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1. An introduction to AC electrical theory: Part 1.
Contents:
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1-1. Basic electrical formulae. 1-2. Resonance. 1-3. Fields. 1-4. Impedance, resistance, and reactance. |
1-5. Vectors and scalars. 1-6. Phasors. 1-7. Voltage magnification and Q. 1-8. Power factor and scalar product. |
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1-1. Basic electrical formulae: The table below summarises some of the basic electrical formulae (or formulas) as they appear in standard textbooks: |
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The first point to note about the contents of this table is that
only the entries in the left-hand column contain fundamental
scientific information. The uppermost entry is a mathematical
statement of Ohm's law; which is that the electrical
current which flows in a conductor is proportional to the electrical
pressure (voltage) exerted on the conductor, the constant of
proportionality being known as the resistance. The entry below
it is the power law; which represents the observation that a
conductor (resistor) heats up as a consequence of an electric
current flowing in it (i.e., it dissipates energy) and the power
consumed (i.e., the energy delivered per unit-of-time) is the
product of the current flowing and the voltage applied, i.e.,
"P=I´V". Also, by
using the substitutions "I=V/R" and "V=IR",
we obtain two alternative power laws: "P=V²/R"
and "P=I²R". The expression "P=I²R"
is known as Joule's law [1],
and is a statement of the fundamental relationship between electricity
and thermodynamics. One important point to note about the standard
power and resistance formulae however, is that they are all derived
from experiments with DC electricity. They represent incomplete
statements of Ohm's law and Joule's law, because they can
only be applied to AC circuits when the load on the generator
is a pure resistance. Later-on in this chapter we will show how
to state these laws in a completely general way, but some groundwork
will be required before that can be done. In the case of inductors and capacitors, the entries in the left-hand column tell us that they also obey Ohm's law when connected to a generator of alternating voltage but, insofar as we can construct them without inadvertently including resistance, they consume no power. In the case of an inductor, the AC resistance or reactance X (measured in Ohms) is directly proportional to the inductance (in Henrys) and to the frequency f (in Hertz, i.e., cycles per second)) of the applied voltage. The quantity 2pf is known as the angular frequency (i.e., the frequency in radians per second, where 2p radians corresponds to 360°) and in other texts it is often given the symbol w (the Greek lower-case letter "omega"). In the case of a capacitor, the reactance is inversely proportional to the angular frequency, and also inversely proportional to the capacitance (in Farads). Note also that capacitive reactance is shown as being negative; because we know that when capacitors and inductors are connected to form tuned circuits, the reactance of the inductor, in some sense, cancels the reactance of the capacitor. This means that one of the types of reactance has to be considered to be negative and, as will be explained later, we choose it to be the capacitive variety in order to be consistent with the conventions of trigonometry. All of the other entries in the table are derived from the entries in the left-hand column, using only Ohm's law and a basic electrical rule known as Kirchhoff's first law [1][2] (pronounced: "kir-khov"). Kirchhoff's law tells us that the sum of all the currents flowing into a given point in a circuit is equal to the sum of all the currents flowing out. This law is correct for DC electricity, because direct current turns out to be a measure of the amount of electrical charge passing a given point in a conductor per unit-of-time (1 Ampere = 1 Coulomb per second, where 1 Coulomb = 6.241460122´10  
electrons). In DC theory therefore, Kirchhoff's law is a statement
of the principle of conservation of electrical charge (what goes
in must come out). The simple idea of electrons flowing in a
pipe will not suffice at radio-frequencies however, because the
average drift velocity of electrons in a conductor is extremely
slow in comparison to the velocity of propagation of electromagnetic
radiation, and ultimately we can only understand electrical circuits
by regarding them as waveguides or 'transmission lines'. The
concept of current can be made to hold good at any frequency
however, even though it cannot be associated with electron flow.
Electromagnetic energy, which is the actual substance in transit,
defies Kirchhoff by leaking out of the circuit all over the place
(and also by being absorbed), but we will later account for the
discrepancy by invoking invisible circuit elements such as stray
capacitance, magnetic induction, and radiation resistance, which
fix Kirchhoff's law and allow us to extend the theory of DC electricity
to cover AC. Therefore we can take Kirchhoff's law to be true,
provided that we are aware that accurate circuit analysis at
high frequencies requires the inclusion of the stray or 'parasitic'
components into the circuit to produce an equivalent circuit
which describes the actual behaviourThe series and parallel combination formulae may all be regarded as examples of simple mathematical models; a mathematical model being an equation or a set of equations which can be used to describe a physical system. Of these, the formula for resistances in series is the simplest of all, and tells us that whenever we encounter two resistors in series, we can treat them as a single resistor with a value equal to the sum of the two resistances. That this statement is derived from existing physical laws can be seen by applying basic techniques of circuit analysis to the circuit shown below: |
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Resistors is Series To analyse this circuit, we first observe that, as a requirement of Kirchhoff's first law, the current flowing in the two resistors must be the same. Ohm's law then tells us that V1=I´R1 and V2=I´R2. Now, since voltage is a measure of electrical pressure, common sense (otherwise known as Kirchhoff's second law [1][2]) tells us that the total pressure-drop is equal to the sum of the pressure-drops across the two resistors, i.e., V = V1 + V2 Putting these ideas together we have: V = I R1 + I R2 = I (R1 + R2) |
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Now, if we postulate a hypothetical resistance R which represents
the series combination of R1 and R2, it must be possible to replace R1
and R2 with this resistance and obtain
the same current for a given voltage, i.e., V = I R = I (R1 + R2) Hence:
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Resistors in Parallel In the case of two resistors in parallel, the voltage across the two resistors is the same. Hence Ohm's law tells us that: |
I = I1 + I2 Now, if we postulate a hypothetical resistance R which represents the parallel combination of R1 and R2, we have: V = I R = (I1 + I2) R We can eliminate I1 and I2 by using equation (1-1.1) above, i.e., I1=V/R1 and I2=V/R2, hence: V= [ (V/R1) + (V/R2) ] R |
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The voltage can then be factored out and cancelled to give: 1 = [ (1/R1) + (1/R2) ] R and dividing each side of the equation by R gives:
R = 1 / [ (1/R1) + (1/R2) ] We then arrange the terms inside the square brackets to have a common denominator (multiply the 1/R1 term by R2/R2 and multiply the 1/R2 term by R1/R1), i.e., R = 1 / [ {(R2/(R1R2)} + {R1/(R1R2)} ] hence: R = 1 / [ (R1 + R2 ) / R1 R2 ] which, upon inversion, gives:
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| The formulae for inductors and capacitors in series and parallel may also be derived by using exactly the same arguments as were used above; the only difference being that inductive reactance XL=2pfL, or capacitive reactance XC=-1/(2pfC), is substituted in place of resistance. The 2pf factors and any minus signs disappear by cancellation, leaving formulae involving only inductance or capacitance. Note incidentally, that the inductors in the illustrations in the table are shown orientated at right-angles to each other, this being done as a reminder that the formulae are only true when there is no magnetic coupling between the coils. Note also, that the capacitor formulae take on the opposite forms of their resistance counterparts; this being due to the reciprocal (inverse) relationship between capacitance and capacitive reactance. |
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1-3. Fields: In any explanation of the physical world, it is necessary to evoke mysterious entities called 'fields'. In radio, of course, we talk freely of electric fields, magnetic fields, and electromagnetic fields; but many people find themselves at a loss for words when asked to give a definition of the basic term [3]. In any endeavour, the lack of clear understanding can lead to the wrong conclusions; and so we will attempt to forestall the problem by stating exactly what we mean. The term 'field' has a vernacular meaning: "region of influence", and in physics, the conception is pretty much the same. Specifically, in the context of physical forces, a field can be defined as "a volume of space under the influence of some agent which acts without physical contact"; the agents in question being gravity, electricity, magnetism, and for those interested in nuclear physics; the weak force and the strong force. Fields are mysterious in the sense that they beg the question, "why or how does action at a distance occur?"; but as soon as we accept that they exist, we find them reassuringly subservient to the principle of conservation of energy. Energy is the currency of the universe: it is neither created nor destroyed but merely converted from one form into another. The point is that the storage and exchange of energy is mediated by fields, and what is meant by this in the context of static (unchanging) fields is something purely mechanical. If a body or weight is held above the ground, it is said to have potential energy; i.e., it has the potential to do some work. We might, for example, attach it to a chain and sprocket and use it to power a clock. Few people would find such a device perplexing, and yet the clock is running from a source of gravitational energy. In a universe without gravitational fields, there would have to be a stretched spring attached between the weight and the floor to draw the two together, and yet we may pass a hand under the weight to verify that no invisible spring is there. The situation with other fields is exactly analogous, and we may take the static electric field in a capacitor as an example. When a capacitor is charged, there exists a surfeit of positive charges on one plate and a surfeit of negative charges on the other. The two plates therefore experience a strong force of attraction, and would fall towards each other were it not for mechanical constraints on their movement. The act of charging the capacitor therefore gives it real mechanical potential energy, and this is indicated by the voltage which is present across the capacitor terminals. It is for this reason that 'voltage' and 'potential' are synonymous in any discussion of electricity (although we would need to engage in arguments involving point-charges to work out exactly why). A capacitor stores energy by creating an electrical attraction between two objects which are physically separated (in this case we refer to a simple capacitor without intervening insulator or 'dielectric'). The drive-mechanism of a long-case clock stores energy by creating a physical separation between two objects which are gravitationally attracted (i.e., the weight and the Earth). Fields are involved in the storage of energy because they mediate the attractions or repulsions between objects which give those objects the potential to perform work. This is what we need to know in order to understand the DC properties of coils and capacitors; but a certain subtlety arises when we note that fields can be created by putting energy into a system and destroyed by removing it. If that is the case, then energy can be considered to be stored in the field itself. This might have proved to be a mere semantic distinction, were it not for the work of James Clerk Maxwell (1831-1879), who realised that oscillating electric and magnetic fields could exist independently of matter. He showed that an oscillating electric field could be combined with an oscillating magnetic field of the same frequency in such a way that a constant amount of energy could be stored by continual transfer of energy from one field to the other. Such an object, made of pure energy, would propagate through space as a wave with a velocity given by the expression: v = 1/Ö(me) where m (mu) is the inductance per unit-of-length or magnetic permeability and e (epsilon) is the capacitance per unit-of-length or electric permittivity of the region in which the wave is travelling. Permeability m is a constant of proportionality obtained from the relationship between the physical dimensions of a coil of wire and its inductance. Permittivity e is a constant of proportionality obtained from the relationship between the physical dimensions of a capacitor and its capacitance. Maxwell observed that the best available measurements for the permeability and permittivity of empty space, m0 and e0 ("mu nought" and "epsilon nought") gave his electromagnetic waves a velocity v=1/Ö(m0e0) which turned out to be the same as the speed of light. From this he deduced that light is electromagnetic radiation, a theory which he held to be "great guns", and so it proved to be. From Maxwell's deduction that oscillating electric and magnetic fields can form electromagnetic waves, there follows a converse deduction that oscillating electric and magnetic fields cannot exist independently; i.e., all electric or magnetic energy in transit is in the form of electromagnetic waves. This means that all electrical devices, contrary to the ideas of DC theory, are in reality optical devices which can be combined to guide and manipulate electromagnetic energy. This is the most authoritative explanation for the behaviour of electrical circuits, and we must be mindful of it in the process of developing an AC electrical theory. What we will find is that an explanation of AC circuit behaviour can be obtained by extending DC theory, but it will often throw up inconsistencies which can only be resolved by recourse to electromagnetic theory. In this chapter we will derive the basic AC theory, which deals with notionally separate resistances, capacitances and inductances, these being known as idealised components. In later chapters we will show that capacitance and inductance cannot exist independently, and resistance cannot exist independently of them; but we can largely correct for the non-ideal behaviour of practical components by representing then as equivalent circuits composed of ideal components. This technique is known as lumped component circuit analysis. It is inherent in the lumped component approach that the speed of light is infinite, i.e., that all parts of a circuit are in instant communication. This simplification will sometimes let us down, and when it does, we will need to represent parts of our circuits as transmission lines, these being devices, either hypothetical or actual, which have a time-delay associated with them. |
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1-4. Impedance, Resistance, and Reactance: The basic electrical laws discussed earlier tell us that resistors consume power when connected across a generator but that perfect inductors and capacitors do not. The combination formulae then tell us how to deal with resistances or reactances in series and parallel; but they do not tell us how to deal with combinations of resistance and reactance. This is a serious limitation, which can only be overcome by introducing the generalised concept of impedance, i.e., the theory of two-terminal devices which obey Ohm's law but do not necessarily consume all of the electrical power delivered to them. A concept which needs to be introduced at this point is that of a linear, passive, two terminal network. An electrical device is linear if the graph of voltage across it against current passing through it is a straight line, i.e., if it obeys Ohm's law; and it is passive if it contains no sources of energy. The general term 'network' is used because, although the theory we are about to develop still covers 'simple' devices like capacitors and resistors, it also covers any combination (actual or hypothetical) of resistances and reactances in series and parallel connected to a single pair of terminals. A network can be hypothetical in the sense that it behaves in the same way as (i.e., serves as a model for) a real two terminal device; and, in particular, when an antenna system is connected as a load to a transmitter, we can treat it as a hypothetical network of resistances and reactances. An antenna, incidentally, is not completely passive, because it also receives radio signals, but we can model the receiving case by considering it to be exactly the same network as in the transmitting case, but with a generator connected in series with it. Any linear passive two-terminal network can be regarded as an impedance. This means that its electrical behaviour at a particular frequency can be explained by invoking two (and only two) mutually independent properties; namely resistance and reactance. Resistance R is that property of the network which enables it to dissipate (i.e., consume or dispose of) energy, and reactance X is that property which enables it to store energy. It is also a special property of our universe that, while energy dissipation is always a one-way trip, energy storage mechanisms tend to come in pairs, positive and negative. This means that resistance is always positive (a statement which we will qualify later), but there are two opposing types of reactance, which of course we know as inductive reactance, XL=2pfL, and capacitive reactance, XC=-1/(2pfC). Inductive reactance arises through the storage of energy in a magnetic field, and capacitive reactance through the storage of energy in an electric field. When inductance, capacitance, and resistance are combined within the same two-terminal 'black box', the opposing reactances will always tend to cancel-out to some degree, and so the two types of reactance make only a single contribution to the impedance at any particular frequency. There is however, no way in which resistance and reactance can be combined to form a single numerical quantity, because the physical processes they represent are mutually exclusive. It so happens that the independence of resistance and reactance is all we need to know in order to derive the whole of AC electrical theory, i.e., we have found the fundamental quantities which define impedance, and the mathematical fix which enables us to generalise Ohm's law to cover networks containing both reactance and resistance. For DC circuits, we can write Ohm's law as "V=IR". For AC circuits therefore, we must suspect that we can write something along the lines of "V=IZ", as long as we recognise that impedance, Z, our generalised form of resistance, must be represented by some composite quantity containing two distinct elements R and X. In circumstances such as these, it is traditional to see if anyone has developed a branch of mathematics which enables us to deal with the problem, and the clue regarding where to look lies in the independence of R and X. If two quantities are completely independent, they must in some sense exist in different dimensions (i.e., they always move at right-angles to each other). This means that impedance cannot be represented by an ordinary number, i.e., a one-dimensional quantity lying on a line between -¥ and +¥, it must be represented by a point on a two-dimensional plane, which is another way of saying that Z can be plotted as a point on a graph of R against X. With regard then to solving problems involving impedance, it so happens that we are spoilt for choice, because there are no less than two appropriate branches of mathematics, namely vectors and complex numbers. The vector approach traditionally preferred by engineers is that of making sketches or graphs, and using trigonometry to work out the actual numbers; whereas the complex number approach is algebraic, in that it allows equations involving two-dimensional objects to be written-down and re-arranged. Both approaches are equivalent however, and sometimes one can clarify the other, and so we will adopt a notation and a way of thinking which enables us to switch freely between them. |
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1-5. Vectors and Scalars: A vector is, by definition, a mathematical object which must be described by two or more independently variable numbers. Impedances, as we have noted, fall into this category; and so vectors can be used to describe them. One very useful property of vectors is that they can be mixed with ordinary numbers and manipulated using the normal rules of arithmetic, provided that the rules are generalised to accommodate them. Because vectors are different from ordinary numbers however, it is helpful to note each one as a letter in a bold typeface (or in handwriting by putting a little arrow above the symbol), and an optional comma-separated list in brackets may be included to denote its extent in its various dimensions. Thus we can represent an impedance as Z(R,X), by which we mean that Z is characterised by an amount R in the resistance dimension and an amount X in the reactance dimension. In the context of vectors, ordinary numbers are known as scalars, because the effect of multiplying a vector by a scalar is to scale it (i.e., magnify or shrink it) without otherwise changing it. Thus, if s is a scalar, we may write: sZ(R, X) = Z'(sR, sX) Note also, a widely used mathematical notation, which is to use an apostrophe ( ' ) to indicate that an object has been modified (the apostrophe is pronounced "prime"). We can immediately deduce a rule for adding vectors by observing that two quantities will only add together if they exist in the same dimension (you can't increase the length of an object by adding to its width). Thus, if we want to add two impedance vectors Z1(R1,X1) and Z2(R2,X2), i.e., find out what happens when the impedances are placed in series, all we have to do is add the R parts and the X parts separately to find the new impedance Z(R1+R2,X1+X2). This operation is indicated by the '+' symbol, just as in ordinary arithmetic, i.e. if Z(R, X) = Z1(R1, X1) + Z2(R2, X2) then R=R1+R2 and X=X1+X2 One point in treating impedances as vectors, is that it enables us to draw diagrams in order to visualise what is going on. We can do this by representing an impedance as a line in a plane, with a particular length and orientation. In this sense, a vector diagram is like a navigation chart, with the distances, in this case, measured in Ohms. Mathematicians calls such maps 'spaces', by analogy with ordinary space; and a space in which distance is measured in Ohms is called impedance space. Now observe that although the R and X parts of an impedance exist in different dimensions, they both exist in the same space because they are connected by the fact that they are measured using the same units (i.e., Ohms). We may therefore deduce that the difference between a space and a graph is that all of the axes in a space must be labelled in the same units; whereas the axes of a graph can have different units, e.g., temperature vs time. You may, of course, have heard of four-dimensional space-time, which appears to disobey the rule just stated, but in fact the fourth physical dimension is not time but the speed of light multiplied by time, i.e., c´t. The units of ct are metres per second ´ seconds, i.e., metres, and so ordinary space has four dimensions with units of length. |
Working in impedance space; if
we adopt the standard convention that resistance increases to
the right and reactance increases upwards, we can obtain the
line representing an impedance by plotting a point, then moving
right by a distance R and upwards by a distance X (or downwards
if X is negative), and plotting another point. The length of
the line which joins the two points is called the magnitude
or 'modulus' of Z, and is written '|Z|' (and pronounced
"mod Z"). The magnitude is always positive by definition,
and is obtained by using Pythagoras' theorem (the square on the
hypotenuse of a right-angled triangle is equal to the sum of
the squares of the other two sides). Hence:
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Notice also that the definition of magnitude has a meaning for
scalars because a scalar can be regarded as a one-dimensional
vector, hence, if s is any scalar: |s| = +Ö(s²) i.e., the effect of taking the magnitude of a scalar is simply to remove the sign (+ or -). The direction of Z is given by the angle f (lower-case "phi") it makes with the horizontal (resistance) axis, which is the angle whose tangent is X/R, i.e., |
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X/R=Tanf Hence:
'). Note that f can be positive or negative; and in particular, if we adopt the standard trigonometric convention that a positive angle is obtained by going anti-clockwise from zero (see diagram right), f will be positive for an impedance with an inductive reactance and negative for an impedance with a capacitive reactance. |
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Notice also that |Z| and
f, taken together, provide a complete
characterisation of a two-dimensional vector and so give us an
alternative way of recording its properties. The form introduced
earlier: Z(R,X) is known as the rectangular form
because it contains a list of values in dimensions chosen to
be at right-angles to each other. The alternative: Z(|Z|,f) is known as the polar form,
because it uses polar co-ordinates (distance and bearing). The
polar form uses different units in its two dimensions (Ohms,
degrees or radians); whereas the rectangular form has the same
units in both dimensions (Ohms, Ohms). There is no ambiguity
between the rectangular and polar forms because the list in brackets
is optional, and a vector has the same properties regardless
of how it is defined. Also, if a specific vector quantity is
to be noted by putting actual numbers into the brackets, a degrees
(°) symbol next to the angle will indicate that the polar
form is intended. We can now regard equations (1-5.1) and (1-5.2) as the transformations which take a two-dimensional vector from the rectangular to the polar form. The reverse transformations are obtained from the standard trigonometric relations: Cosf=R/|Z| and Sinf=X/|Z|, i.e., R = |Z| Cosf X = |Z| Sinf The full set of transformations is summarised in the table below: |
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| If we want to add two impedances graphically, we simply place the beginning of the second against the end of the first, and draw a new line from the beginning of the first to the end of the second. Thus we get a new impedance, with a new magnitude and a new direction. This might all seem rather unnecessary in view of the simple addition rule given earlier, but the meaning of vector addition is (hopefully) obvious when it is visualised in this way. Whatever the method used in performing the arithmetic however, the point in doing it, as we shall see, is that it allows us to keep track of the relationship between the voltage applied across an impedance and the current which flows in it. |
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1.5a. Balanced Vector Equations: It is here that we must observe the principal property of the 'equals' symbol, which is that if a given type of mathematical object lies on one side of it, then exactly the same type of object must lie on the other. Thus, now that we know that impedances are vectors, we must re-write Ohm's law in such a way that equality is never violated. There are numerous ways in which this can be done, but for the moment we will examine three possibilities:
Since impedance is a vector, then either voltage is a vector, or current is a vector; or (presuming that the product of two vectors is also a vector) both current and voltage are vectors. In fact both voltages and currents are vectors because they each have associated with them a magnitude, a frequency, and a phase; the phase being defined as: the time at which a chosen event in the wave cycle occurs (e.g., the time of zero-crossing from negative to positive in the illustration below). The generator frequency is not an independent variable in the definition of impedance, because it already appears in the reactance (XL=2pfL, XC=-1/[2pfC]), and so we may deduce that the direction of the impedance vector (f) constitutes phase information, i.e., it gives us the time difference (in degrees or radians) between corresponding events in the voltage and current waveforms. Hence f is known as the phase angle, and can be converted into a time difference in seconds by dividing a complete cycle of the waveform into 360° or 2p radians and noting that the time-per-cycle or period of the waveform is 1/f. |
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Now note that since I and V are vectors, we can
write them in polar or rectangular forms using the transformations
(1-5.3) given earlier, i.e.,
I( |I| , f ) = I( |I|Cosf , |I|Sinf ) V( |V| , f ) = V( |V|Cosf , |V|Sinf ) In general, it is natural to think of currents and voltages in their polar forms; but the rectangular form is important for understanding what happens when the phase angle is either 0° or 180°. Taking a current vector as an example: I( |I| , 0° ) = I( |I|Cos0°, |I|Sin0° ) = I( +|I| , 0 ) and I( |I| , 180° ) = I( |I|Cos180°, |I|Sin180° ) = I( -|I| , 0 ) When a two dimensional vector lies along the 0° direction, either pointing with it or in opposition, its extent in one of its spatial (i.e., rectangular form) dimensions is zero; and, as in our interpretation of negative frequency given in section 2, the minus (-) symbol is associated with a 180° phase shift (or phase reversal). So, now that we know that both current and voltage are vectors, we must conclude that V=IZ is the general statement of Ohm's law. It transpires however, that we may admit the validity of the other possibilities I=V/Z and V=IZ under certain circumstances. The point is that, in AC theory, we are usually interested not in the absolute phases of the voltages and currents (i.e., the phases relative to some external reference), but in their phases relative to each other. This means that we are often at liberty to choose the direction of one of the vectors in order to learn the directions of the others relative to it. The direction chosen for this special reference vector is in principle arbitrary; but a simplification occurs if we choose it to be either 0° or 180° because Sinf goes to zero in either case, and a vector which is zero in one of its spatial dimensions behaves, in this context, as though it has one less dimension. A two-dimensional vector which drops a dimension in this way, of course, becomes a one-dimensional vector, i.e., a scalar. Hence, whenever a voltage or current appearing in a mathematical expression is written as a scalar, the symbol can be (and, as we shall see later, must be) interpreted to mean that the corresponding vector is lying along the 0° axis. A vector which transforms as a scalar in some specific context is called a pseudoscalar. A pseudoscalar has the property that when its space co-ordinates are reflected with respect to the origin (0,0) it changes sign, whereas an true scalar remains unchanged [4]. Hence voltages and currents become pseudoscalars when we choose their directions to be 0° or 180°. Another electrical pseudoscalar is resistance; a special kind of impedance which can be treated as a scalar, but which becomes negative if the co-ordinates of impedance space are reversed. If we choose the current in Ohm's law to be our reference vector, and set its phase angle to 0°, it becomes a pseudoscalar of value equal to its extent in the 0° direction; i.e., it is identifiable as the quantity |I|Cos0° or +|I|. Thus, in the relationship V=IZ, we can recognise I as the reference vector against which the phase of V will be determined: I = I( |I| , 0° ) = +|I| and V = I Z = (+|I|) Z The pseudoscalar current I is therefore equal to the current magnitude |I|, the latter being the quantity registered by an ordinary AC ammeter (an device ignorant of phase). It is however not identical to the magnitude because it can be negative in principle, even if not usually in practice; i.e, if, for some reason, the reference phase is chosen to be 180°, then: I = I( |I| , 180° ) = -|I| An AC ammeter must be considered to register magnitude |I| rather than pseudoscalar current I because swapping the connections makes no difference to the reading, i.e, the instrument can never give a negative indication (and putting a minus sign in front of each of the numbers on the scale won't help, because then it will never be able to give a positive indication). We can however equate the meter reading |I| with the reference vector I if we want to know the phase of the voltage relative to 0°. A similar logic applies in the case of the relationship I=V/Z, where, on the (correct) assumption that the reciprocal of a vector (i.e., 1/Z) is also a vector,. we can identify the pseudoscalar voltage V as the reference vector against which the phase of I will be determined: V = V( |V| , 0° ) = +|V| |V| is, of course, the quantity registered by an ordinary AC voltmeter, and we can equate it to V if we want to know the phase of the current relative to 0°. So, what we have seen here is that if one of a set of voltage or current vectors is replaced by its magnitude, it becomes a reference vector pointing at 0°. We may also deduce the converse, which is that if a vector should happen to be pointing at 0° by virtue of a choice made elsewhere, then it too can be replaced by its magnitude. It is however important to understand that there is a difference between a vector which has dropped a dimension and a magnitude, because there will be many circumstances in which we will want to use the magnitudes of vectors which are free to point in any direction. In particular, we will need this distinction later in order to generalise Joule's law. It will however become apparent that adoption of the convention that vectors written as scalars (i.e., un-bold) are pointing at 0° (or 180°) preserves the meaning of most of the DC and pure-resistance-only formulae which appear in standard textbooks. The correspondence arises because, whenever a vector is written as a scalar, a statement is made to the effect that the phase of that vector (except for the sign) can be ignored. A DC formula works for AC when the circuit contains only pure resistance because, in that case, rotating one vector to point at 0° rotates all of the others to point at 0°, and so they can all drop a dimension. Hence V=IZ becomes V=IR (for example). One consequence of all of this is that, in formulae, we should avoid writing voltages and currents as scalars unless we really mean them to be pointing at 0° or 180°. We must however permit a common convention, without which the notation will appear very cumbersome; which is that whenever we refer to a current or a voltage without mentioning phase we mean magnitude, i.e., the observable quantity which can be measured with a two-terminal meter. In other words, a measurement taken from a voltmeter may be written in isolation (say) Vout=27V, but as soon as it is inserted into a formula with other vectors it acquires a phase, even if we don't need to know what that is, and must then be identified as |Vout|. So, mindful of the warning that reference vectors and magnitudes are not quite the same thing, the expression V=IZ, now tells us that if we multiply an impedance by the magnitude of the current flowing through it, we will obtain a vector representing the magnitude of the applied voltage and its phase relative to the phase of the current. This is an extremely useful result, and stems from the fact that the vector representation has captured the physics of the situation exactly. In effect, having observed that resistance and reactance act independently on the current flowing through an impedance, and that inductive and capacitive reactances act in opposition; we have elected to represent pure inductive reactance as a vector pointing at +90°, pure resistance as a vector pointing at 0°, and pure capacitive reactance as a vector pointing at -90°. This relates to what happens in an actual circuit because; when a pure resistance is connected across a generator, it dissipates all of the electrical energy it receives, and this condition corresponds to the voltage across the resistance being exactly in phase with the current flowing through it (0° phase difference). When a pure reactance is connected across a generator however, all of the energy it receives while the magnitude of the applied voltage is rising, it gives back while the magnitude of the applied voltage is falling. The only way in which this can happen is if the voltage and current are exactly 90° out of phase (a situation known as 'quadrature'), and there are two possibilities for this condition, +90° and -90°. It follows, that when some mixture of resistance and reactance is connected across a generator, the angle for the voltage-current phase difference will lie at some intermediate value, and the use of vectors allows this angle, the phase angle f, to be determined from simple geometry. More to the point however, a phase angle of 0° implies that an impedance will absorb all of the power delivered to it, and a phase angle of ±90° implies that an impedance will not accept any power. Thus we can observe that the phase angle represents not only the relationship between voltage and current for an impedance, but also the effectiveness with which power can be delivered into it. Conversely, had we only ever discovered impedance, and that V-I quadrature means no net power delivery and V-I in-phase means complete power delivery, we could have deduced the existence of resistance and reactance, and that reactance must come in male and female versions. |
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1-6. Phasors: As we have just shown; one of the interpretations of Ohm's law is that, if an impedance vector is scaled by a current magnitude, it is transformed into a voltage vector. Since the act of scaling a vector does not change its direction, it transpires that both the impedance vector and the voltage vector contain the same phase information, and that this information is conserved after multiplication by a scalar. Put in plain language, this means that, although the current flowing in an impedance will change according to Ohm's law as the applied voltage is changed, the V-I phase relationship will not change provided that the frequency is held constant. It is for this reason that vectors used in impedance related applications are known as 'Phasors' (i.e., phase-vectors, or 'carriers of phase'), and diagrams involving them as 'Phasor Diagrams'. The special properties of phasors (as distinct from vectors in general) are as follows:  The phase co-ordinate
is defined in relation to other phasors rather than to an absolute
time reference. Time is measured
in degrees or radians relative to one cycle of the frequency
at which the analysis is being carried out. A phasor deemed
to be pointing at 0° may be replaced by its magnitude, and
a phasor deemed to be pointing at 180° may be replaced by
the negative of its magnitude. Phasors are strictly
two-dimensional. Readers familiar with general vector theory
should note that the vector product or "cross product"
(which produces a new vector at right angles to the original
two) has no meaning for phasors.Shown below is a phasor diagram illustrating what happens when an impedance consisting of a resistance, an inductance, and a capacitance in series is connected across a generator. We can easily deduce the total impedance by inspection in this case; but notionally, it is obtained by regarding the individual series elements as phasors: ZR(R,0), ZL(0,XL), and ZC(0,XC); and adding them together. Thus: ZR(R, 0) + ZL(0, XL) + ZC(0, XC) = Z(R, XL+XC) We can draw the resultant phasor Z by moving along by a distance R and moving up by a distance XL+XC (or down if XL+XC is negative), but notice that in the diagram, the resistances and reactances have all been scaled by a reference phasor I, which is equal to the magnitude of the current. By so doing, all of the quantities have been turned into voltages, and so the diagram has become a voltage phasor diagram. |
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With regard to the physical phenomena represented here; observe
that, since R, C, and L are in series, they must all carry the
same current. We can deduce the magnitudes of the voltages across
across the three components using Ohm's law, i.e., |VL|=I´XL, |VC|=I´XC, and VR=I´R (the
latter being written as a scalar because it is in phase with
I and therefore pointing at 0°). We also know the relative
phases of these voltages because they are all linked to the phase
of the common current; i.e., the voltage IR across the resistance
is in phase with the current, the voltage IXL
across the inductance is at +90° relative to the current,
and the voltage IXC across the capacitance
is at -90° relative to the current. We can therefore add
these three voltages as vectors to obtain the magnitude of the
generator voltage and its phase relative to the phase of the
current; although in the diagram the voltages across the two
reactances have been added first to produce the more diagrammatically
convenient quantity IXL+IXC
(this being the voltage across the total reactance in the system).
Note that the voltages across the two reactances always
tend to cancel because there is a fixed 180° phase difference
between them, and so the magnitude of the voltage across the
total reactance is always smaller than the magnitude of either
IXL or IXC . The relationship between the phase-angle obtained from a phasor diagram and the waveforms which can be observed using a two-channel oscilloscope is shown below: (">" means "greater than", "<" means "less than"): |
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Here we have obtained a waveform which is exactly in phase with
the current by measuring the voltage across the resistive component
(bottom trace). When this is compared against the waveform of
the total voltage V (using the upward zero-crossing as
an arbitrary reference point), we find that V is advanced
in time (i.e., leading) relative to I when the impedance
is inductive (XL+XC>0),
and V is retarded (lagging) relative to I when
the impedance is capacitive (XL+XC<0). If we call the time difference observed
on the oscilloscope Dt (where 'D' is upper-case "Delta", a symbol
normally used to mean "the difference in"), then the
ratio of Dt to the time of a complete
cycle is the same as the ratio of the phase angle f
to a complete circle. The time-per-cycle (also known as the period
of the waveform) is of course the reciprocal of the frequency
(1/f), hence, if f is measured in
radians: Dt/(1/f) = f/(2p) i.e.,
Note incidentally, that it is impossible (neglecting the use of superconductors) to make a series LCR network from which all of the resistance can be isolated; because practical inductors and capacitors always have some internal resistance. A measurement made across any part of the total series resistance will however always produce a voltage which is in phase with I. A device which measures current by sampling the voltage across a resistance is, of course, an ammeter. |
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1-7. Voltage Magnification and Q: One of the curious properties of the series LCR network discussed above is that the voltages across the reactances can be much larger than the applied voltage. Take for example, an impedance consisting of a 1W resistance in series with an inductance having XL=100W and a capacitance having XC=-100W, all connected across a generator giving 1V output. In this case, the system is resonant because XL+XC=0; and so the voltage across the total reactance, |VX|=0 and the phase angle, f=0°. Because the two reactances have cancelled each other out, the impedance looks like a pure 1W resistance, but there is a current of 1A flowing through each reactance, and so each has a voltage of 100V across it. This voltage magnification (100:1) is also, by definition, the Q or "quality" of the series tuned circuit formed by L, C, and R, i.e., Q=XL/R and also Q=-XC/R (or |XC|/R). The smaller the series resistance, the better the quality. In the case of an impedance such as an antenna, of course, we cannot get inside it and measure the voltages across the individual components (and a simple series LCR combination is nowhere near complicated enough to account for the way in which antenna impedance varies with frequency). When maximising the power delivered to an antenna however, we frequently need to place the antenna impedance in series with another impedance in such a way as to create a pure resistance into which the transmitter can deliver all of its power. We would of course, like to cancel the reactance of the antenna by placing a pure opposite reactance in series with it, but pure reactance is unattainable, and so our compensating (or conjugate) reactance always brings some extra (loss) resistance with it. In this case, although the voltage appearing across the terminals of the combined impedance may be very low, the voltage across the antenna terminals may be enormous, and we must choose the voltage ratings of our matching network components accordingly. For an illustration of the voltage magnification effect, consider the short vertical antenna system depicted in the diagram below: |
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In order to avoid misconceptions it is important to be aware
that the vertical rod itself is not the antenna. To use the rod
as a radiator, we must apply a voltage to the pair of terminals
formed by it and the ground-plane, and so the antenna is the
combination of the rod and the ground-plane. The input
impedance of an electrically short (less than a quarter-wavelength
long, i.e., <l/4) vertical antenna
looks predominantly like a very small capacitor, which is essentially
the capacitance which exists between the vertical section and
the ground. A small capacitor has a large negative reactance
(recall Xc=-1/2pfC),
and so we need to place a large inductive reactance (XL=2pfL) in series with the antenna to make
the whole thing look like a resistor. If we now fill-in some
of the details about the antenna and the loading coil, we are
in a position to calculate the voltages across the antenna terminals
and the loading coil for a given generator power, and also the
overall efficiency of the complete antenna system (i.e., the
proportion of the applied power which is actually radiated).
Any mechanism which dissipates energy, i.e., consumes power, must look electrically like a resistance. The resistive part of the antenna input impedance is shown to contain two components Ra and Rr. Ra is the electrical losses of the antenna, due mainly to the RF resistance of the metal conductors used to make the rod and the ground plane and the dielectric losses (RF heating) of any insulating materials used. Rr is the radiation resistance of the antenna, i.e., a resistive component associated with energy radiated into space. Both Ra and Rr are in some sense distributed over the whole antenna, but they appear as a single resistive component (Ra+Rr) in the antenna input impedance. Take for example an antenna with a radiation resistance of 2W and an input reactance of -3000W. These are the approximate values to be expected for vertical section and ground-plane radial lengths of about 7% of the wavelength at the frequency of operation, i.e., 0.07l (graphs for estimation of radiation resistance for short antennas are given in references [5] and [6]). If we are very careful about the materials used in the antenna system, we might keep the loss component Ra down to about 0.5W, so that the input impedance of the antenna will look like 2.5W of resistance and -3000W of reactance. To cancel the antenna reactance (Xa=-3000W), we need to place a coil having XL=+3000W in series with it. Such a 'loading coil' is normally placed out with the antenna, mainly because coils inside metal boxes have more losses than coils mounted in wide-open spaces, but even so, the coil will not be perfect and will have a distributed RF resistance which looks like another resistive component in the antenna input impedance. The amount of coil resistance is given by the Q of the coil, which is the ratio of reactance to loss resistance, i.e., QL=XL/RL. A well-made loading coil might have a Q of about 400, and so RL=XL/QL=7.5W. With the reactance of the antenna now cancelled by the coil, the input impedance of the whole antenna system now looks like a pure resistance of 2.5+7.5=10W. Suppose we now decide to deliver 10 Watts (10W) from a generator (transmitter) into this 10W resistance. Knowledge of the power level enables us to calculate the current flowing in the antenna and hence the voltages which appear between the various terminals, but a word of caution is in order before using the standard power formulae for this purpose. The expressions: "P=IV", "P=I²R", and "P=V²/R" are all deeply suspect because, if we try to convert them into vector expressions simply by changing the voltages and currents into vectors, then the equations which result will be nonsense because power (energy per unit-of-time) is a scalar. We need to develop some additional ideas on the subject of vectors before this matter can be fully resolved, but the standard expressions will balance if all of the vectors involved can drop a dimension. Hence we can concede that the expressions are true for voltage and current magnitudes, provided that the generator is driving a purely resistive load (a somewhat restrictive condition, but we happen to satisfy it here). Thus, using the standard formulae, we can work out that the current in the antenna will be |I| = I = Ö(P/R) = Ö(10/10) = 1A and the voltage at the generator will be |V| = V = Ö(PR) = Ö(10´10) = 10V. Note however that the current will result in a voltage of 3000V (IX) across both reactances, and although these voltages are cancelled at the generator, we can certainly experience them as real by touching the junction between the rod and the loading coil (not recommended). In fact, the electric field-strength at the top of the loading coil is so great that a neon lamp or a small fluorescent tube held there will light without any wires connected to it (see photograph below). The actual voltage appearing across the coil, |VL|, can be obtained by using Pythagoras' Theorem, i.e., |VL| = IÖ(XL²+RL²) = IÖ(3000²+7.5²) = 3000.01V and the voltage across the antenna terminals (i.e., the voltage between the bottom of the rod and the ground plane), |Va| = IÖ(Xa²+[Ra+Rr]²) = IÖ(3000²+2.5²) = 3000.001V. These voltages are barely different from the voltages across the (theoretical) pure reactances, and reflect the fact that reactance dominates the impedances of both the coil and the antenna; but despite the reactive input impedance of the antenna we have nevertheless turned it into an effective radiator. One way to look at this is to say that by resonating the antenna with a loading reactance, we exploit the voltage magnification of the resulting tuned circuit in order to force power into a reactive load. We could, of course, do the same by sheer brute force; but that would involve using a generator with an output of just over 3000V to get a measly 1A to flow in the antenna. One further point to note here is that the voltages calculated by the above methods are all RMS voltages. RMS stands for 'square-Root of the Mean of Squares', this being a mathematical trick to find an equivalent constant (DC) voltage or current which gives the same heating effect as an alternating voltage or current (for a discussion of the RMS average see reference [7] and [A1.1]). The voltages and currents referred to in the theory of impedance must be RMS values by definition, because that is the only way in which Ohm's law can be generalised to include both DC and AC (DC becomes a special case of AC with f = 0). The need for an RMS average arises because the ordinary average of a sinusoidal alternating voltage is zero (the voltage spends as much time being positive as it does negative). If however, we square the instantaneous voltage, we obtain a function which is proportional to the power it will deliver to a resistance. If we average that power function (i.e., take the mean of the squares) and then take the square root, we will obtain the equivalent direct voltage, i.e., the constant voltage which will deliver the same amount of power to a given resistance. The RMS average of a sine wave is the instantaneous peak value divided by the square-root of 2, i.e., VRMS=VPk/Ö2, and VPk=VRMS´Ö2. Thus if we calculate a voltage of 3000VRMS across the antenna terminals, the maximum instantaneous (peak) voltage will be 3000´1.4142=4247V, and it is this higher figure which must be used in calculating the voltage ratings of the components used. |
| The final piece of information we can extract from the vertical antenna example under discussion is the efficiency of the system. In this case, all we have to do is note that the total input resistance was 10W, whereas the radiation resistance was 2W. This gives an efficiency of 2/10 or 20%, i.e., 2W radiated for 10W in, 20W radiated for 100W in. This incidentally, is not a disaster, and represents a good figure for a short loaded-vertical; around 10W of SSB radiated from a reasonably high location being sufficient for worldwide short-wave communication in suitable atmospheric conditions. |
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Voltage magnification in action: For the antenna in the illustration on the right, the frequency of operation is 1.84MHz and the physical length of the antenna assembly is 1.45m from the bottom of the rubber mounting base to the top of the neon lamp soldered to the tip. The section above the loading coil is 0.76m long (0.0047l). The clamp holding the fluorescent tube is made from acrylic, and there is no electrical connection between the tube and the whip. The glow from both lamps is visible at an input level of 1W, but since the photograph was taken on a bright summer's day at about noon (albeit in the shade), the power input to the ATU was turned up to 100W to overcome the daylight. The antenna is one of the author's old 160m mobile whips from the early 1970s. It is not an optimal design, but it gave useful service (a range of several miles using 1W of AM) despite having an efficiency of considerably less than 1%. The long thin shape of the coil does not give maximum Q, but it does cause the coil to radiate to some extent (some of its 'loss' resistance is actually radiation resistance). The 6W fluorescent tube was added for this demonstration, but the neon at the tip was always used as a tuning aid. The generator in the photograph is a Kenwood TS430s, with its mains power supply; and the ATU is an MFJ989C T-network. The input to the whip is resistive when the length is adjusted correctly (about 25W, mainly due to the coil), and the ATU was used simply to transform this resistance to 50W, as required by the generator. |
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| Note that the TS430s is not rated for continuous 100W output, and so to avoid overheating the transmitter, the author operated a Morse key briefly with his foot while taking the photograph. Those wishing to reproduce this demonstration should note that, apart from the mains lead, there is no proper ground-plane for the setup, and the author had to tune-up wearing rubber gloves in order to avoid getting his fingers burnt. Mounting the antenna on a car is much safer. |
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1-8. Power Factor and Scalar Product: In the preceding discussion, we observed that reactance acts as an impediment to the delivery of power into an impedance, and that the applied voltage must be increased in order to overcome it. This means that the DC power formula "P=IV", if we interpret it to mean the product of the current and voltage magnitudes, is not generally true for impedances because, except in the special case that X=0 (i.e., when the impedance is a pure resistance) it will give a result which is larger than the actual power delivered. As mentioned earlier, we can deduce what is wrong with the equation in a purely abstract way by noting that I and V are phasors, whereas P is a scalar. Now we will fix the problem by finding a method of vector multiplication which produces a scalar. The first step in doing so is to refer to the product |I||V| as the apparent power: Papparent = |I| |V| The true power, on the other hand, is the power dissipated in the resistive part of the impedance, which can be determined from the magnitude of the current passing through it, i.e., using a properly balanced version of the DC formula: P = |I|² R and if we choose I as a 0° reference vector: P = I² R Earlier in this chapter, we showed how an impedance phasor diagram can be scaled by a reference phasor I to obtain a voltage phasor diagram (i.e., every resistance or reactance in the diagram is multiplied by I). The phasor diagram below has been scaled by I² to obtain a 'power phasor' diagram. Here we should be aware that the phrase 'power phasor' is an oxymoron (i.e., a contradiction in terms like "encrypted broadcast") because power is scalar; but apparent power is not power, and we can think of it as a vector. In particular, having set the phase of the current to be 0°, the 'phase' of the apparent power is given by the expression: Papparent = I V and since V = I Z(R,X), then Papparent = I² Z(R,X) which gives the definition of apparent power as: Papparent(I²R, I²X) Thus the phase of V relative to I is the 'phase' of the apparent power relative to the true power (P=I²R); and the magnitude of the apparent power is the diagonal of the 'phasor' diagram shown below: |
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From this, we can determine a correction factor for the |I||V|
power formula, particularly by observing that the cosine (adjacent
/ hypotenuse) of the phase angle is P/(|VI|), i.e.;
" P = V I Cosf " but unfortunately, there is nothing we can do to salvage this traditional version. We will prove later that the un-bold symbols V and I, when used in an equation, must be interpreted as phasors pointing at 0° or 180° because that is the only way in which we can incorporate DC and AC into the same theory. V and I however can only point in the same direction when f=0°. The standard formula is therefore internally inconsistent (a mathematical oxymoron). The best that can be said for it is that there is little choice but to assume V and I to be magnitudes in this instance, since the expression is nonsense otherwise. The quantity ' |V||I|Cosf ' is known as the scalar product or dot product of the two vectors, and is defined in the same way for all vectors (regardless of the number of dimensions): A·B = |A| |B| Cosf It is the component (shadow length) of B when projected onto the direction of A multiplied by the length of A (and vice versa, i.e., A and B are interchangeable). Had we attacked the DC power formula "P=IV" with a foreknowledge of vector theory, we would have failed it on the grounds of dimensional inconsistency (P has too many dimensions) and deduced that the scalar product is required, i.e.,
In the context of impedance, Cosf is known as the power-factor (PF), and is of particular interest to electricity generating companies, which prefer their customers to place pure resistances across the supply so that they do not have to run their generators into reactive loads. Thus if a load such as an electric motor is inductive as well as resistive, a suitable capacitor placed across it or in series with it can be used to cancel the reactance and bring the power-factor to unity, i.e., f=0° and Cosf=1. This brings the apparent power into coincidence with the actual power consumed, and has the effect of minimising the consumer's electricity bill as well as minimising the stress on the generators and power transmission equipment. Thus power-factor correction, in relation to electricity distribution, is equivalent to the business of bringing an antenna system into resonance. The reactance-cancelling step in antenna matching, and the insertion of a loading coil into a vertical antenna, can both perfectly well be regarded as a forms of power-factor correction. Now, since power can only be delivered to the resistive part of an impedance, only that part of voltage-multiplied-by-current which corresponds to true power (i.e., the |I|²R component) can be measured in Watts. The reason is that power (the amount of energy delivered or work done in unit time) establishes the relationship between electricity and thermodynamics, and the connection is through energy dissipation. It is therefore the convention in electrical engineering, to express apparent power in volt-amps (VA) and only true power in Watts. Many readers will already be aware that mains transformers and portable electric generators (for example) are rated in VA; the implication being that to get the full power output without over-stressing the device, it is necessary to make the apparent power in VA equal to the true power in Watts, i.e., to provide the transformer or generator with a resistive load. Since maximum power output will be associated with a particular value of load resistance; it transpires that all generators, not just radio transmitters, require impedance matching if maximum allowable output is to be obtained. One further point which we will make on the subject of power, is that true power is always positive. In DC circuits, this means that, if the voltage applied to a resistance is taken to be positive, then the current which flows in the resistance must also be taken to be positive; and if the voltage is taken to be negative, then the current is negative. In AC circuits, of course, power is calculated from RMS voltages and currents; but since power itself is always positive, there is never a need to compute the RMS value (it can be done, out of mathematical curiosity, but it is numerically not the same as |VRMS||IRMS|Cosf. For a full explanation see ref [7]). Thus the term "RMS Watts", commonly seen in the Hi-Fi literature, is nonsense and should be avoided. 
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© D W Knight 2007.
David Knight asserts the right to be recognised as the author of this work.
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