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Part 2 |
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1. An introduction to AC electrical theory: Part 3.
Contents:
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1-21. Phasor Theorems. 1-22. Generalistion of Ohm's Law 1-23. General statement of Joule's law. 1-24. Bandwidth. 1-25. DeciBels and Logarithms. |
1-26. Bandwidth of a series resonator. 1-26a. Centre frequency. 1-26b. A sensible definition for Q0. 1-26c. Bandwidth function in terms of Q0. |
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1-21. Phasor theorems: Early in this chapter we observed that the standard electrical formulae represent incomplete statements of Ohm's Law and Joule's Law. We then went on to generalise Ohm's law, but have yet to state all of its implications; and we repaired the VI power law by introducing the scalar product, but have yet to analyse Joule's law. We also introduced the the idea that if a phasor is pointing at 0° or 180° it can be treated as a scalar, a trick which obviously works, but for which we offered no convincing mathematical proof. All of these discomforts arise because of a narrative expediency, which is that of delaying the introduction of complex numbers until after that of vectors. We will now resolve all of the residual issues, with the aid of a handful of simple theorems which require complicated geometrical arguments if they are to be proved using vectors, but are easy to prove using complex numbers. These theorems incidentally are also true for real numbers, which are effectively one-dimensional vectors. |
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1-21.2 Magnitude reciprocal theorem:
The magnitude of the reciprocal of a (complex) number is equal to the reciprocal of its magnitude.
Let N1 = 1+j0 = 1 Now |N1/N2| = |N1| / |N2| Therefore: |1/N2| = |1| / |N2| |1/N2| = 1 / |N2| |
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1-21.3 Magnitude product theorem:
The magnitude of the product of two (complex) numbers is equal to the product of their magnitudes.
Let N1=A1+jB1 and N2=A2+jB2 Then: N1 N2 = ( A1 +jB1 )( A2 +jB2 ) = A1A2 - B1B2 +j(A1B2 + A2B1) |N1 N2| = Ö[(A1A2 - B1B2)² + (A1B2 + A2B1)²] = Ö[(A1A2)² + (B1B2)² -2A1A2B1B2 + (A1B2)² + (A2B1)² +2A1A2B1B2 ] = Ö[A1²(A2² + B2²) + B1²(A2² + B2²)] = Ö[(A1² + B1²)(A2² + B2²)] = [Ö(A1² + B1²)][Ö(A2² + B2²)] = |N1| |N2| |
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1-21.4 Scaling theorem: The magnitude of the product of a scalar and a complex number is equal to the product of the scalar and the magnitude.
Let s be a scalar, and N=A+jB. sN = sA +jsB |sN| = Ö[(sA)² + (sB)²] = Ö[s²(A² + B²)] = sÖ(A² + B²) = s|N| i.e., a scalar can be factored out of or multiplied into a magnitude bracket in the same way that it can be done with any other type of bracket. |
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1-21.5 Drop-dimension theorem: A phasor with a phase angle of 0° or 180° transforms as a scalar:
A phasor pointing at 0° can be represented as a complex number with a positive real part and a zero imaginary part. A phasor pointing at 180° can be represented as a complex number with a negative real part and a zero imaginary part. Hence if: N = A+j0 then: N = A and |N| = +Ö(A² + 0²) = |A| Hence: N = N = ±|N| where N is a pseudoscalar equal in value and sign to the real part of N. This may appear trivial, but it shows that our assumption that a phasor which has dropped a dimension can be treated as a scalar is universal, rather than a special interpretation of a particular phasor expression. A further implication however is that N is not identical to the magnitude of N, because magnitudes are always positive whereas N can be positive or negative. We can force N to become equal to |N| by stipulating that f = 0°. We can also drop a dimension, i.e., set the imaginary part to zero, by choosing f = 180°, but in that case we get N =-|N|. Thus the alleged scalar which results from dropping a dimension is not a magnitude, but it is a quantity which is equal in magnitude to a magnitude, and if f = 0° it is positive. This may seem a pedantic distinction, but the point in making it is that if we restrict the scope of our phasor algebra through erroneous interpretation, we lose the ability to include DC electricity in our theory, and we lose the ability to explore exotic ideas such as negative resistance. The pseudoscalar we obtain by dropping a dimension can be negative, even if usually it isn't. |
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1-21.6 Square magnitude theorem:
The product of a complex number and its complex conjugate is the square of the complex number's magnitude.
Let N=A+jB and N*=A-jB N N* = A² + B² but |N| = Ö(A² + B²) therefore N N* = |N|² Hence the product of a complex number and its complex conjugate is a true scalar. It is also literally a scalar product. Recall that the definition of a scalar product is: A·B = |A| |B| cosf but if A and B are identical, then f=0° and cosf=1. Hence: N·N = |N|² = N N* |
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1-21.7 Conjugate product theorem: The complex conjugate of the product of two complex numbers is the product of the complex conjugates.
Let N1=A1+jB1 and N2=A2+jB2 Then: N1 N2 = (A1 +jB1)(A2 +jB2) = A1A2 - B1B2 +j(A1B2 + A2B1) Therefore: (N1 N2)* = A1A2 - B1B2 -j(A1B2 + A2B1) = A1(A2 -jB2) -jB1(A2 -jB2) = (A1 -jB1)(A2 -jB2) = N1* N2* |
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1-21.8 In-phase quotient theorem: If two phasors are in phase, their ratio can be treated as a scalar.
Using the polar to complex transformation (1-9.4): N1(|N1|, f) = |N1|(cosf + jsinf) N2(|N2|, f) = |N2|(cosf + jsinf) where the phase angle f is the same in both cases. Therefore: N1 / N2 = |N1|(cosf + jsinf) / [ |N2|(cosf + jsinf) ] = |N1| / |N2| |
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1-21.9 Magnitude Caveat: The mathematical operation of 'taking a magnitude' destroys information. Specifically, it is important to be aware that if |A| = |B|, then it is not necessarily true that A=B. The magnitude operation discards the directional information of a vector, and the sign information of a scalar. Note for example that although: |A| = |A*| = |-A| = |-A*| any one of the quantities inside magnitude brackets is definitely not identical to any of the others. The magnitude retains only the length of the object. In so doing however, it does retain the unit of measurement; i.e., a magnitude is a length in impedance space, or voltage space, or current space, etc., and so has the units of the space in which it exists. |
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1-21.10 Magnitude Equivalence: As noted above, for any complex number Z : |Z| = |Z*| = |-Z| = |-Z*| When designing electrical circuits, it is not unusual to meet situations in which the magnitude of a voltage or current needs to be determined, but the phase is unimportant. As the theorems above show, when only the magnitude is needed, all of the impedances involved in the calculation can be replaced by their magnitudes (provided that the impedances are factors, i.e., multipliers or divisors, not terms in a summation). What is less obvious however, is that an impedance Z enclosed between magnitude brackets can then be replaced by one of the alternatives having the same magnitude, namely Z*, -Z and -Z*. This principle of Magnitude Equivalence allows us to deduce alternative networks which will produce the same outcome. In particular, it allows us to identify situations in which inductance can be replaced by capacitance and vice versa. |
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1-21.11 About these Theorems: The theorems given above do not appear in standard engineering textbooks. Hence, it is legitimate to ask: 'Why have they been stated here when everyone else manages without them?' The answer to the question is this: By sticking to the mathematical rules: particularly by ensuring that we always use properly balanced vector equations, and by using any simplifications which can be proved in a general way; we eliminate the need for phasor diagrams. Essentially, we can let the algebra do all of the reasoning. We can still use phasor diagrams for the purpose of explaining what is going on, but they become merely illustrative and make no difference whatsoever to the outcome of a problem solving exercise. The traditional role of the phasor diagram has been to help in sorting out the ambiguities caused by unrigorous mathematical definitions. But the mathematics is self-consistent. If the problem is defined correctly, the hand-waving becomes unnecessary. |
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1-22. Generalisation of Ohm's law: We have already arrived at a general statement of Ohm's law in section 1-5a by observing that it can be written as a phasor equation: V=IZ. We have also observed that we are at liberty to treat either I or V as a scalar equal in value to its own magnitude in order to learn the phase of the other relative to it. The drop-dimension theorem (1-21.5) gives our justification for doing so, but also allows us to generalise our phasor Ohm's law to include DC. We do this by noting that if a circuit has any series capacitive reactance, then as the frequency goes to zero (f®0), then XC=1(2pfC) goes to infinity (XC®¥), hence the magnitude of the impedance goes to infinity ( |Z(R,X)|®¥) and the impedance becomes an open circuit. The only type of reactance which gives electrical continuity for DC is, of course, inductive reactance, and at f=0, XL=2pfL=0. Thus Z(R,X) drops a dimension and becomes Z(R,0)=R. Hence we can write V=IR for DC (or for pure resistance and AC), but since both V and I are then in phase, they can drop dimensions also. Thus we obtain V=IR if we drop dimensions at f=0°; but more to the point, we are also at liberty to drop dimensions at f=180° and obtain the perfectly valid alternative: (-V) = (-I) R i.e., we have a theory which covers all aspects of AC electricity and also allows us to have the negative voltages and currents required for the analysis of DC circuits. This is why we must insist that the un-bold symbols V and I are not magnitudes, they are pseudoscalars (or, if you prefer, complex numbers in the form a+j0) which can point in either a positive or a negative direction. It is only resistance which can never be negative in a passive network, and that is for thermodynamic rather than for mathematical reasons. An additional interpretation of Ohm's law is also given to us by the magnitude ratio and product theorems (1-21.1 - 1-21.3). These allow that if V=IZ, then: |V| = |I Z| = |I| |Z| (and all possible rearrangements). This says that the need for complex arithmetic is removed if all you want to know is a magnitude, i.e., if the left hand side of an equation is a magnitude, then all of the phasors on the right can be replaced by their magnitudes. This observation simplifies some problems enormously, since failure to apply the magnitude ratio and product theorems when the situation allows results in unwitting repetition of the working used in the proofs in sections 1-21.1 to 1-21.3. Shown below are some of the possible interpretations of Ohm's law which stem from the discussion in this chapter. There is no need to memorise these formulae because they are all derived from the statement "V=IZ". What they show (hopefully) is that all manner of complicated arguments involving phasor diagrams are in fact trivial and can be deduced by inspection of the master equation. |
| V = I Z | Z = V / I = V I* / |I|² | I = V / Z = V Z* / |Z|² |
| V = I (R+jX) | R + jX = V / I | I = V / (R+jX) = V (R-jX) / (R² + X²) |
| V = I Z | Z = V / I = VI* / |I|² | I = V / Z = V Z* / |Z|² |
| V = I Z | Z = V / I | I = V / Z = V Z* / |Z|² |
| |V| = |I| |Z| | |Z| = |V| / |I| | |I| = |V| / |Z| |
| |V| = I |Z| | |Z| = |V| / I | I = |V| / |Z| |
| V = |I| |Z| | |Z| = V / |I| | |I| = V / |Z| |
| V = I R | R = V / I | I = V / R |
| (-V) = (-I) R | R = (-V) / (-I) | (-I) = (-V) / R |
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Where: V I and Z are phasors, V* I* and Z* are complex conjugates, |V| |I| and |Z| are magnitudes, V and I are phasors pointing at 0°, (-V) and (-I) are negative values of V and I, and thus are phasors pointing at 180°, and un-bold Z is not normally used, because an impedance pointing at 0° already has the symbol R. |
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1-23. General statement of Joule's law: At the very beginning of the chapter, we gave Joule's law in its standard form: P = I² R This, now that we know that I should be interpreted as a phasor pointing at 0° or 180°, proves to be a correctly balanced vector equation; but it is only so by an accident of notation and, as we shall see shortly, the restriction on the phase of the current is unnecessary and limits the scope of the formula. Joule's law, even in its standard form, is a profound philosophical statement; because the squaring of the current prevents the direction of the current from having any effect on the direction of the power. It therefore tells us that resistance is positive because power (energy per unit-of-time) is positive, i.e., the dissipation of energy is a uni-directional process. The direction in question is that of entropy, the general spreading out and cooling down of the universe, which is associated with the irreversibility of time. Thus our relationship with impedance space is curiously skewed, in that we are not allowed to venture into the regions where resistance is negative; and one far-reaching consequence is that we cannot devise electrical networks which will give an output before receiving an input, i.e., we cannot build circuits which violate causality. There are however non-linear electronic devices which have a negative resistance characteristic (such as the Esaki diode or tunnel diode [10][11]), but this is only in the sense that there is a region in the graph of I vs V where the current goes down as the voltage is increased. Thus negative resistance devices can go from a particular level of power dissipation to a lower level as the applied voltage is increased, but they can never achieve a state of negative power dissipation. To put it circuit modelling terms: you only ever get negative resistance when it's in series with a larger positive resistance. In order to generalise Joule's law completely, we must write it in a way which allows the current phasor I to adopt an arbitrary phase but which gives an explicitly scalar result. The obvious candidate expression is:
P = I I* R or more to the point:
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Consider the system shown on the
right. In this case, V and I are not necessarily
in phase, but we can easily obtain an expression for I
in terms of V, and since we now appear to have a version
of Joule's law which allows I to point in any direction,
there is no need to impose a restriction on any of the phasors
involved. Thus we can write: I = V / (R+jX) |
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Now, putting the reciprocal impedance into the a+jb form
by multiplying numerator and denominator by the complex conjugate
of the denominator we obtain: I = V (R -jX) / (R² + X²) and, using the conjugate product theorem (1-21.7): I* = V* (R +jX) / (R² + X²) Now we can insert these definitions into equation (1-23.2): P=I*RI, thus: P = V V* R (R -jX)(R +jX)/[(R² + X²)²] and using the square magnitude theorem (1-21.6) we obtain:
Here we use the power factor (V I scalar product) rule (1-8.1). P = V·I = |V| |I| Cosf |
Now, using the diagram on the right, we can see that Cosf (adjacent / hypotenuse) is R/Ö(R²
+ X²). Hence:
I = V / (R+jX) and using the magnitude ratio theorem (1-21.1) we obtain: |I| = |V| / |(R+jX)| |
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i.e.: |I| = |V| / Ö(R² + X²) Now, substituting this into expression (1-23.4) we have: P = |V|² R / (R² + X²) Which is the same as equation (1-23.3) and so demonstrates that the power-factor rule is already embedded in Joule's law when we write the latter as a properly balanced and un-restricted vector equation. The universal electrical power laws can therefore be summarised as follows: |
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| Where |I| is the reading obtained from an ammeter, and |V| is the reading obtained from a voltmeter. If the impedance has no reactive component, or if the frequency is 0Hz (DC), the general formulae above revert to their standard textbook forms: |
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| Where V and I are the readings from AC or DC instruments and can be positive or negative; but if V is negative then I must be negative, so the V I product formula is best written as a magnitude to ensure that this requirement obtains. |
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1-24. Bandwidth: We are often interested the way in which the gain or loss of a network or circuit varies over a particular band of frequencies. We will introduce this type of analysis shortly in connection with resonant networks, but before doing so it is necessary to give a proper definition of the term 'bandwidth'. Most readers will be aware that an amplifier will generally show a fall-off of gain at low frequencies, this often being due to the increasing magnitudes of the reactances of coupling capacitors in series with the signal path; and it will also show a fall-off of gain at high-frequencies, this being due to a variety of factors including the falling magnitudes of the reactances of any stray capacitances in parallel with the signal path. Consequently amplifiers, and indeed many other types of circuit, usually show a hump-like frequency response; and will only pass signals usefully over a particular frequency range. The problem in defining bandwidth therefore lies in the definition of what we mean by 'useful' and, since this will vary according to the particular application, there is really no resolution to this issue. We must therefore eschew vague concepts like 'usefulness' in favour of a definition on which everyone can agree; and so it is universally accepted that bandwidth, unless stated otherwise, is defined in terms of what are known as the 'half-power points', i.e., the upper and lower frequency points at which the power delivered by the system (for a constant input) has fallen to half of that which is delivered at the frequency at which the maximum response occurs. The half-power points are chosen, as we shall see, because they have special mathematical significance; and for simple networks at least, knowledge of where they lie provides a complete definition of the frequency-response function of the system. Now, if we call the power delivered at the frequency of maximum response Pmax, then the power delivered at the half-power points is: P = Pmax / 2 and P / Pmax = ½ We can express this ratio in deciBels using the general definition: Ratio in dB = 10Log10( P / Pref ) (where Pref is the reference power level against which power P is being compared). i.e. 10Log10(½) = -3.010299957 Hence the half-power points are also known as the ' -3dB ' points, and the frequency interval between the lower point and the upper point is also often called the ' -3dB bandwidth ' It is a good idea to be specific in this way, because when the term 'bandwidth' is used without qualification, there is always the fear that it may involve some non-standard definition. |
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1-25. deciBels and Logarithms: In using deciBels, the basic approach is to consider the power levels at two points in a circuit or power transmission system and thereby define the gain. It is also useful however, to express power in relation to some external reference or standard, and this leads to an extension of the notation, the most commonly encountered variants being as follows: |
´
B
= B
= 10
. As
an alternative to using tables, the same operations can be achieved
by using two identical engraved logaritmic scales and sliding
one relative to the other, the device for so doing (invented
by William Oughtred in 1622) being known as a slide rule
[
= 1, always, regardless
of B.|
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= Ö(PR) |
= Ö(P/R) |
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dB relative to 1mW in 50W* |
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dB relative to 1mW in 600W |
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dB relative to 1W |
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dB relative to 1V |
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| * in old audio publications and service manuals, 'dBm' may be used to mean 'dB relative to 1mW in 600W'. |
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By extending the definition in this way, the dB notation may
be used to express an absolute power (rather than a relative
power); and if a reference resistance is specified, an absolute
voltage or current as well. For example, if the line output level
from an audio recorder is specified as -10dBu, then the output
voltage is obtained by rearranging the expression: -10 = 20Log(Vout/Vref) where Vref = Ö(0.001 ´ 600) = 774.6mV Hence: Vout = Vref ´ 10 
= Vref / (Ö10)
= 244.9mV RMS.The dBW notation was brought into European Amateur Radio documents some years ago, this being the preference in the field of broadcast and professional radio engineering. Thus a 400W transmitter (for example) becomes a 10Log(400)=26dBW transmitter; and it is possible to determine the effective radiated power (ERP) of a radio installation by adding the transmitter power in dBW to the (negative) gain in dB of the antenna feeder and the gain in dB of the antenna. This is all very well of course, but it does beg the question: 'why, for a group of spectrum users generally only equipped to measure voltage and resistance to a reasonable accuracy, is it necessary to state power restrictions in a way which requires a knowledge of exponential functions in order to work out what they mean?' It would seem equally logical to state road speed restrictions in dBmph or dBkm/h, and so for the sake of any bureaucrats who may be reading this, we will also address the question: 'do speed ratios require the 10Log or the 20Log formula?' This question, perhaps surprisingly, is not meaningless, and can be answered by noting that power is equivalent to energy delivered per unit-of-time. A power ratio is thus an energy per [unit-of-time] compared to a reference energy per [unit-of-time], and since the ' [unit-of-time]s ' will cancel (provided that they are the same - seconds are very popular), a power ratio is also an energy ratio. Hence: N/dB = 10Log10(E / Eref) Newton's laws of motion tell us that the kinetic energy of a moving body is given by E=mv²/2 (where m is the mass and v is the velocity), so energy is proportional to velocity squared as well as to voltage squared and current squared. Hence, a speed in dBmph is given by 20Log(v), so 30mph becomes 29.5dBmph and 70mph becomes 36.9dBmph. Now, having upgraded all of our road signs in line with the preferred notation for Government standards documents, we are only left with the problem of how to measure money in deciBels. Here we may note that currency names are often derived from weights (of silver, but there has been some devaluation since Roman times), and that Newton's and Einstein's laws tell us that mass is proportional to energy. Thus we can deduce that the 10Log formula is the correct one in this case. We might have solved this conundrum without recourse to physics however, by recalling the famous old saying: "money is power". |
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1-26. Bandwidth of a series resonator: It is well known that, when used to make filters for the RF and IF amplifiers of radio receivers, high-Q resonant circuits provide better selectivity than low-Q resonant circuits. Since good selectivity is synonymous with narrow bandwidth, there is evidently a relationship between bandwidth and Q, and as we shall see, this relationship is a particularly simple one if we define the bandwidth as the interval between the half-power (-3.01dB) points. The question we must address next therefore is: "what is this power to which we must relate the bandwidth?" The answer in the case of an amplifier driving a resistive load is obvious, it is the power dissipated in the load. In the case of a resonant circuit however, there may or may not be a load in the normal sense, and we are left with the uncomfortable conclusion that bandwidth must be defined in terms of the power which would be dissipated in the load, if there were a load. We can crack this riddle for the series resonant case by considering the circuit shown below: |
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This is the circuit of a simple band-pass filter. It has an input
voltage, an output voltage, and a load resistance; and the bandwidth
is very clearly the frequency interval between the points where
the power in the load is half of that which occurs at the frequency
of maximum response (for constant |Vin|).
It is also however, a series resonant circuit, and the Q at resonance
can be defined as: Q0 = X0L / R = -X0C / R where R is the total resistance, i.e., R=RL+RC+RLoad; and the subscript ' 0 ' has been added to the reactances as a reminder that Q0 is defined in terms of their values at the resonant frequency f0. Now, we can easily write an expression for the current which flows from the generator because the impedance connected across the generator is simply: Z = R + j(XL+XC) and so, taking the current as a reference phasor, and using Ohm's law and the magnitude ratio theorem (1-21.1), we obtain: I = |I| = |Vin| / |Z| We can also state that the power delivered to the load is: PLoad = I² RLoad and that maximum power will occur at the resonant frequency because the total reactance will then be zero and the magnitude of the total impedance |Z| will be at a minimum. Hence we will call the maximum load power P0Load, and the power at the bandwidth limits will be: PLoad = P0Load / 2 i.e., I² RLoad = I0² RLoad / 2 where I0 is the current at resonance. Hence the bandwidth limits lie at the points where I² = I0² / 2 i.e., where I = I0 / Ö2 So the load resistance, having served to allow us to define the bandwidth, has promptly vanished; and the bandwidth becomes the interval between the points where the current has fallen to 1/Ö2 of its value at resonance. Furthermore, we can observe that we will always obtain this result regardless of which resistance we define as the load. RL, RC, and RLoad are only symbols, and since the corresponding resistances are connected in series, we can swap their designations at will. We can also consider any combination of these resistances to be the load, including the total resistance R, and this will always cancel and tell us that the half-power points occur when I=I0/Ö2. Thus to define the bandwidth of a series resonant circuit, we do not need to designate any resistance as a load, we need only to consider the current. So it transpires that we can choose any resistance in a series network and analyse the power dissipated in it to determine the bandwidth; and since we are interested here in the relationship between bandwidth and Q, the obvious resistance to choose is R, the total resistance. We can always isolate a portion of R to determine the power delivered to it or the voltage across it if we so wish, this is a trivial matter of proportions; but for a general analysis, the problem simplifies to that of understanding the behaviour of the simple series LCR network shown below. The first part of the analysis is to determine the frequency response function for this circuit and plot it as a graph to see what it looks like. A good function to plot for this purpose is the ratio P/P0 vs frequency, because this ratio has a value of 1 at f0 and is also in the correct form for conversion into deciBels. The power ratio is equal to the square of the current ratio: P/P0 = I²/I0² = (I/I0)², because P=I²R and P0=I0²R. Hence we will start by obtaining an expression for the current ratio. |
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The general expression for the
current is: I = |I| = |V| / |Z| where Z=R+j(XL+XC) At the resonant frequency however, the impedance is purely resistive, so: I0 = |V| / R Hence: I / I0 = ( |V| / |Z| ) / ( |V| / R ) = R / |Z| = R / { Ö(R² + [XL+XC]²) } which, by writing the reactances explicitly, gives: |
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 , i.e.,
C=L/10. Values
for L and C were then obtained by solving the resonance formula
for L, i.e., 10  = 1/[2pÖ(LC)]
= 1/[2pÖ(L²/10)]
= 100/[2pL]L = 100/[2p´10 ]
= 1.59154943mHC = L/10 =
159.154943pFSince Ö(L/C)=100W is also the value of XL and -XC at resonance, resonant Qs of 100, 10 and 1 correspond to total resistances (R) of 1W, 10W and 100W respectively. Notice in the graphs below how the squaring pushes the curve of P/P0 downwards in comparison to I/I0. Notice also that the half power level is 1/Ö2=0.7071 for I/I0 and 1/2 for P/P0, and that the deciBel scales on the right differ accordingly. |
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So, having shown how the shape of the frequency response function
varies with resonant Q, we will now derive an expression for
the relationship between Q and bandwidth, the bandwidth being
defined as the interval between the upper and lower half-power
points. The procedure is to write a general expression for the
current and solve it for the frequencies at which I=I0/Ö2. Notice that the word "frequencies"
is plural: the expression will be a quadratic equation. As we have already determined, I=|V|/|Z|, and I0 =|V|/R. Hence, at the -3dB bandwidth limits: |
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I = |V|/|Z| = I0/Ö2
= |V|/(RÖ2) Thus the bandwidth limits occur at the frequencies where: |Z| = RÖ2 i.e., Ö(R² + X²) = RÖ2 R² + X² = 2R² X² = R² which, taking the square root of both sides and noting that there are two possibilities from so doing, gives: X = ±R Now, writing X explicitly we obtain the expression: |
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±R = 2pfL -1/(2pfC) which we must solve for f. We may proceed by putting the right hand side onto a common denominator (i.e., by multiplying top and bottom of the 2pfL term by 2pfC): ±R = [(2pf)²LC -1]/(2pfC) i.e., ±2pfCR = (2pf)²LC -1 This rearranges to: (2pf)²LC ±2pfCR -1 = 0 Which is a quadratic equation in the form af²+bf+c=0, with a=4p²LC, b=±2pCR and c=-1. Notice however, that this particular equation will have four solutions, rather than the usual two, because the b term has a '±' symbol attached to it. The reason for this is that there are both positive and negative frequency solutions for each of the band-edges. To obtain all four of these frequencies we apply the general solution for quadratic equations (1-9.1): f = [-b ±Ö(b² - 4ac) ] / 2a f = { ±2pCR ±Ö[(2pCR)² +4´4p²LC] }/(2´4p²LC) and using the substitution C=C²/C to obtain a cancellation of C from all but one term: f = {±CR ±Ö[(CR)² +4LC²/C] }/(4pLC)
f+ = {[+Ö(R² +4L/C)] + R}/(4pL) and the lower (positive) bandwidth limit is: f- = {[+Ö(R² +4L/C)] - R}/(4pL) and the bandwidth is: fw = f+ - f- = {[Ö(R² +4L/C)] + R - [Ö(R² +4L/C)] + R}/(4pL) i.e., fw = R/(2pL) . . . . . (1-26.3) Now recall that the resonant Q is defined as Q0=XL/R=2pf0L/R. Hence: Q0/f0 = 2pL/R Hence:
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1-26a. Centre frequency. One additional matter which might be of interest is that, although some authors refer to f0 as the "centre frequency", the frequency interval between the half-power points is not symmetrical about f0. We can find the mid-point or median frequency by taking the average of the upper and lower band limits, i.e., fm = (f+ + f-)/2 = [Ö(R² +4L/C) + R +Ö(R² +4L/C) - R]/(2´4pL) = [2Ö(R² +4L/C)] /(2´4pL) fm = [Ö(R² +4L/C)] /(4pL) . . . . . . (1-26.5) This quantity is only equal to f0 when R®0, i.e. (noting that (ÖL)/L=1/ÖL): fm ® [Ö(4L/C)] /(4pL) = 1/(2pÖLC) = f0 This limiting condition (or boundary condition) arises because the bandwidth of the resonant circuit is infinitely narrow when R®0, i.e., the upper and lower band-limits become coincident with f0 in the limit that R®0. It is therefore an essential property of a correct expression for the mid-band frequency, but it does give us a simplification for equation (1-26.5). Squaring (1-26.5) gives: fm² = (R² +4L/C)]/(4pL)² = (R/4pL)² + f0² but from expressions (1-26.3) and (1-26.4) given earlier, f0/Q=R/(2pL), hence: fm² = (f0/2Q)² + f0² = f0²(1 + [1/2Q]²)
Although the bandwidth function is not symmetric about its peak if we choose frequency as the horizontal (x) axis, it can be made symmetrical if we instead plot it against an appropriately chosen function of frequency. In particular, we need a frequency function such that any resonance peak is always just as far from zero-frequency as it is from infinite frequency, i.e., we need an infinity to the left of the resonance, and an infinity to the right, and both infinities must be of the same type. Such a requirement is satisfied by, and indeed is one of the principal properties of, the logatirhmic function: x = Log(f) the choice of base being arbitrary. Hence the peak can be said to be symmetric about its logarithmic centre frequency x0 = Log(f0) Since frequency can be scaled arbitrarily without affecting the shape of the bandwidth function (units of Hz are not mandatory); this matter can be "proved" numerically by plotting P/P0 against Log(f) with f0=1 and noting that the function is symmetric about x0=Log(1)=0 for any value of Q0. It should also be noted that our universe has an infinty of scales in all dimensions (microscopic to macroscopic), and it is often more natural to think in logarithmic dimensions than in linear ones. In the case of frequency, this can be seen by considering the classic representation of the electromagnetic spectrum as illustrated below. |
There is no theoretical minimum frequency on the logarithmic
scale; although the lowest electromagnetic frequency which can
be encountered in practice is the reciprical of the age of the
universe, about 1/(13.7´10 ´365.242199´24´60²)
= 2.31´10  Hz by current reckoning.
Practical DC electrical systems come nowhere close to this because
even the best stop working after a few years. Hence zero frequency
is impossible; we employ the concept merely as a mathematical
convenience for the purpose of circuit analysis. |
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1-26b. A sensible definition for Q0: Now that we have established that Q0 is an important circuit parameter, we will take the opportunity to have another look at at its definition. The point is that there is something horribly unsatisfying about writing:
Q0 = [ +Ö(-X0C X0L) ] / R which is the same as multiplying the two standard definitions and taking the positive square root (i.e., taking the geometric mean of the two definitions)? Of course: -X0C X0L = 2pf0L /(2pf0C) = L/C (i.e., the L/C ratio) hence:
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1-26c. Bandwidth function in terms of Q0:
Now, if we forcibly factorise the quantity L/C from the right hand term in the denominator we obtain:
which, noting that L/ÖL=ÖL and (ÖC)/C=1/ÖC, simplifies to:
2pÖ(LC) = 1/f0 Hence:
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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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Part 2 |
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