G3YNH info: AC theory part 5.

1. An introduction to AC electrical theory: Part 5.
Contents:

1-36. Admittance, conductance, susceptance.
1-36a. Reciprocal space counterparts.
1-37. Parallel resonator bandpass filter.

1-38. Unloaded Q of a parallel resonator.
1-38a. Current magnification.
.

1-36. Admittance, conductance, and susceptance:
The linear circuit analysis technique demonstrated so far consists of breaking the circuit down into two-terminal networks and treating those networks as impedances. This approach has allowed us to attack a wide range of problems; but it results in extremely messy algrebra when impedances in parallel are involved. Utlimately, we need a way of dealing with arbitrarily large numbers of impedances in parallel, just as we can already deal with any number of impedances in series; and it transpires that this can be achieved by defining the properties of our component two-terminal networks not in terms of impedance, but in terms of the reciprocal of impedance, this being called admittance. By so doing, we move the problem out of what we so-far think of as its natural space (impedance space) and into what is known as its reciprocal space; and the re-definition, trivial though it is in the case of phasors, is known as a reciprocal-space transformation.
     The reciprocal space transformation is a mathematical invention of James Clerk Maxwell. Its most far-reaching application is in the field of X-ray crystallograpy, it being the means by which the X-ray diffraction patterns of crystals are traced back to the inernal arrangement of atoms. Here however, we need only a simplified version, because the problems we wish to solve are strictly two-dimensional.
     The reciprocal of impedance space is known as admittance space. A pair of two-dimensional reciprocal spaces has the property that straight lines in one appear as circles in the other (a correspondance which we will use in chapter 5); but the real power of the transformation lies in the fact that phasor problems requiring the double-slash product in one space, become problems of addition in the other.
     Converting an impedance into an admittance is simply a matter of taking the reciprocal. Admittance is usually given the symbol Y (and here we put it in bold because it is complex), hence: Y=1/Z. Now, if Z=R+jX this gives:
Y = 1/( R +jX )
which can be put into the a+jb form by multiplying the numerator and denominator by the complex conjugate of the denominator, i.e.:

Y =	R - jX (R + jX) ( R - jX)

hence:

Y =	R - jX (R² + X²)

This expression can be written:

Y = G +jB

where the real part of the admittance, G, is called the conductance, and the imaginary part, B, is called the susceptance (of the network under consideration). From the above, we can extract definitions for conductance and susceptance which are:

Conductance,	G = R / (R² + X²)
Susceptance,	B = -X / (R² + X²)

Now observe that when the impedance of a network is purely resistive, the conductance is 1/R, and G=1/R is the definition of conductance in DC electrical theory. When an impedance is purely reactive, the susceptance B=-1/X (susceptance has no DC counterpart).
Admittance, conductance, and susceptance, of course, have units; and the modern unit in this case is the Siemens, which is given the dimension symbol capital S (as opposed to the second, which has a small s). In old textbooks and papers, the unit of admittance is often given as the 'Mho' (Ohm spelt backwards), but in either case, the actual dimensions are in reciprocal Ohms, i.e., /W or W

. A pure resistance of 50W therefore corresponds to a conductance of 1/50 Siemens, i.e., 20 milli-Siemens or 20mS. A pure reactance X=100W corresponds to a susceptance B=10mS, and so on. "Siemens", incidentally, is a name, like "Jones". The singular of Jones is not Jone, and so Siemens keeps its final s in both singular and plural forms (one Siemens, several Siemens). The plural "Siemenses" is not recommended (but is a lot less embarrassing than the quasi-singular "Siemen").

The double-slash product was previously defined (in section 1-14.5) as:
a // b = ab/(a+b)
We can demonstrate that addition is the reciprocal-space counterpart of the double slash operator by transforming the parallel impedance formula; i.e., if:
Z = Z1 Z2 / (Z1 + Z2)
then
Y = (Z1 + Z2) / Z1 Z2.
If we let Y1=1/Z1 and Y2=1/Z2, then
Y = Y1 Y2 [ (1/Y1) + (1/Y2) ]
Which rearranges to:

Y = Y1 + Y2

i.e., when two networks are placed in parallel, their admittances are added.
Recall that the formula for resistances in parallel, R=R1R2/(R1+R2) is a rearrangement of the expression:
1/R = (1/R1) + (1/R2)
It should now be apparent, that what the formula really says is:
G = G1 + G2

The formula for impedances in parallel is of course a rearrangement of:
1/Z = (1/Z1) + (1/Z2)
and this expression can be extended to cover any number of impedances in parallel by adding more terms, i.e.:
1/Z = (1/Z1) + (1/Z2) + (1/Z3) + . . . . . + (1/Zn)
This is a sum of admittances, and may be re-written as:
Y = Y1 + Y2 + Y3 + . . . . . + Yn
We can express this result using the double slash notation:

1/( Z1 // Z2 // Z3 // . . . // Zn ) = Y1 + Y2 + Y3 + . . . . . + Yn
where Yx = 1/Zx (x being any subscript).

The admittance representation of an electrical circuit is no less authoritative than the impedance representation, and is no more difficult to use. Admittances are phasors, and all of the phasor techniques we have developed in this chapter will work on them. It is however, helpful to remember that a (numerically) large admittance corresponds to a small impedance and vice versa.

1-36a. Reciprocal-space counterparts:

Impedance space	Admittance space
Impedance Z = R+jX = 1/Y	Admittance Y = G+jB = 1/Z
Resistance R = G/(G²+B²)	Conductance G = R/(R²+X²)
Reactance X = -B/(G²+B²)	Susceptance B = -X/(R²+X²)
Pure resistance R = 1/G	Pure conductance G = 1/R
Pure reactance X = -1/B	Pure susceptance B = -1/X
Inductive reactance XL = 2pfL	Inductive susceptance BL = -1/(2pfL)
Capacitive reactance XC = -1/(2pfC)	Capacitive susceptance BC = 2pfC
// operator	+ operator
+ operator	// operator
Straight line	Circle
Circle	Straight line

Note: The admittance definition used here is taken from Hartshorn [17]. Other authors, including Langford-Smith [8] define admittance as Y = G-jB. This gives B=X/(R²+X²),
BL=1/(2pfL), and BC=-2pfC. The alternative definition results in an expression for the bandwidth of a parallel resonator which is negative relative to the accepted form, i.e., it deviates from the convention that frequency is positive. This, and other considerations of mathematical symmetry, indicate strongly that the definition given in the table above is the one that should be adopted.

1-37. The parallel resonator bandpass filter:
The reader may have noticed that, having determined the relationship between bandwidth and Q for a series resonator, we did not immediately do the same for a parallel resonator, but instead digressed into the subjects of source impedance and impedance transformation. There was a very good reason for doing so, as we shall soon see, which is that there is no satisfactory design procedure for parallel-resonant bandpass filters if the source and load impedances cannot be controlled. This situation prevails because, in order to use the resonator as a filter, we need methods for injecting energy into it and extracting energy from it, and the impedances presented by these input and output networks affect the Q.
The prototype band-pass filter is shown below. The generator and load coupling scheme used is not the only one possible, but all other schemes are equivalent to this one after suitable transformation. Here we inject energy via a source resistance Rs, which is the sum of the generator output resistance and any additional resistance placed in series with it. Rp is the parallel combination of the resonator dynamic resistance and any load resistance which might be placed across it. Notice that we have provided the model with source and load resistances rather than impedances. We are at liberty to do so without affecting the generality of the analysis, because any reactive components in the source and load impedances will turn out to be effectively in parallel with the resonator. This means that these additional reactances will modify the effective values of XCp and XLp (i.e., they will change the resonant frequency), but they will not affect the general circuit behaviour.

If we define V0 as the output voltage at resonance, then the bandwidth function is |V/V0| plotted against frequency. We can write expressions for V and V0 by treating the circuit as a potential divider, thus, noting that XCp // XLp ®¥ at resonance (f = f0):
V0 = Vg Rp / ( Rs + Rp )
and if we choose the generator voltage as our phase reference we can drop dimensions:

V0 = Vg Rp / ( Rs + Rp )
We will also avail ourselves of a useful property of the potential divider formula (equation 1-29.2a), which is that if we multiply it by a unit quantity consisting of the source resistance divided by itself (i.e., Rs/Rs ), it becomes a double-slash product:
V0 = Vg ( Rp // Rs ) / Rs
Similarly, for the output voltage in general:
V = Vg ( Rp // jXCp // jXLp ) / [ Rs + ( Rp // jXCp // jXLp ) ]
and using the associative rule (section 1-14.4):
V = Vg ( Rs // Rp // jXCp // jXLp ) / Rs
So we can write the ratio V/V0 as:
V/V0 = ( Rs // Rp // jXCp // jXLp ) / ( Rp // Rs )
The bandwidth function is the magnitude of this expression; but with all of the components represented as impedances, anyone attempting to expand and simplify it, or isolate part of it as the load, is in for a hard time. We will therefore convert it into an admittance problem, using the relationship:
1/( Z1 // Z2 // Z3 // . . . // Zn ) = Y1 + Y2 + Y3 + . . . . . + Yn

Hence:

1 / ( Gs + Gp + jBCp + jBLp )

1 / ( Gp + Gs )

Where G stands for conductance and B for susceptance, and Gs=1/Rs , Gp=1/Rp , BCp=-1/XCp and BLp=-1/XLp. The expression above can be re-written:

Gs + Gp

Gs + Gp + jBCp + jBLp

and the magnitude is:

|V|

Ö[ (Gs + Gp)² ]

Ö[ (Gs + Gp)² + (BCp + BLp)² ]

i.e.,

|V|

Gs + Gp

Ö[ (Gs + Gp)² + (BCp + BLp)² ]

This can be plotted against frequency by substituting BCp=2pfCp and BLp=-1/(2pfLp), but we will not bother to do so here because it is identical in appearance to the graph of |I|/I0 for a series resonator given in section 1-26. We will instead go on to determine the half-power points by noting that, whatever proportion of the parallel resistance Rp is designated as the load, power will always be delivered to it in proportion to |V|², so the half-power points occur when |V|=V0/Ö2. Hence, at the half-power points:

(Gs + Gp)²

(Gs + Gp)² + (BCp + BLp)²

Which, upon inversion gives:

(BCp + BLp)²

(Gs + Gp)²

+ 1 = 2

i.e.:

(BCp + BLp)² / (Gs + Gp)² = 1
and taking the square root:
(BCp + BLp) / (Gs + Gp) = ±1
Thus:
BLp + BCp = ±(Gs + Gp)
and if we define the sum Gs+Gp as GQ (i.e., the conductance which determines the Q):
BLp + BCp = ±GQ
Now, using the substitutions BCp=2pfCp and BLp=-1/(2pfLp), we obtain:
[ -1/(2pfLp) ] + 2pfCp = ±GQ
and by factoring out 1/(2pfLp) from the left hand side and re-arranging:
±2pfLpGQ = -1 + (2pf)²LpCp
i.e.,
[4p²LpCp]f² ±[2pLpGQ]f -1 = 0
This is a quadratic equation in f with a=4p²LpCp , b=±2pLpGQ , and c=-1. It has four solutions as was the case for the series resonator (section 1-26), these being the upper and lower bandwidth limits for positive and negative frequencies. To solve it we apply the standard formula:
f = [-b ±Ö(b² - 4ac) ] / 2a
Hence:
f = { ±2pLpGQ ±Ö[(2pLpGQ)² + 4´4p²LpCp] } / (2´4p²LpCp)
and using the substitution Lp=Lp²/Lp to obtain cancellation of Lp from all but one term:
f = { ±LpGQ ±Ö[(LpGQ)² + 4(Lp²/Lp)Cp] } / (4pLpCp)

f = { ±GQ ±Ö(GQ² + 4Cp/Lp) } / (4pCp)

Now, since Ö(GQ² + 4Cp/Lp) will always be larger than GQ, we can identify the positive frequency upper bandwith limit as:
f+ = { [Ö(GQ² + 4Cp/Lp)] + GQ } / (4pCp)
and the positive frequency lower bandwidth limit as:
f- = { [Ö(GQ² + 4Cp/Lp)] - GQ } / (4pCp)
and the bandwidth is:
fw = f+ - f- = GQ/(2pCp)
This is the admittance counterpart of the result obtained at this stage in the derivation of the Q of a series resonator (equation 1-26.3) and so we will deduce that the bandwidth of the parallel resonator BPF is f0/Q0, and use this deduction to find a definition for Q.
f0/Q0 = GQ/(2pCp)
2pf0Cp = BCp0 = Q0 GQ
Q0 = BCp0 / GQ
Now let RQ=1/GQ, where RQ is "the resistance which determines the Q ". Also observe that BCp0=-1/XCp0., and at resonance -XCp0=XLp0 Hence:

Q0 = - RQ / XCp0 = RQ / XLp0

or, in keeping with the definition of resonant Q given in section 1-26b:

Q0 = RQ / Ö(Lp/Cp)

1-37.1

RQ is simply the parallel combination of the source resistance, the load resistance, and the dynamic resistance of the resonator, i.e.:
RQ = Rs // Rp0 // RLoad

This result gives us the theoretical information we need in order to be able to design parallel resonant bandpasss filters. Firstly, we may observe that the source and load impedances are effectively in parallel with the resonator, which is why any minor source and load reactances can be lumped with the resonator reactances and cause only a detuning effect (if such reactances are very large however, they will cause a significant change in the dynamic resistance and the problem is best re-analysed from scratch). The source, load, and dynamic resistances however, are critical in determining the Q, and we need to obtain high values for all of them in order to obtain a high Q. We can of course adjust the source and load resistances using transformers; and as we shall see shortly, we can replace the resonator coil with a transformer so that the inductor and the transformer become one and the same. Before we look at such coupling schemes however, we must draw attention to a particularly misleading inference of the formula, which is that high Q can be obtained by making the ratio Lp/Cp as small as possible. Some authors state this to be the case, but they are wrong. If the reactive components are of reasonable quality, the parallel form L/C ratio (Lp/Cp ) is only slightly different from the series form L/C ratio, and as we showed in section 1-18, imaginary resonance can occur if the L/C ratio becomes too low. The imaginary resonance condition is entirely a function of the series (loss) resistances of the coil and the capacitor. It is nothing to do with the source and load resistances because avoidance of imaginary resonance is a matter of ensuring that the +90° component of the coil current at resonance is sufficiently large to cancel the -90° component of the capacitor current (or vice versa, but in practice coils are more lossy than capacitors). Consequently, the design procedure for a paralell resonator BPF is to make the L/C ratio large enough to obtain a good strong resonance (without making the inductance so large that the coil self-resonance occurs in the frequency range of interest), and then to make RQ even larger in order to obtain a useful working Q.

1-38. Unloaded Q of a parallel resonator:
The expression for Q obtained in the previous section is sometimes referred to as the loaded Q of the resonator, because it is the Q which results when the source and load impedances are taken into account. We may also imagine that the resonator has an unloaded Q, which is that which obtains when the source and load are disconnected. It is not immediately obvious why we should wish to employ such a concept, because it is impossible to use the resonator without connecting it to something; but it is nevertheless useful because it sets an upper limit on the Q which can be obtained in a practical circuit. It is obviously obtained by substituting the dynamic resistance in place of RQ in equation (1-37.1), i.e.,
Q0u = Rp0 / Ö(Lp/Cp)
but it would be a lot more useful intuitively if we could express it in terms of the coil and capacitor impedances in their series (R+jX) forms. We can do so by using the series-to-parallel transformation (section 1-16); and using the definition of Q from section 1-26b as precedent, we expect a result in the form:
Q0u = [Ö(L/C)] / R . . . (1-38.1)
the point being to find out what is meant by R in this case.

The translation from parallel to series form is indicated in the set of equivalent circuits shown below:

Here we identify Rp0 as RCp//RLp, i.e.:
Rp0 = RCp RLp / ( RCp + RLp )
and an expansion in terms of the series forms of the impedances has already been given as equation (1-17.5):

Rp0 =

(RC² + XC²) (RL² + XL²)

RL(RC² + XC²) + RC(RL² + XL²)

The unloaded Q is defined as:
Q0u = Rp0 / Ö(-XCp XLp)
and, from the series-to-parallel transformation (equation 1-16.3b), we have:
XCp = (RC² + XC²) / XC . . . (1-38.2)
and
XLp = (RL² + XL²) / XL . . . (1-38.3)
Putting all of this together we have:

Q0u

(RC² + XC²) (RL² + XL²) / [ RL(RC² + XC²) + RC(RL² + XL²) ]

Ö[ -(RC² + XC²) (RL² + XL²) / XC XL ]

and noting that Ö(-XC XL )=Ö(L/C), this rearranges to:

Q0u

[Ö(L/C)] {Ö[ (RC² + XC²) (RL² + XL²) ] }

[ RL(RC² + XC²) + RC(RL² + XL²) ]

So, at this point we have extracted Ö(L/C) as required by equation (1-38.1), and the resistance by which Ö(L/C) must be divided in order to obtain Q0u is:

[ RL(RC² + XC²) + RC(RL² + XL²) ]²

(RC² + XC²) (RL² + XL²)

1-38.4

Which, upon expanding the numerator gives:

R²

RL²(RC²+XC²)²

(RC²+XC²)(RL²+XL²)

RC²(RL²+XL²)²

(RC²+XC²)(RL²+XL²)

2RLRC(RC²+XC²)(RL²+XL²)

(RC²+XC²)(RL²+XL²)

The simplification we require here comes from noting that the terms (RC²+XC²) and (RL²+XL²) occur in the expressions for XCp and XLp given above (equations 1-38.2 and 1-38.3), and that at resonance -XCp=XLp. Hence:
(RC² + XC²) / (-XC) = (RL² + XL²) / XL
i.e.:
(RC² + XC²) / (RL² + XL²) = -XC / XL and (RL² + XL²) / (RC² + XC²) = XL /-XC
Hence:
R² = RL²(-XC/XL) + RC²(XL/-XC) + 2RLRC
which can be factorised:
R² = { RL[Ö(-XC/XL)] + RC[Ö(XL/-XC)] }²
Hence:
R = RL[Ö(-XC/XL)] + RC[Ö(XL/-XC)] . . . (1-38.5)
(strictly ±R, but resistance is always positive, so we will ignore the negative solution) and so:

Q0u = [Ö(L/C)] / { RL[Ö(-XC/XL)] + RC[Ö(XL/-XC)] }

This is an exact solution provided that RL and RC do not vary; and once again may be assumed exact for normal engineering purposes because RL and RC will not vary significantly in the vicinity of the resonant frequency. Note however that XL=-XC to an extremely good approximation when the L/C ratio is reasonably large, and this relationship is exact when RL=RC. Hence, for most practical purposes:

Q0u = [Ö(L/C)] / (RL + RC)

1-38.6

Which means that the unloaded Q of the parallel resonator is the same as that of the series resonator, it is the square root of the L/C ratio divided by the total series resistance. In other words, we can estimate the unloaded Q of the parallel resonator by considering it to be a series resonator connected as a loop.

1-38a. Current magnification:
There is another way to determine the unloaded Q of a parallel resonator, which stems from the observation, that just as a series resonator exhibits the phenomenon of voltage magnification, the parallel resonator exhibits current magnification. In effect, the parallel resonator is a series resonator connected in a different way, because its characteristics at resonance are principally determined by a large circulating current, and the current it draws from the generator is small in comparison (Q times smaller than the circulating current in fact).

In the diagram on the right, the current I flowing into the resonator is IL+IC, where:
IC = V /(RC +jXC) = V(RC -jXC)/(RC²+XC²)
and
IL = V /(RL +jXL) = V(RL -jXL)/(RL²+XL²)
but at reasonance I is real, which means that the imaginary parts of IL and IC add up to zero, and the total current at resonance becomes:

I0 =

(RC²+XC²)

(RL²+XL²)

(V and I0 are now in phase and will be taken as real). Putting the expression onto a common denominator yields:

I0 =

RC(RL²+XL²) + RL(RC²+XC²)

(RC²+XC²) (RL²+XL²)

1-38.7

where the term inside the square brackets is the reciprocal of the dynamic resistance (see equation 1-17.5), i.e., I0=V/Rp0.

Now, the current circulating in the resonator can be determined from either branch as the total current flowing in the branch, less the current drawn from the generator. Hence the circulating current is simply the imaginary part of the current in the branch. The resonant condition (and the concept of circulation) also implies that the circulating current is of the same magnitude for both branches, but of opposite sign. Hence, if we call the circulating current IQ, then (taking the imaginary parts of the expressions for IC and IL above) we have:
IQ = jVXL/(RL²+XL²) = -jVXC/(RC²+XC²)
and the magnitudes are:
|IQ| = VXL/(RL²+XL²) = V(-XC)/(RC²+XC²)
We can also create a definition involving both branches by taking the geometric mean:
|IQ| = V Ö{ (-XCXL) / [ (RC²+XC²) (RL²+XL²) ] }
which allows us to extract the L/C ratio:
|IQ| = V Ö{ (L/C) / [(RC²+XC²) (RL²+XL²)] }

Now, let us define the unloaded Q of the resonator as the ratio of the circulating current to the through current:
Q0u = |IQ| / I0
Which can be expanded using equation (1-38.7):

Q0u' =

(L/C) / [(RC²+XC²) (RL²+XL²)]

[RC(RL²+XL²)+RL(RC²+XC²)]² / [(RC²+XC²)(RL²+XL²)]²

and rearranged:

Q0u' = [Ö(L/C)]

(RC²+XC²)(RL²+XL²)

[ RC(RL²+XL²) + RL(RC²+XC²) ]²

The rightmost square-root bracket is simply the reciprocal of R as defined in equation (1-38.4); hence:
Q0u = [Ö(L/C)] / R
and we have proved that the current-magnification definition for unloaded Q is identical to that obtained on the assumption that Q is the magnitude of the resonant frequency divided by the bandwidth of the resonator.

The only residual issue is that of why the exact expression for R is (as given by equation 1-38.5):
R = RL[Ö(-XC/XL)]+RC[Ö(XL/-XC)]
rather than simply R=RL+RC. This however can be understood by noting that the (real) current flowing through the resonator will be very slightly biased in favour of the branch with the lowest resistance. This difference is very small for practical resonators of moderate unloaded Q, and may normally be ignored.

TX to Ae

Ch 1 Contents

Part 4