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Part 1 |
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1. An introduction to AC electrical theory: Part 2.
Contents:
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1-9. Complex numbers. 1-10. Complex arithmetic. 1-11. Impedances in parallel. 1-12. Parallel resonance. 1-13. Dynamic resistance. 1-14. Double slash notation. 1-15. Parallel to series transformation. |
1-15a. f and Q of parallel impedance. 1-15b. Magnitude of parallel impedance. 1-16. Series to parallel transformation. 1-17. Parallel resonator in parallel form. 1-18. Imaginary resonance and critical resistance. 1-19. Phase analysis. 1-20. Resistance tuning? |
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1-9. Complex Numbers: Although the graphical 'phasor diagram' approach outlined in the previous sections is suitable for problems involving phasor addition and scaling (i.e., series networks); it is somewhat intractable for solving problems involving phasor multiplication and division, one of particular importance being that of how to analyse networks involving impedances in parallel. In section 1-1 we derived an expression for resistances in parallel, and also by inference an expression for reactances in parallel, i.e., R = R1 R2 / (R1 + R2 ) X = X1 X2 / (X1 + X2) It should come as no surprise that, if we repeat the exercise with impedances instead of reactances or resistances, applying ordinary arithmetical operations to the phasors without knowing what they mean, we end up with the expression:
Complex numbers were first discovered as a 'necessary evil' in solving quadratic equations, i.e., equations which can be written in the form: ax²+bx+c=0. They were once considered to be the work of the Devil, but in fact, they merely indicate that ordinary numbers are not the whole story. Those who studied quadratic equations at school, but never got as far as complex numbers, may be surprised to learn that all of the examples they were given were deliberately chosen so as not to involve complex numbers; and that education systems in general expend more effort trying to protect students from the knowledge of complex numbers than they expend trying to teach the subject. A derivation of the general solution for all quadratic equations is shown in the box below, and results in the well known formula:
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General Solution for Quadratic Equations: Obtaining the general solution to all quadratic equations is a matter of re-arranging the general form ax²+bx+c=0 so that x is all alone on one side of the equation. We therefore start by subtracting c from both sides, so that: ax² + bx = -c and then divide both sides by a, so that: x² + (bx/a) = -(c/a) . . . . (1-9.2). We now need to find a substitution for the term x²+(bx/a) such that x is on its own. We can do this by observing that x²+(bx/a) looks similar to part of the expansion of a quantity in the form (x+p)², (where 'p' is just an arbitrarily chosen symbol) i,e, (x+p)² = x²+2px+p² . . . . (1-9.3) To use this substitution, we equate the term 2px in equation (1-9.3) with the term bx/a in equation (1-9.2), i.e., we put p=b/2a and rewrite equation (1-9.3) thus: [ x + (b/2a) ]² = x² + (bx/a) + (b²/4a²) which can be rearranged by subtracting (b²/4a²) from both sides to give: x² + (bx/a) = [ x + (b/2a) ]² - (b²/4a²) Substituting this into expression (1-9.2) gives: [ x + (b/2a) ]² - (b²/4a²) = -(c/a) and adding (b²/4a²) to both sides gives: [ x + (b/2a) ]² = (b²/4a²) -(c/a) We then put the terms of the right-hand side onto a common denominator, thus: [ x + (b/2a) ]² = (b² - 4ac)/4a² We can now take the square root of both sides to get x on its own, but note that when a square-root is taken, there are two possibilities because q´q is the same as (-q)´(-q), i.e., Ö(q²)=±q. Hence: x + (b/2a) = ±Ö[ (b² - 4ac)/4a² ] = [ ±Ö(b² - 4ac) ]/2a finally, we subtract b/2a from both sides to obtain:
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The formula looks innocuous enough, but what happens when 4ac
is larger than b²? In that case, the solution for x has
a term containing the square-root of a negative number (i.e.,
a number which is negative when multiplied by itself) even though
the basic rules of arithmetic demand that when a number is squared,
the answer must always be positive. Take, for example, the seemingly innocent quadratic equation x²-x+1=0. In this case: a=1, b=-1, and c=1, and the solution is: x = (1/2) ±(Ö(-3) )/2 The best simplification we can manage is to factor out the square root of -1, i.e., x = 0.5 ±0.866Ö(-1) Thus there are two solutions, x = 0.5+0.866Ö(-1) and x = 0.5-0.866Ö(-1), both of which contain a part which is a real number, and a part which is not a real number. That which is not real is imaginary, and so the oddball quantity ' Ö(-1) ' was given the symbol ' i ', (by Leonhard Euler, 1707-1783) and this symbol is still used by mathematicians. When it became apparent to scientists researching into electricity that this branch of mathematics might be useful however, the symbol ' i ' had already been allocated to represent current, and so the next letter in the alphabet, ' j ', was allocated for use in conjunction with electrical problems (here we will write the symbol in bold, to make it easier to spot). Thus we can write the un-simplifiable solution to the previous example as: x = 0.5 ±0.866j That which is not simplifiable is complex, and so in this case, x is a complex number. ' j ' is called the imaginary operator, because it operates on a number in such a way as to make it impossible to add it to a real number. Once ' j ' (or ' i ' ) was discovered, mathematicians went on to find general solutions for cubic equations, and quartic equations (i.e., equations involving x³ and x  ),
and it was proved that no other type of imaginary operator was
required. This means that all numbers can be reduced to
the sum of a real part and an imaginary part, and expressed in
the general form:x = a + jb with the proviso that sometimes b=0 and the number is purely real, and sometimes a=0 and the number is purely imaginary. Thus it is not so much that complex numbers are peculiar, but that real numbers are a special class of complex numbers which just happen to have the imaginary part equal to zero. Once it was understood that numbers are in general complex, the next step was to work out what this means. The clue comes from our earlier discussion of vectors. Firstly, we may observe that all real numbers must lie on a line stretching between -¥ and +¥. Secondly we may observe that j causes imaginary numbers to exist in a dimension separate from real numbers. Therefore the effect of j is to rotate the number-line through 90°. Thirdly, we may observe that the numbers 0 and 0+j0 are the same, so that the real and imaginary number-lines must cross at 0. The upshot is that complex numbers (i.e., all numbers) can be represented as points in a plane, which is the same as saying that the number a+jb can be plotted as a point on a graph of a vs b. This graph is, of course, number space, and maps in this space are known as Argand diagrams. |
We must observe, at this point, that complex numbers are so like
impedances that had they been discovered by electrical engineers,
they might well have been named after impedances. Naturally,
since complex numbers are the general class of numbers to which
all numbers belong, they are essential for solving all kinds
of mathematical problems, but nowhere is the association so direct
and so profound that all we have to do to convert an impedance
into a complex number is to write:
It follows also, from the relationships implicit in Ohm's law, that if impedances can be treated as complex numbers, then so too can voltages and currents. This does not mean that these objects have somehow ceased to be vectors however, far from it. The complex number form is just another two-dimensional vector representation, which complements the rectangular and polar forms we have already met. In fact, it is merely a version of the rectangular form in which the 90° difference between the dimensions is imposed by the j operator; and a vector always behaves in the same way regardless of how it is defined. This minor change makes a huge difference however, because it allows a phasor to be written as an ordinary algebraic sum. An expression with j in it might not seem ordinary of course; but it is so in the sense that the existence of j is required by the rules of common arithmetic, and so j is by definition subject to those rules. The complex form of a phasor makes the rectangular form effectively redundant. The transformations from the complex to the polar form are given below, and are very similar to the transformations given in table 1-5.3. |
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| Notice also that j can be regarded as a phasor operator, because its effect on an algebraic expression is to turn that expression into a phasor (another good reason for writing j in bold). Hence, in the matter of writing properly balanced vector equations, we may note that if a live phasor (i.e., one which has not been turned into a scalar by taking a magnitude or a scalar product) exists one one side of the '=' symbol, then there must be a live phasor or an expression with j in it (i.e., a live phasor) on the other side. |
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Euler's Formula: For those familiar with exponents, note that: Cosf + jSinf = e  This equation is known as Euler's formula, and defines the relationship between algebra and trigonometry; where 'e' is sometimes referred to as Euler's number and is, to more decimal places then you'll probably ever need: 2.718 281 828 459 045 235 360 287 471 352 662 497 757 247 093 699 959 574 966 967 627 724 076 630 353 547 594 571 382 178 525 166 427 427 466 391 932 003 059 921 817 413 596 629 043 572 900 334 295 260 595 630 738 132 328 627 943 490 763 . . . . etc., etc. |
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1-10. Complex arithmetic: Complex numbers can be added in the same way as vectors, i.e., (R1 +jX1) + (R2 +jX2) = (R1 + R2) +j(X1 + X2) and they can be scaled in the same way as vectors, i.e., s(R + jX) = sR + jsX (it is traditional to move j to the beginning of the term it operates on, to make its presence more obvious). The real power of the representation however, comes from the fact that we know immediately how to perform multiplication involving complex numbers because, although expressions having non-zero real and imaginary parts cannot be reduced to a single number, we can deal with the multiplication cross-terms by observing that j²=-1. Hence: (R1 + jX1)(R2 + jX2) = R1R2 + jX1R2 + jX2R1 +j²X1X2 = (R1R2 - X1X2) +j(R1X2 + X1R2) Thus we can multiply two complex numbers and always obtain a result which can be re-arranged into the form 'a+jb'. This outcome demonstrates also that the ordinary algebraic product of two phasors, AB, is another phasor; and is not the same as the dot (scalar) product A·B. The ordinary product is known as the complex product, or the phasor product, (and is also not the same as the cross product used in general vector theory). The statement j²=-1 incidentally, is the same as saying that rotation of a number through 90° followed by another rotation through 90° has the effect of reversing its original direction, i.e., multiplying it by -1. We now have part of the solution of how to interpret the expression: Z=Z1Z2/(Z1+Z2). One further trick is required in order to cope with the division part of the problem however, and this comes from noticing what happens when the complex number a+jb is multiplied by the complex number a-jb: (a +jb)(a -jb) = a² +jab -jab -j²b² = a² + b² a-jb is called the complex conjugate of a+jb, and vice versa. An asterisk is normally used to denote the complex conjugate of a number, e.g., if Z=R+jX, then Z*=R-jX (Z* is pronounced "Z-star"). When a number is multiplied by its complex conjugate, the result is always real. Thus if j appears in the denominator (the bottom part) of a fraction, we can multiply both the numerator (the top part) and the denominator by the complex conjugate of the denominator. Multiplying both the top and bottom of a fraction by the same number makes no difference to the value, but the operation makes the denominator real, so that the fraction can then be rearranged into a form which looks, once again, like a+jb. We now have a complete set of definitions for mathematical operations involving phasors, and thus armed, we are in a position to attack the parallel impedance problem. |
| The formula (and variants) given above for impedances in parallel, while not exactly memorable, has the advantage of being completely general. First note that if we put X1=0 and X2=0, then all of the reactive terms vanish and we are left with the formula for resistors in parallel, i.e., R=R1R2/(R1+R2). Similarly, if we put R1=R2=0, we end up with the parallel reactance formula X=X1X2/(X1 +X2). More to the point however, we can put only X2=0 and so find out what happens when a resistance is placed in parallel with an impedance, and we can put R2=0 and find out what happens when a pure reactance is placed in parallel with an impedance. The latter operation, as we shall see in later chapters, is of particular importance in the matter of analysing antenna matching networks. |
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1-12. Parallel resonance: In an earlier section, we said that it is not possible to calculate the exact resonant frequency of a parallel tuned circuit, nor the impedance which it presents at resonance, without taking the resistances of the coil and capacitor into account. Now, of course, having derived a general equation for impedances in parallel, we are in a position to rectify this omission. The network we need to analyse is shown on the right; where RC is the so-called equivalent series resistance (ESR) of the capacitor, and RL is the loss resistance of the coil, which we previously defined as RL=XL/QL (here Q is given the subscript 'L' to indicate that it is the Q of the coil, not the overall Q of the tuned circuit). For the purposes of this discussion, we will assume that both RC and RL are predominantly due to the RF resistance of the wires and other conducting materials used to make the components, and for reasons which will be explained in the next chapter, are considerably larger than the DC resistances. For the types of components used in HF antenna matching applications, RC will be of the order of 0.1W, and RL typically a few ohms. |
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In the general electronic literature,
several different definitions are used for the resonant frequency
of a parallel tuned circuit; the alternatives being the frequency
at which the impedance of the circuit has its largest magnitude,
and the frequency at which XL=-XC (Refs [2][8]). Here however, we will adopt the
most straightforward definition, which is the frequency at which
the impedance is purely resistive (also known as the 'unity power-factor
frequency'). We can find this frequency by setting the imaginary
part equal to zero in equation (1-11.1)
above, i.e.: X = [XCXL(XC +XL) +XCRL² +XLRC² ] / [ (RC + RL)² +(XC + XL)² ] = 0 (Where the subscripts 1 and 2 have been changed to C and L as befits the current problem). Now notice, that to make the reactance equal to zero, we only need to make the numerator of this expression equal to zero, i.e., we may ignore the denominator. Hence: XCXL(XC +XL) +XCRL² +XLRC² = 0 . . . (1-12.1) We now need to make the frequency dependence of this expression explicit by using the substitutions: XC=-1/2pf0C, and XL=2pf0L, i.e.: -(2pf0L / 2pf0C )( 2pf0L - 1/ 2pf0C ) - ( RL² / 2pf0C ) + ( 2pf0L RC² ) = 0 The resonant frequency can now be found by re-arranging this expression to get f0 on its own. Also, since we know that the series-resonance formula is an approximation for the expression we are about to derive, we expect the result to look like the series-resonance formula with an additional correction term or factor. We can begin by multiplying-out the first two brackets. Hence: -( 2pf0L² / C ) + ( L / 2pf0C² ) - ( RL² / 2pf0C ) + ( 2pf0L RC² ) = 0 Now we will put all of the terms containing 2pf0 on one side, and the terms containing 1/(2pf0) on the other. 2pf0 ( LRC² - L²/C ) = (1 / 2pf0 ) [( RL²/C ) - ( L/C² )] Then multiply both sides by 2pf0, and divide both sides by (LRC²-L²/C): (2pf0)² = [( RL²/C ) - ( L/C² )] / ( LRC² - L²/C ) and factor-out 1/LC from the right-hand side: (2pf0)² = ( 1 / LC ) ( RL² - L/C ) / ( RC² - L/C ) Here we will also multiply top and bottom by -1 to put the L/C terms first, L/C generally being much larger than the resistance-squared terms, hence: (2pf0)² = ( 1 / LC ) ( L/C - RL² ) / ( L/C - RC² ) which rearranges to:
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Example: A 3mH coil is connected in parallel with a 42pF capacitor. The approximate resonant frequency is: 1/(2pÖLC)=1/(2pÖ3´10   ´42´10  )=14.178649MHz. In the region of 14MHz, the coil has a loss resistance of 2W and the capacitor has an equivalent series resistance (ESR) of 0.1W. Thus L/C=71428.57, RL²=4, and RC²=0.01. Hence the correction factor is: Ö[(71428.57-4)/(71428.57-0.01)]=0.99994414. The precise resonant frequency (to the nearest 1Hz) is therefore 0.999944´14.178649=14.178253MHz. |
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The quantity L/C is called the "L C Ratio"
of the tuned circuit (and it has units of 'Ohms squared'). Note
that: L/C = - XL XC = |XL XC| It will turn out that the L/C ratio is an important parameter of resonant circuits. Also, there is some precedent for referring to the square root of the L/C ratio as the characteristic resistance of the tuned circuit, by analogy with the characteristic impedance of a lossless transmission line, which is R0 = Ö(L/C) where L is inductance per unit of length and C is capacitance per unit of length; but the lengths cancel and so the characteristic resistance of an ideal transmission line is the square root of its L/C ratio. In the example given above, the resonant frequency differed from the ideal case by only 0.0028% or 396Hz, the reason being that the L/C ratio was very large in comparison to the squares of the loss resistances. In HF radio applications, the L/C ratios of tuned circuits are generally in the order of several tens of thousands of W², while the value tolerances of radio components are seldom better than 1% and often considerably worse. In order to obtain an exact resonant frequency, it is necessary to make either the coil or the capacitor adjustable; and the required adjustment range will easily swallow any deviation caused by using the ideal-case formula f0=1/[2pÖ(LC)]. We may therefore conclude that, in normal circumstances, the assumption of zero losses may be perfectly acceptable when calculating the resonant frequency of a parallel-tuned circuit; but, as we shall see in the next section, it is not acceptable when calculating the impedance at resonance. |
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1-13. Dynamic Resistance: For an ideal parallel tuned circuit (i.e., RL=0 and RC=0), the impedance becomes infinite at resonance. This, of course, does not happen in practice; but provided that the loss resistances of the components are small, it does rise to a high value. Since we have defined resonance as the frequency at which the reactance goes to zero, this impedance is also purely resistive, and it is known as the dynamic resistance of the parallel tuned circuit. Here we will give it the symbol Rp0 (effective parallel resistance when f = f0). It is, of course, given by the real part of equation (1-11.1) (the parallel impedance formula given earlier); i.e.:
Rp0 = [ 0.42 +2(71432.56399) +0.1(71424.57907) ] / [ (2.1)² +(-0.0149384)² ] Rp0 = 150008.0059 / 4.410223156 Rp0 = 34.0137 KW The only problem with equation (1-13.1) is that it is very cumbersome (and resistant to simplification). We might therefore be inclined to look for some simplifying assumptions; and the most obvious of these is to note that since XL is very nearly equal to -XC, we might as well assume the term XL+XC to be zero. This also implies that XC²=XL²=-XCXL=L/C, hence equation (1-13.1) becomes: Rp0 = [ RLRC(RL+RC) +(L/C)(RL+RC) ] / (RL+RC)² i.e.
Rp0 = (2´0.1/2.1) + 71428.57/2.1 Rp0 = 0.095 + 34013.61 W Rp0 = 34.0137 KW The approximation is almost exact for components of moderate Q. Also we may observe that the term RLRC/(RL+RC) is much smaller than the term (L/C)/(RL+RC), and given that we are unlikely to know the component resistances very accurately, we might as well drop the first term. Hence the appropriate formula for calculating the dynamic resistance is:
Rp0 = RL XC² / RL² i.e., Rp0 = XC² / RL If we apply this formula to our example data we obtain: Rp0 = 71432.56399 / 2 = 35.7163KW In this case the deviation from the true value is 1702.6W, or 5%, which may be a reasonable approximation for many purposes, but needs to be treated with caution. Also, the failure to eliminate reactance from the formula makes computation more difficult. |
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1-14. Double slash notation: In geometry, the expression: ' AB//CD ' means: "the line drawn from point A to point B is in parallel with the line drawn from point C to point D". Hence, by existing convention, the symbol ' // ' means "in parallel with". In electrical engineering, of course, we are frequently interested in circuits in which components are connected in parallel, and so we can usefully adapt the double slash notation to have a non-geometric meaning. We can, for example, re-state our basic parallel component formulae as follows: R1 // R2 = R1 R2 / ( R1 + R2 ) X1 // X2 = X1 X2 / ( X1 + X2 ) Z1 // Z2 = Z1 Z2 / ( Z1 + Z2 ) L1 // L2 = L1 L2 / ( L1 + L2 ) and possibly, but best avoided: C1 // C2 = C1 + C2 This convention is often convenient, because it saves the bother of having to define a temporary variable to represent the parallel combination; i.e., instead of writing: "Let R represent the parallel combination of R1 and R2" and then having to remember what "R" is; we simply work with the quantity "(R1//R2)", which can be expanded or calculated when necessary, but more to the point is just a resistance with an obvious definition.. While straightforward however, the use of the // notation involves a subtlety which lies in the distinction between physical and mathematical objects. In describing a test procedure, for example, we might put an entry in a table: "Test load: 68W // 100pF". The item "68W // 100pF" is a physical object, a capacitor in parallel with a resistor, but it is not a complete mathematical statement of impedance and cannot be treated as an impedance in any calculation. In order to turn the parallel combination into a mathematical object; we must ensure that the quantities on either side of the // symbol are of the same type and that they are expressed in the same units. In this case we can fix the problem by noting that, if the report is to have any useful meaning, a test frequency must be stated somewhere. If that frequency is, say, 14MHz, then the reactance of the capacitor becomes -1/(2pfC) = -113.7W, and its impedance (assuming that losses are negligible) is 0-j113.7W. Hence we can re-state the test load as "(68 // -j114)W" . This is the same as saying "(68+j0 // 0-j114)W"; and is, of course, a complete statement of the load impedance in the form Z1//Z2 which can be converted into the R+jX form if so desired. . A particular logic emerges from these observations and it is important to be aware of it: |
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1-14.1) A resistance
is an impedance. Resistances and impedances are the same type of object. A resistance in parallel with an impedance is an impedance. A resistance is simply an impedance which happens to have its imaginary part equal to zero. This means, incidentally, that the pseudoscalar symbol for Z is R, rather than Z. 1-14.2) A reactance is not an impedance. The statement: Z = 68 // 114 has a completely different meaning to the statement: Z = 68 // -j114 (the first is a resistance in parallel with a resistance, the second is a resistance in parallel with a reactance). Mathematically, a reactance cannot be combined directly with an impedance, but a reactance can be converted into an impedance by multiplying it by j. Looking at this another way: impedance and reactance have reference directions which are 90° apart. To make them compatible, it is necessary to rotate one of them through 90°. 1-14.3) Scalability is preserved. When the double slash notation is used to create a mathematical object, i.e., the same type of phasor exists on both sides of the // symbol, it has the useful property that a common factor can be multiplied-in or divided-out of the parallel object, i.e.: k Z1 // k Z2 = k (Z1 // Z2) Proof: k Z1 // k Z2 = kZ1 kZ2 / (kZ1 + kZ2 ) = k Z1 Z2 / (Z1 + Z2 ) = k (Z1 // Z2) 1-14.4) The associative rule. The double slash notation can be extended to represent any number of impedances in parallel: Z1 // Z2 // Z3 //....// Zn = 1 / [ (1/Z1) + (1/Z2) + (1/Z3) + . . . . . + (1/Zn) ] and the associative rule of arithmetic (and linear electrical devices in parallel) is obeyed, i.e.: (Z1 // Z2) // Z3 = Z1 // Z2 // Z3 1-14.5) Double-slash product definition. The // notation implies a specialised kind of phasor multiplication, which we might call the double-slash product or the parallel product of a pair of phasors. Since its use in conjunction with parallel capacitors is pointless, we will adopt the following strict mathematical definition:
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1-15. Parallel-to-series transformation: In the discussion so far, we have adopted the habit of representing every impedance as a resistance in series with a reactance. It makes good sense to do so in most circumstances, because it allows the impedance to be written directly in the form R+jX. There are many situations however, in which circuit analysis can be simplified by representing an impedance as a resistance in parallel with a reactance. The two possible representations are equally valid; but it should be obvious from the parallel impedance equation (1-11.1) derived earlier , that the parallel representation for a particular impedance requires a different combination of resistance and reactance to that of the series representation. In the next two sections we will explore the relationships between the two representations, beginning with the transformation of an impedance from its parallel to its series form: |
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To derive this transformation, we simply regard the parallel
elements as two separate impedances Rp+j0
and 0+jXp, and apply the formula
for impedances in parallel (i.e., [Z1//Z2]=Z1Z2/[Z1+Z2] ). Hence: R +jX = ( Rp // jXp ) i.e.: R +jX = jXp Rp / ( Rp + jXp ) and R and X are simply the real and imaginary parts of the right hand side of this expression once it has been put into the form a+jb. We proceed as usual by multiplying the top (numerator) and bottom (denominator) by the complex conjugate of the denominator, thus:
which rearranges to:
Further pieces of information which we can extract from the parallel-to-series transformation, and which will be useful later, are the phase-angle, magnitude and Q of an impedance in its parallel form. 1-15a. Phase angle and Q of an impedance in parallel form: Recall from the earlier section on vectors, that the phase angle for an impedance in its normal series form is given by the expression (1-5.2): f = Arctan(X / R) By using expression (1-15.1) above we can substitute for X and R to obtain: f = Arctan(Xp Rp² / Rp Xp²) i.e.,
1-15b. Magnitude of an impedance in parallel form: The magnitude of an impedance in its series form is given by (1-5.1): |Z| = Ö(R² + X²). Substituting for R and X using expression (1) we obtain: |Z| = Ö[{ (Rp Xp²)² + (Xp Rp²)² }/ (Rp² + Xp²)² ] = Ö[{ Rp² Xp² ( Xp² + Rp²) }/{ (Rp² + Xp²)² }] We can take the square root of the Rp²Xp² term and so factor it out of the square-root part of the expression, provided that we only use the positive result (magnitudes are always positive). Hence:
|Z| = | Rp Xp / { Xp Ö [ (Rp²/Xp²) + 1 ] } | Now, since Rp and Rp²/Xp² are always positive, we can drop the magnitude brackets to obtain:
1 / Ö [ (Rp/Xp)² + 1 ] which goes to unity (®1) when the reactance is large in comparison to the resistance. |
| 1-16. Series-to-Parallel transformation: |
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From expression (1-15.1) in the
previous section, we have: R = Rp Xp² / ( Rp² + Xp² ) . . . (1-16.1) and X = Xp Rp² / ( Rp² + Xp² ) . . . (1-16.2) Obtaining the series-to-parallel transformation is a matter of using these two equations to obtain equations for Rp and Xp. This would prove to be a somewhat tricky problem, had we not noticed from the preceding derivation of the magnitude (equation 1-15.2) that: |Z|² = R² + X² = Rp² Xp² / ( Rp² + Xp² ) The right hand side of this equation can also be obtained by multiplying expression (1-16.1) by Rp, or by multiplying expression (1-16.2) by Xp. Hence:
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1-17. Parallel resonator in parallel form: Having derived the series to parallel transformation, we are now in a position to analyse the parallel resonator in a different way. The outcome should be mathematically unsurprising, because we are bound to obtain the same results as before, but the technique will give us a new way of thinking about the circuit. |
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(The symbol " º " means:
"is equivalent to". The symbol " // " means
"in parallel with".). As the diagram above illustrates; the parallel impedance representation allows us to visualise the circuit as an ideal parallel resonator with a resistance connected across it. This separates the reactive and the resistive parts of the problem and tells us immediately that unity power-factor resonance occurs when XLp=-XCp, and that the dynamic resistance is given by the value of RLp//RCp at f0. We can, of course, relate the parallel impedance form of the resonator to the series impedance form by using the transformations given in the previous section (equations 1-16.3), i.e.,
Using the appropriate transformations (1-17.2 and 1-17.4), the resonance condition XLp=-XCp becomes: (RL² + XL²) / XL = -(RC² + XC²) / XC which can be rearranged to: XC (RL² + XL²) + XL (RC² + XC²) = 0 and then to: XCXL(XC +XL) +XCRL² +XLRC² = 0 We have seen this expression before as equation (1-12.1), and so the derivation may continue as in section 1-12 to give the parallel resonance formula (1-12.2). f0 = {1/[ 2p Ö(L C) ] }{ Ö[(L/C - RL²)/(L/C - RC²)] } The dynamic resistance Rp0 is given by: Rp0 = RLp//RCp Hence using the transformations (1-17.1) and (1-17.3) we have:
which simplifies to:
and the denominator has dimensions of W³.
In equation (1-17.6) however,
the numerator has dimensions of W³
and the denominator W². Hence
the numerator of (1-17.6)
is of lower degree than that of (1-17.5),
and the denominators likewise. This means that (1-17.5)
can be simplified further and ultimately transformed into (1-17.6); although for anyone who
cares to try it, the manipulations required are laborious, and
require the use of equation (1-12.1)
as a substitution.Something more tractable happens however when we multiply equations (1-17.5) and (1-17.6) and take the square root to obtain a new expression for Rp0, i.e., we take the geometric mean of the two formulae. In this case the denominator of (1-17.5) cancels the numerator of (1-17.6) and we obtain:
1) Since XL² is normally much greater than RL² in radio circuits, RL² can be deleted from the numerator without making much difference. 2) Since XC² is also usually much greater than RC², RC² can be deleted from the numerator without making much difference. 3) If the Qs of the resonator components are reasonably high, (XL+XC) is very nearly zero at resonance and can therefore be deleted from the denominator without making much difference. The result is:
+Ö(X²)=|X| This rule must be strictly applied, because simply deleting the superscripts and the square root symbol would have given us a negative value for Rp0 because XC is negative. Hence: Rp0 = |XL||XC| / (RL + RC) We have noted before that: |XL||XC| = L/C hence:
While it is instructive to attack a derivation from several directions and verify that all approaches lead to the same conclusion however, the point of the parallel impedance representation is that it often makes problems easier to solve. The parallel resonator is a good example because the parallel representation gives a direct separation of the resistive and reactive parts of the problem. A further and very important point however, is that we do not use the parallel representation with a view to converting it into the series form at the earliest opportunity. It is simply another way of expressing impedance; and it is no less authoritative than the series form. Hence if we have data for an inductor or capacitor in series form, we can transform it into the parallel form and use it like that. The parallel form may seem less authoritative than the series form because the expression for Rp (equation 1-16.3a) has reactance in it, and so explicitly varies with frequency. In reality however, the resistive component in the series form also varies with frequency, due to a variety of frequency dependent losses such as, skin effect and dielectric absorption (see chapter 2), capacitive and inductive coupling to resistive materials in the vicinity of the component, and of course our old friend radiation. Thus, when solving problems using simple circuit models, we need to be aware that resistances inserted to represent losses are expected to vary with frequency, regardless of representation. Thus the practical problem of finding the dynamic resistance of a parallel resonator becomes that of measuring the impedances of the components at a frequency reasonably close to the desired resonance, transforming the losses into parallel resistances, and taking the parallel combination of those. |
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1-18. Imaginary resonance and
critical resistance: It was observed, in section 1-2, that the series-resonance formula: f0= 1/[2pÖ(LC)] always gives two solutions for the resonant frequency, one positive and one negative. The parallel resonance formula (1-12.2) does the same, but presents us with a further conceptual challenge, in that it also allows imaginary solutions. If we inspect the formula:
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The answer to this conundrum can
be obtained by considering the parallel resonator as two separate
impedances connected across a generator (see diagram right).
Real resonance implies a condition where the current flowing
out of the generator is in phase with the voltage it produces,
i.e., it occurs at a frequency where the resonator constitutes
a resistive load. Now, the output current I is the vector
sum of the currents flowing in the two branches of the resonator,
i.e.; I = IL + IC |
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and so real resonance occurs when IL+IC is real. Real resonance can only occur however,
if the current in one branch can become large enough for its
imaginary component to cancel the imaginary component of the
current in the other branch. Notice that if we allow RL
to become extremely large, then practically no current will flow
in the inductive branch and the circuit will never resonate.
The same argument applies, of course, to the capacitive branch.
The parallel resonance formula can therefore be seen to tell
us that true (i.e., real) resonance cannot occur if the resistance
in either branch rises above a certain critical value, that value
being the square-root of the L/C ratio (the characteristic
resistance), i.e., R0 = Ö(L/C) If the resistance in one branch rises above Ö(L/C), then the current in that branch will always be too feeble to bring the system into resonance. What happens instead is that the phase of the total current I can approach and move away from the phase of the generator voltage as the frequency is varied, but it is never able to reach it. The 'resonant frequency' is simply the imaginary frequency of closest approach (and it does not exist on the real frequency line). It is imaginary because the combined impedance of the two branches can never become real (i.e., resistive) by cancellation; and being imaginary, it cannot be found by tuning a signal generator and watching an ammeter. Note however, by inspecting the circuit, that the combined impedance does become resistive at zero and infinite frequencies, but that this is not due to cancellation: At 0Hz, XL=0 and XC®¥ ('®' means "approaches"), so the impedance is simply RL; and at infinite frequency, XL®¥ and XC=0, so the impedance is RC. If a real resonant frequency does not exist therefore, what will be obtained is a network which has a voltage-current phase relationship always on one side or the other of zero degrees, only approaching 0° at zero or infinite frequency. It was mentioned earlier that resonant circuits used in HF radio applications tend to have large L/C ratios, often greater than 10000W². In the case of parallel resonance, one reason for this policy should now be apparent; i.e., we need to obtain a high characteristic resistance ( R0 = +Ö[L/C] ) in order to ensure that the circuit will function properly with practically realisable inductors. A parallel resonator with an L/C ratio of 100W², for example, will not work if the RF resistance of the inductive branch is greater than 10W at the expected resonant frequency, and it is by no means impossible for a practical inductor to exceed such a limit. We may conclude, from this discussion, that a parallel tuned circuit will only resonate usefully if Ö(L/C) is made larger than the resistance in either of the branches. The qualification 'usefully' must be applied however, because if the resistance in both branches is allowed to become larger than the critical value, then both the numerator and the denominator of the term inside the square-root bracket will become negative, and so the term itself will be positive. Thus there will be a real resonance, but the current in both of the branches will be feeble, and so the resonance will also be feeble and of no practical use. One final significance of the characteristic resistance which is worth remembering is that it is equal to the magnitudes of the reactances in the circuit at the 'ideal case' resonant frequency, i.e., the resonant frequency when the resistance in both branches is equal. This frequency, as was mentioned earlier, is given by the series resonance formula, i.e., f0s = 1/[ 2p Ö(L C) ] or in radians / sec: 2pf0s = 1/[Ö(L C) ] Now, if we call the inductive reactance at this frequency XL0s, then: XL0s = 2pf0s L = L / [Ö(L C) ] and, since any number is the square of its own square root:
XC0s = -1/[ 2pf0s C ] = -[ Ö(L C) ] / C
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1-19. Phase analysis: We can visualise the phase relationship between voltage and current in a parallel resonant circuit by deriving an expression for the I-V phase angle and plotting it as a graph against frequency for various values of included resistance. This is only one of the many situations in which graphs of phase vs frequency are instructive, and so this section will serve as a general introduction to the technique of phase analysis as well as a specific investigation of the parallel resonator. |
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The circuit to be analysed is shown
on the right, and we can use Ohm's law straight away to write
an expression for the current: I = V / Z where Z is the parallel combination of the impedances in the two branches of the resonator, and we choose the phase of V to be 0° and treat it as a scalar. If we also define: ZL=RL+jXL, and ZC=RC+jXC, then: Z = ZL ZC / (ZL + ZC) hence: I = V (ZL + ZC) / (ZL ZC) |
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Now, we noted earlier that the phase angle of a complex expression
a+jb is given by: f = Arctan(b/a) so in order to obtain the I-V phase difference we first write: I / V = (ZL + ZC) / (ZL ZC) then split the right hand side of the equation into its real and imaginary parts, divide the imaginary by the real, and take the inverse tangent. Expanding the expression above we get:
and multiplying out the terms in the denominator gives:
Now we multiply numerator and denominator by the complex conjugate of the denominator:
then multiply out the numerator, crossing out equal and opposite terms, to get:
This is in the form a+jb, so the phase angle is given by:
becomes:
which rearranges to:
We will now use the expression above to evaluate the effect of resistance in a fairly representative parallel resonator. For this example we will use an inductance of 1mH and a capacitance of 100pF. This combination gives an L/C ratio of 10000W² and hence a critical resistance R0=Ö(L/C)=100W. The 'ideal' resonant frequency, i.e., the resonant frequency when RL=RC is: f0s = 1/[ 2p Ö(L C) ] = 15.91549431MHz (i.e., 2pf0s = 100M radians/sec), and at this frequency, XL=-XC=Ö(L/C)=100W. Shown below is a set of graphs of the I-V phase relationship for our example resonator with various values of RC and RL between 1W and Ö(L/C). These graphs were produced using the Open Office Calc spreadsheet program (available free from OpenOffice.org), the procedure being to create columns for frequency, XL, and XC, and use the calculated reactance values in the Arctangent (inverse tangent) of equation (1-19.1) given above. Note that spreadsheets often give the results of inverse trigonometric functions in radians, and so it is necessary to multiply the expression by 180/p=57.29577951 to get the result in degrees (there are 2p radians in 360°), i.e.: f = -57.29577951Arctan{ [XL(RC²-L/C)+XC(RL²-L/C)]/[RL(RC²+XC²)+RC(RL²+XL²)] } The plotted curves below were created using the spreadsheet "chart" tool. |
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Of the curves shown, only the example with RL=1
and RC=1 constitutes a good healthy resonance.
The choice of 1W in each of the branches
incidentally was made simply so that the resonant frequency would
coincide with f0s. Any curve with at total
resistance RL+RC=2W will have an almost identical appearance.
The Q of the resonance (as will be explained in section
1-38) is 50 in this case (i.e., Q0=XL/[RL+RC] ), which is fairly high; and so the phase
of the current lags the voltage by nearly 90° at frequencies
a few percent below the resonant frequency, and leads it by nearly
90° at frequencies a few percent above. Hence the circuit
provides the generator with a nearly pure inductive load below
resonance, and a nearly pure capacitive load above. In the case where RL=50 and RC=50, the Q of the resonance is 1. A large resistive component is present in the impedance at all frequencies, and so the I-V phase difference never approaches 90° in either direction. The curves for RL=50 and RC=1, and RL=1 and RC=50, are included to show that the resonant frequency (the point where the curve crosses the zero phase-difference axis) moves to low frequency when RL exceeds RC, and vice versa. The curves for RL=100 and RC=1, and RL=1 and RC=100 show that the 'resonant frequency' goes to zero when RL=Ö(L/C), and goes to infinity when RC=Ö(L/C). These results seem to indicate that the parallel resonator is infinitely tunable by means of a variable resistor, a proposition which warrants careful examination. |
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1-20. Resistance tuning? The parallel resonator shown on the right was offered as a "circuit idea" in Electronics World [9]; it being pointed out in the article that the tuning range is 0 to ¥ if R=Ö(L/C), and the Q of the circuit is stable because the total resistance is constant. Both of these claims are true, within the scope of the model; but there are a couple of fatal flaws in the concept and we will address them lest people should start to believe that the circuit will work. |
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| The author of the article was perhaps a little unsure of the 0 to ¥ claim, and so concluded that a variable resistor can give a "much wider" frequency range than a variable capacitor or inductor. We however, can straight-away dispense with the infinite upper limit by drawing the circuit model on the right. We might describe the original circuit as "what you try to build", whereas this makes some attempt to simulate "what you actually get". |
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A capacitor is simply two pieces
of electrically conducting material in proximity. The conductors
do not have to be plates. Capacitance appears whenever two conductors
have the ability to be at different relative voltages (i.e.,
capacitance is made by not shorting things together), and so
there will always be some 'stray capacitance' across the coil.
This precludes resonance at infinite frequency, but in fact,
a coil behaves as though it has considerably more parallel capacitance
than simple consideration of strays would predict. The reason
is that it takes a finite amount of time for an electromagnetic
wave to make its helical journey along the wire in the coil,
and the resulting phase shift has to be represented by placing
a hypothetical capacitance, the coil's self-capacitance
CL in parallel with the the idealised
pure inductance. Self-capacitance is dependent on the length
of the winding wire and the effective velocity for a wave travelling
along it. This propagation velocity (the so-called phase velocity)
is frequency dependent (see
chapter 3), but most radio coils are operated in a regime
where the velocity is reasonably constant and so the self-capacitance
appears to take on a definite value. In the constant velocity
regime, the apparent self-capacitance turns out to depend only
on the external dimensions of the coil (the turn-to-turn spacing
and the number of turns are practically irrelevant). The inclusion
of self-capacitance in the model allows for the fact that the
coil has a self-resonant frequency (SRF) even when
there is nothing whatsoever connected to it, and it is part of
the HF resonator design procedure to ensure that the SRF is outside
the frequency-range of interest. A physically small resonator coil suitable for radio receiver applications might have a self-capacitance of about 1pF. Let us suppose therefore that this applies to the 1mH coil from the previous example. This amount of unavoidable capacitance places an upper limit on the maximum attainable resonant frequency somewhere very roughly around 1/[2pÖ(LCL)]=160MHz. Stray capacitance between the connecting wires will reduce this frequency, so if we construct the circuit carefully we should expect the inductive branch to self-resonate somewhere in a range from about 40 to 160MHz. All electrical conductors have inductance (a coil is simply a structure designed to enhance inductance by causing the magnetic fields developed by adjacent turns to add together). Hence the wires and plates involved in making up the capacitive branch of the resonator will constitute an additional series inductance, which we can model to a good approximation by imagining a small inductor LC in series with the capacitor. For the 100pF capacitor of our previous example, it will be ver |