Effects at High Frequencies

A direct current flows constantly and uniformly throughout the cross-section of a uniform wire. An alternating current of any frequency is forced away from the wire's center, toward its outer surface. This is because the acceleration of an electric charge in an alternating current produces waves of electromagnetic radiation that cancel the propagation of electricity toward the center of materials with high conductivity. This phenomenon is called skin effect.

At very high frequencies the current no longer flows in the wire, but effectively flows on the surface of the wire, within a thickness of a few skin depths. The skin depth is the thickness at which the current density is reduced by 63%. Even at relatively low frequencies used for high power transmission (50–60 Hz), non-uniform distribution of current still occurs in sufficiently thick conductors. For example, the skin depth of a copper conductor is approximately 8.57 mm at 60 Hz, so high current conductors are usually hollow to reduce their mass and cost.

Since the current tends to flow in the periphery of conductors, the effective cross-section of the conductor is reduced. This increases the effective AC resistance of the conductor, since resistance is inversely proportional to the cross-sectional area in which the current actually flows. The AC resistance often is many times higher than the DC resistance, causing a much higher energy loss due to ohmic heating (also called I^2R loss).



Techniques for reducing AC resistance

For low to medium frequencies, conductors can be divided into stranded wires, each insulated from one other, and the relative positions of individual strands specially arranged within the conductor bundle. Wire constructed using this technique is called Litz wire. This measure helps to partially mitigate skin effect by forcing more equal current flow throughout the total cross section of the stranded conductors. Litz wire is used for making high-Q inductors, reducing losses in flexible conductors carrying very high currents at lower frequencies, and in the windings of devices carrying higher radio frequency current (up to hundreds of kilohertz), such as switch-mode power supplies and radio frequency transformers.

Techniques for reducing radiation loss

As written above, an alternating current is made of electric charge under periodic acceleration, which causes radiation of electromagnetic waves. Energy that is radiated represents a loss. Depending on the frequency, different techniques are used to minimize the loss due to radiation.

Twisted pairs

At frequencies up to about 1 GHz, pairs of wires are twisted together in a cable, forming a twisted pair. This reduces losses from electromagnetic radiation and inductive coupling. A twisted pair must be used with a balanced signalling system, so that the two wires carry equal but opposite currents. Each wire in a twisted pair radiates a signal, but it is effectively cancelled by radiation from the other wire, resulting in almost no radiation loss.

Coaxial cables

At frequencies above 1 GHz, unshielded wires of practical dimensions lose too much energy to radiation, so coaxial cables are used instead. A coaxial cable has a conductive wire inside a conductive tube, separated by a dielectric layer. The current flowing on the inner conductor is equal and opposite to the current flowing on the inner surface of the tube. The electromagnetic field is thus completely contained within the tube, and (ideally) no energy is radiation or coupling outside the tube. Coaxial cables have acceptably small losses for frequencies up to about 20 GHz. For microwave frequencies greater than 20 GHz, the losses (due mainly to the dissipation factor of the dielectric) become too large, making waveguides a more efficient medium for transmitting energy.

Waveguides

Waveguides are similar to coax cables, as both consist of tubes, with the biggest difference being that the waveguide has no inner conductor. Waveguides can have any arbitrary cross section, but rectangular cross sections are the most common. Because waveguides do not have an inner conductor to carry a return current, waveguides cannot deliver energy by means of an electric current, but rather by means of a guided electromagnetic field. Although surface currents do flow on the inner walls of the waveguides, those surface currents do not carry power. Power is carried by the guided electromagnetic fields. The surface currents are set up by the guided electromagnetic fields and have the effect of keeping the fields inside the waveguide and preventing leakage of the fields to the space outside the waveguide.

Waveguides have dimensions comparable to the wavelength of the alternating current to be transmitted, so they are only feasible at microwave frequencies. In addition to this mechanical feasibility, electrical resistance of the non-ideal metals forming the walls of the waveguide cause dissipation of power (surface currents flowing on lossy conductors dissipate power). At higher frequencies, the power lost to this dissipation becomes unacceptably large.

Fiber optics

At frequencies greater than 200 GHz, waveguide dimensions become impractically small, and the ohmic losses in the waveguide walls become large. Instead, fiber optics, which are a form of dielectric waveguides, can be used. For such frequencies, the concepts of voltages and currents are no longer used.

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Utility Frequency (Power Frequency)

The line frequency (American English) or mains frequency (British English) is the frequency at which alternating current (AC) is transmitted from a power plant to the end user. In most parts of the world this is 50 Hz, although in the Americas it is typically 60 Hz. Precise details are shown in the list of countries with mains power plugs, voltages and frequencies.

During the development of commercial electric power systems in the late 19th and early 20th centuries, many different frequencies (and voltages) had been used. Large investment in equipment at one frequency made standardization a slow process. However, as of the turn of the 21st century, places that now use the 50 Hz frequency tend to use 220-240 V, and those that now use 60 Hz tend to use 100-120 V. Both frequencies co-exist today (some countries such as Japan use both) with no technical reason to prefer one over the other and no apparent desire for complete worldwide standardization.


Figure 1. The waveform of 230 volt, 50 Hz compared with 110 V, 60 Hz.

Unless specified by the manufacturer to operate on both 50 and 60 Hz, appliances may not operate efficiently or even safely if used on anything other than the intended frequency.

Operating factors

Several factors influence the choice of frequency in an AC system. Lighting, motors, transformers, generators and transmission lines all have characteristics which depend on the power frequency.

All of these factors interact and make selection of a power frequency a matter of considerable importance. The best frequency is a compromise between contradictory requirements. In the late 19th century, designers would pick a relatively high frequency for systems featuring transformers and arc lights, so as to economize on transformer materials, but would pick a lower frequency for systems with long transmission lines or feeding primarily motor loads or rotary converters for producing direct current. When large central generating stations became practical, the choice of frequency was made based on the nature of the intended load. Eventually the improvements in machine design allowed a single frequency to be used both for lighting and motor loads; a unified system improved the economics of electricity production since system load was more uniform during the course of a day.

Lighting

The first applications of commercial electric power were incandescent lighting and commutator-type electric motors. Both devices operate well on DC, but DC cannot be easily transmitted long distances at utilization voltage and also cannot be easily changed in voltage.

If an incandescent lamp is operated on a low-frequency current, the filament cools on each half-cycle of the alternating current, leading to perceptible change in brightness and flicker of the lamps; the effect is more pronounced with arc lamps, and the later mercury-vapor and fluorescent lamps.

Rotating machines

Commutator-type motors do not operate well on high-frequency AC since the rapid changes of current are opposed by the inductance of the motor field; even today, although commutator-type universal motors are common in 50 Hz and 60 Hz household appliances, they are small motors, less than 1 kW. The induction motor was found to work well on frequencies around 50 to 60 Hz but with the materials available in the 1890s would not work well at a frequency of, say, 133 Hz. There is a fixed relationship between the number of magnetic poles in the induction motor field, the frequency of the alternating current, and the rotation speed; so, a given standard speed limits the choice of frequency (and the reverse). Once induction motors became common, it was important to standardize frequency for compatibility with the customer's equipment.

Generators operated by slow-speed reciprocating engines will produce lower frequencies, for a given number of poles, than those operated by, for example, a high-speed steam turbine. For very slow prime mover speeds, it would be costly to build a generator with enough poles to provide a high AC frequency. As well, synchronizing two generators to the same speed was found to be easier at lower speeds. While belt drives were common as a way to increase speed of slow engines, in very large ratings (thousands of kilowatts) these were expensive, inefficient and unreliable. Direct-driven generators off steam turbines after about 1906 favored higher frequencies. The steadier rotation speed of high-speed machines allowed for satisfactory operation of commutators in rotary converters.

Direct-current power was not entirely displaced by alternating current and was useful in railway and electrochemical processes. Prior to the development of mercury arc valve rectifiers, rotary converters were used to produce DC power from AC. Like other commutator-type machines, these worked better with lower frequencies.

Transmission and transformers

With AC, transformers can be used to step down high transmission voltages to lower utilization voltage. Since, for a given power level, the dimensions of a transformer are roughly inversely proportional to frequency, a system with many transformers would be more economical at a higher frequency.

Electric power transmission over long lines favors lower frequencies. The effects of the distributed capacitance and inductance of the line are less at low frequency.

System interconnection

Generators can only be interconnected to operate in parallel if they are of the same frequency and wave-shape. By standardizing the frequency used, generators in a geographic area can be interconnected in a grid, providing reliability and cost savings.

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AC Generation Analysis

Analysis of the AC power generation process and of the alternating current we use in almost every aspect of our lives is necessary to better understand how AC power is used in today’s technology.

Effective Values

The output voltage of an AC generator can be expressed in two ways. One is graphically by use of a sine wave (Figure 1). The second way is algebraically by the equation e = Emax sin wt, which will be covered later in the text.


Figure 1 Voltage Sine Wave

When a voltage is produced by an AC generator, the resulting current varies in step with the voltage. As the generator coil rotates 360°, the output voltage goes through one complete cycle. In one cycle, the voltage increases from zero to Emax in one direction, decreases to zero, increases to Emax in the opposite direction (negative Emax), and then decreases to zero again. The value of Emax occurs at 90° and is referred to as peak voltage. The time it takes for the generator to complete one cycle is called the period, and the number of cycles per second is called the frequency (measured in hertz).

One way to refer to AC voltage or current is by peak voltage (Ep) or peak current (Ip). This is the maximum voltage or current for an AC sine wave.

Another value, the peak-to-peak value (Ep-p or Ip-p), is the magnitude of voltage, or current range, spanned by the sine wave. However, the value most commonly used for AC is effective value. Effective value of AC is the amount of AC that produces the same heating effect as an equal amount of DC. In simpler terms, one ampere effective value of AC will produce the same amount of heat in a conductor, in a given time, as one ampere of DC. The heating effect of a given AC current is proportional to the square of the current. Effective value of AC can be calculated by squaring all the amplitudes of the sine wave over one period, taking the average of these values, and then taking the square root. The effective value, being the root of the mean (average) square of the currents, is known as the root-mean-square, or RMS value. In order to understand the meaning of effective current applied to a sine wave, refer to Figure 2.

The values of I are plotted on the upper curve, and the corresponding values of I^2 are plotted on the lower curve. The I2 curve has twice the frequency of I and varies above and below a new axis. The new axis is the average of the I^2 values, and the square root of that value is the RMS, or effective value, of current. The average value is ½ Imax^2. The RMS value is then, 2^(1/2)Imax^2/ 2 or 2^(1/2)/2 x Imax, which is equal to 0.707 Imax.
There are six basic equations that are used to convert a value of AC voltage or current to another value, as listed below.

1. Average value = peak value x 0.637……………….(1)
2. Effective value (RMS) = peak value x 0.707………(2)
3. Peak value = average value x 1.57…………………(3)
4. Effective value (RMS) = average value x 1.11…….(4)
5. Peak value = effective value (RMS) x 1.414………(5)
6. Average value = effective (RMS) x 0.9……………(6)

The values of current (I) and voltage (E) that are normally encountered are assumed to be RMS values; therefore, no subscript is used.


Figure 2 Effective Value of Current

Another useful value is the average value of the amplitude during the positive half of the cycle.
Equation (7) is the mathematical relationship between Iav , Imax , and I.

Iav = 0.637 Imax = 0.90 I………….(7)

Equation (8) is the mathematical relationship between Eav , Emax , and E.

Eav = 0.637 Emax = 0.90 E……….(8)

Example 1: The peak value of voltage in an AC circuit is 200 V. What is the RMS value of the voltage?
Solution:
E = 0.707 Emax = 0.707 (200 V) = 141.4 V

Example 2: The peak current in an AC circuit is 10 amps. What is the average value of current in the circuit?
Solution:
Iav = 0.637 Imax = 0.637 (10 amps) = 6.37 amps

Phase Angle

Phase angle is the fraction of a cycle, in degrees, that has gone by since a voltage or current has passed through a given value. The given value is normally zero. Referring back to Figure 1, take point 1 as the starting point or zero phase. The phase at Point 2 is 30°, Point 3 is 60°, Point 4 is 90°, and so on, until Point 13 where the phase is 360°, or zero. A term more commonly used is phase difference. The phase difference can be used to describe two different voltages that have the same frequency, which pass through zero values in the same direction at different times. In Figure 3, the angles along the axis indicate the phases of voltages e1 and e2 at any point in time. At 120°, e1 passes through the zero value, which is 60° ahead of e2 (e2 equals zero at 180°). The voltage e1 is said to lead e2 by 60 electrical degrees, or it can be said that e2 lags e1 by 60 electrical degrees.


Figure 3 Phase Relationship

Phase difference is also used to compare two different currents or a current and a voltage. If the phase difference between two currents, two voltages, or a voltage and a current is zero degrees, they are said to be "in-phase." If the phase difference is an amount other than zero, they are said to be "out-of-phase."

Voltage Calculations

Equation (9) is a mathematical representation of the voltage associated with any particular orientation of a coil (inductor).

e = Emax sin teta........(9)

where
e = induced EMF
Emax = maximum induced EMF
 (teta)= angle from reference (degrees or radians)

Example 1: What is the induced EMF in a coil producing a maximum EMF of 120 V when the angle from reference is 45°?
e = Emax sin teta= 120 V (sin 45°)= 84.84 V

The maximum induced voltage can also be called peak voltage Ep. If (t) is the time in which the coil turns through the angle, teta (), then the angular velocity, w () of the coil is equal to teta/t and is expressed in units of radians/sec. Equation (10) is the mathematical representation of the angular velocity.
teta = wt………….(10)

where
,teta= angular velocity (radians/sec)
t = time to turn through the angle from reference (sec)
,w= angle from reference (radians)

Using substitution laws, a relationship between the voltage induced, the maximum induced voltage, and the angular velocity can be expressed. Equation (11) is the mathematical representation of the relationship between the voltage induced, the maximum voltage, and the angular velocity, and is equal to the output of an AC Generator.

e = Emax sin (wt)……………(11)

where
e = induced EMF (volts)
Emax = maximum induced EMF (volts)
, w = angular velocity (radians/sec)
t = time to turn through the angle from reference (sec)

Current Calculations

Maximum induced current is calculated in a similar fashion. Equation (7-12) is a mathematical representation of the relationship between the maximum induced current and the angular velocity.

i = Imax sin (wt)…………(12)

where
i = induced current (amps)
Imax = maximum induced current (amps)
= ang, wular velocity (radians/sec)
t = time to turn through the angle from reference (sec)

Frequency Calculations

The frequency of an alternating voltage or current can be related directly to the angular velocity of a rotating coil. The units of angular velocity are radians per second, and 2radians is a full revolution. A radian is an angle that subtends an arc equal to the radius of a circle. One radian equals 57.3 degrees. One cycle of the sine wave is generated when the coil rotates 2phi (2 x 3.14) radians. Equation (13) is the mathematical relationship between frequency (f) and the angular velocity, w () in an AC circuit.

w = 2phi.f --> 2 x 3.14 x f………….(13)
where
, w= angular velocity (radians/sec)
f = frequency (HZ)

Example 1: The frequency of a 120 V AC circuit is 60 Hz. Find the following:
1. Angular velocity
2. Angle from reference at 1 msec
3. Induced EMF at that point

Solution:
1. w= 2phi.f = 2 (3.14) (60 Hz) = 376.8 radians/sec
2. teta = wt = (376.8 radian/sec) (.001 sec) = 0.3768 radians
3. e = Emax sin teta= (120 V) (sin 0.3768 radians)= (120 V) (0.3679)= 44.15 V

Summary

Voltage, Current, and Frequency Summary

• The following terms relate to the AC cycle: radians/second, the velocity the loop turns; hertz, the number of cycles in one second; period, the time to complete one cycle.
• Effective value of AC equals effective value of DC.
• Root mean square (RMS) values equate AC to DC equivalents:
- I = 0.707 Imax = Effective Current
- E = 0.707 Emax = Effective Voltage
- Iav = 0.636 Imax = 0.9 I = Average Current
- Eav = 0.636 Emax = 0.9 E = Average Voltage
• Phase angle is used to compare two wave forms. It references the start, or zero point, of each wave. It compares differences by degrees of rotation. Wave forms with the same start point are "in-phase" while wave forms "out-of-phase" either lead or lag.

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AC Generation

An understanding of how an AC generator develops an AC output will help us analyze the AC power generation process.

The elementary AC generator (Figure 1) consists of a conductor, or loop of wire in a magnetic field that is produced by an electromagnet. The two ends of the loop are connected to slip rings, and they are in contact with two brushes. When the loop rotates it cuts magnetic lines of force, first in one direction and then the other.

Development of a Sine-Wave Output


Figure 1 Simple AC Generator

At the instant the loop is in the vertical position (Figure 2, 0o), the coil sides are moving parallel to the field and do not cut magnetic lines of force. In this instant, there is no voltage induced in the loop. As the coil rotates in a counter-clockwise direction, the coil sides will cut the magnetic lines of force in opposite directions. The direction of the induced voltages depends on the direction of movement of the coil. The induced voltages add in series, making slip ring X (Figure 1) positive (+) and slip ring Y (Figure 1) negative (-). The potential across resistor R will cause a current to flow from Y to X through the resistor. This current will increase until it reaches a maximum value when the coil is horizontal to the magnetic lines of force (Figure 2, 90o). The horizontal coil is moving perpendicular to the field and is cutting the greatest number of magnetic lines of force. As the coil continues to turn, the voltage and current induced decrease until they reach zero, where the coil is again in the vertical position (Figure 2, 180o). In the other half revolution, an equal voltage is produced except that the polarity is reversed (Figure 2, 270o, 360o). The current flow through R is now from X to Y (Figure 1).



Figure 2 Developing a Sine-Wave Voltage

The periodic reversal of polarity results in the generation of a voltage, as shown in Figure 2. The rotation of the coil through 360° results in an AC sine wave output.

Summary

• A simple generator consists of a conductor loop turning in a magnetic field, cutting across the magnetic lines of force.
• The sine wave output is the result of one side of the generator loop cutting linesof force. In the first half turn of rotation this produces a positive current and in the second half of rotation produces a negative current. This completes one cycle of AC generation.

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