Phase B by 90°. If the two outputs were combined the output would be Vres and would be the vector sum

phases A and B, and a new phase angle difference of (theta)

Multi-phase systems are constructed by placing

three coils around the stator of a generator. Each of these three coils is displaced from the next by an angle of 120°. The armature produces a rotating magnetic field that passes by each of the coils, in turn, causing an AC voltage to be induced across each coil but with the phase difference between each pair of coils of 120°.

Single-phase power is a two-wire alternating current (AC) power circuit. Typically, there is

one power wire, the phase wire, and one neutral wire, with current flowing between the power wire (through the load) and the neutral wire.

Residential homes are usually served by a single-phase power supply but sometimes this can be two phases, while commercial and industrial facilities, including aircraft, usually use a

three-phase supply.

One key difference between single-phase versus three-phase is that a

three-phase power supply better accommodates higher loads. Single-phase power supplies are most used when typical loads are lighting or heating, rather than large electric motors.

Single-phase systems can be derived from

three-phase systems. Aircraft voltage levels are such that a three-phase system can also serve as three single-phase systems.

A single-phase AC power system consists of a generator connected through a pair of wires (the transmission line) to a load. The first diagram below depicts a single-phase two-wire system. It resembles a simple DC circuit. The second diagram shows what is more common in practice - a single-phase three-wire system. It contains two identical sources (equal magnitude and the same phase) which are connected to two loads by two outer wires and the neutral. For example, the normal household system is a single-phase three-wire system because the terminal voltages have the same magnitude and the same phase. Such a system allows

the connection of both 120-V and 240-V appliances.

if three identical coils are mounted on the same former (rotor) with a mutual displacement of 120° between their axes and rotated anti-clockwise at constant speed in a uniform magnetic field, the emf’s induced into the windings will be

sinusoidal and identical except that there will be mutual phase displacement of 120° between them.

These three coils constitute a three-phase system. The phases are usually identified by colours, RED, YELLOW, and BLUE, and the normal phase sequence is R - Y - B. On aircraft, the phases are usually

A - B - C.

If the phases are kept entirely separate then six wires are required, but if the phases are interconnected then only three or at most four wires are required. On aircraft

the 4 or 6-wire systems are the most common, depending on the supply-demand requirements.

Where three sets of values are being generated by the same AC generator at almost the same time and separated only by the angular difference between them (a three-phase system), they can be connected in either of two ways, known as

Delta and Star. The connection used will determine the output voltage and the effect on the system of a phase line-to-earth or line-to-line fault.

One end of each winding is connected to a common, star or neutral point. The three output lines are taken from the other ends of the windings.

A neutral, fourth wire, or earth connection (if required) is also made at the star point.

in a star connection how many voltages are available

here are six voltages available. Between any line and neutral is the PHASE VOLTAGE ( Vph). There are three phase voltagesVr, Vy and VB, all equal in magnitude with a mutual phase displacement of 120°.

Between any two lines is the LINE VOLTAGE ( Lv). There are three line voltages Vry, Vyb and Vbr, all equal in magnitude with a mutual phase displacement of 120°.

The relationship between Line (L) and Phase Voltages (ph) is:

• The voltage between any two lines is the phasor difference between the phase voltages of those lines.

VL = square root 3Vph and IL= Iph

in a delta connection the windings are connected from the

‘finish of one’ to the ‘start of the next’ to form a closed loop. The three output lines are taken from the junctions

in a delta is is there a neutral

There is no neutral or designated earth

IL = square root 3Iph and VL= Vph

in a delta connection each phase current is the difference between

two line currents, but the voltages do not change.

The animation of a delta connection below shows the flow of current through each part of the circuit. The return path for the current through each load is through the other two loads. Because the loads are balanced, and the phase difference between each phase is 120°,

no current flows in the neutral line.

Two different voltages can be measured in aircraft three-phase systems: one phase referenced to the ground is 115 V, and the voltage measured phase-to-phase is 200 V.

Therefore, an advantage of three-phase arrangements is that there are

two different voltages available.

All three phases are connected to a common return line (neutral line), which, in practice, would be

the airframe

If all three phases have the same loading, and there is no phase shift in the circuits, there will be

here will be no current in the common return line (although there will be a current flowing between the common connections of the three loads).

If there is a complete imbalance in the three loads but they are all purely ohmic loads (real loads, with no capacitive or inductive component), there could be

current flowing in the neutral line, up to that of one phase of the generator.

If there is a completely unbalanced load with a phase shift, the current in the neutral line could be up to the value of

two phase currents.This system allows the designer to place a component between one phase and neutral and get 115 V, or between two phases and get 200 V.

It is not possible to connect all three phases, as this would cause

a short circuit of the generator.