 TX to Ae Z matching - Part 2 >>

Impedance Matching. Part 1: Basic Principles.
By David Knight G3YNH and Nigel Williams G3GFC
Contents:
 0. Introduction. 1. Z-plane operations. 2. Series reactance and resistance. 3. Parallel impedance. 4. Admittance, conductance, susceptance. 5. Equation of a circle. 6. Parallel resistance (constant B). 7. Parallel reactance (constant G) . 8. Transformers (constant φ) 9. Impedance matching strategies. 10. Z-plane regions. Part 2 >>> .

 1. Z-Plane Operations: The impedance matching process is best considered as a set of operations in the impedance plane, or Z-plane. The Z-plane is simply a graph of R against jX on which a given impedance can be plotted as a point. The Z-plane therefore corresponds to a mathematical space; impedance space, in which all impedances can be considered to lie. A matching network can be regarded as a toolkit of components that can be combined with an impedance in some fashion in order to move the resultant impedance to a new position in the Z-plane.To represent this process, we will use the notation: Z → Z' which is pronounced: "Z goes to Z prime", or "Z tends towards Z prime". The prime (single apostrophe) symbol is generally used to indicate modification, i.e., Z' is different from Z but is related to it because it results from an operation performed on Z.      In general, impedance matching is a matter of manipulating the point Z by the addition of some combination of transformers and series and parallel impedances. In practice, for reasons of efficiency, and especially in the case where the impedance to be matched is an antenna system; the manipulation is performed using only high Q coils and capacitors, and any transformers used should preferably be of the transmission-line variety.

 2. Series Reactance and Resistance: The simplest operation that can be carried out on an impedance is to place a resistance or reactance in series with it. As was discussed in chapter 1; if a pure resistance is placed in series with an impedance Z, it simply adds to the R-part of Z without affecting the X-part. If a pure reactance is placed in series with Z it adds to the X-part without affecting the R-part. Thus if an impedance Z=R+jX has a reactance XS placed in series with it, the new impedance is Z'=R+j(X+XS). If Z has a resistance RS placed in series with it, then Z'=(R+RS) +jX.      The effect of these operations in impedance space is shown below: Note that while we can only add resistance (so moving Z to the right), we can add or subtract reactance, because inductors and capacitors have opposite effects, and so move freely up or down in the vertical direction. Vertical lines in the Z-plane (i.e., parallel to the jX axis) are known as lines of constant resistance. Horizontal lines (parallel to the R axis) are known as lines of constant reactance.      Note that if an inductor is used, increasing L (and hence XL) moves Z' upwards. If a capacitor is used however, reducing C (increasing the magnitude of Xc) moves the Z' downwards, i.e., a series capacitor must be initially large to have no effect, and its effect increases as its capacitance is reduced.      Resistance, of course, is not normally used for antenna matching purposes; but in situations where a moderate loss of power is acceptable or desirable, such as in matching the input impedance of a linear amplifier, there may be reason to include it in the matching network. Understanding the effect of resistance is also important for an understanding of the effect of practical components, particularly inductors, which often have substantial loss resistance at radio frequencies.

 3. Parallel Impedance: In [AC Theory, Section 14] a demonstration was given of what might be called the 'brute-force' solution to the problem of finding out what happens when impedances are connected in parallel. The resulting expression is shown below for an impedance Zp=Rp+jXp placed in parallel with an initial impedance Z=R+jX, i.e., it is the expansion of the parallel impedance formula Z'=ZZp/(Z+Zp).

 Z' = RRp(R+Rp) +RXp² +RpX²(R + Rp)² + (X + Xp)² +j XXp(X+Xp) +XRp² +XpR²(R + Rp)² + (X + Xp)²

 This equation, while possessing a certain symmetry, offers a version of what is happening that is extremely difficult to visualise; and even hardened mathematicians like to think in pictures. There is a better way of looking at the problem, which involves re-defining it in terms of the reciprocal of impedance, i.e., 1/Z, this quantity being known as admittance. This alternative approach (which is the basis of the Smith chart) does not particularly simplify the expressions we need to use when writing computer programs or performing spreadsheet analyses, but it scores on two major points: it allows us to view the parallel impedance problem as a simple matter of addition, and it allows a straightforward graphical way of visualising the effect of placing a resistance or a reactance in parallel with an impedance.

 4. Admittance, Conductance, Susceptance. The subject of admittance was introduced in [AC Theory, section 44]. There we defined admittance as Y=1/Z, where, as usual, Z=R+jX. From this we obtained the relationship: Y = (R - jX) / (R² + X²) which can be written Y = G + jB where the real part of the admittance, G, is called the conductance, and the imaginary part of the admittance, B, is called the susceptance (of the network under consideration). Hence the definitions for conductance and susceptance are: G = R / (R² + X² ) B = -X / (R² + X² ) Now observe that if a network is purely resistive, it has no susceptance, i.e., Y=G+j0. Consequently, if we place a pure resistance Rp (with Gp=1/Rp) in parallel with an impedance Z (with Y=1/Z) the resultant admittance is given by: Y' = ( G + Gp ) + jB. This means that, provided that it is greater than zero (i.e., not a short-circuit), a parallel resistance cannot alter the original value of susceptance. The resultant impedance is therefore constrained to follow a path of constant susceptance as the parallel resistance is varied. Similarly, if a network is purely reactive, it has no conductance, i.e., Y=0+jB. Consequently, if we place a pure (non zero) reactance Xp (with Bp=-1/Xp) in parallel with an impedance, the modified admittance is: Y' = G + j( B + Bp ) Which indicates that the parallel reactance cannot alter the original value of conductance, and the resultant impedance must therefore be constrained to lie on a path of constant conductance. It transpires that when susceptance or conductance is held constant, the path on which an impedance is constrained to lie takes on a very simple form. The curve in question was no doubt discovered (many years ago) by someone who plotted a graph of it the hard way (using the parallel impedance equation) and became suspicious of the shape. Here we will pre-empt the discovery, by exploring the properties of the simplest object that can be drawn using a pair of compasses.

 5. The Equation of a Circle. The diagram above shows a circle of radius r, with its centre C placed arbitrarily at a point (Ro, Xo). In order to produce an equation for this circle (an hence any circle), all we have to do is write an expression for the radius that is true regardless of how we measure it; i.e., regardless of where we put the point Z from which the distance from C to Z (the radius) is measured. Since r is the hypotenuse of a triangle with sides X-Xo and R-Ro, the equation is given by Pythagoras' theorem, i.e.:
 r² = (R - Ro)² + (X - Xo)²
This is the general equation of a circle, expressed for our convenience using impedance-related symbols. Notice that it is quadratic (involves powers of 2) in both R and X, reflecting the fact that there are two values of R for every X and vice versa. If the expression is expanded, it becomes:
 R² + X² -2RRo -2XXo + Ro² + Xo² - r² = 0
Now observe that it contains a term R²+X². We should therefore be mindful of the fact that when a complex number Z=R+jX is multiplied by its complex conjugate Z*=R-jX, the result, ZZ*=R²+X². Recall also that the magnitude of an impedance, |Z|=√(R²+X²). The equation of a circle is also an expression of magnitude, it being the curve traced out when a vector of fixed magnitude r is allowed to point in any direction. It appears that there may be a relationship between circles and complex numbers (but perhaps we would not have engaged in this exercise if there wasn't). In fact, any expression that contains the structure R²+X² describes a circle provided that it does not also contain cross-terms (i.e., terms containing RX) , and provided that the value for the radius is not zero or complex (i.e., it must be measurable in the plane in which it is defined, and since the radius is a magnitude it must be real).

 6. Parallel Resistance: Earlier in this article, from a derivation given in [AC Theory, Section 14], a general expression was given for the impedance that results when two impedances are placed in parallel. If one of the impedances is purely resistive, the expression reduces to:

 Z' = Rp²R +RpR² +RpX²(R + Rp)² +X² +j X Rp²(R + Rp)² +X²

6.1

where Z' is the impedance that results when an initial impedance Z=R+jX is modified by the connection of a parallel resistance Rp. Notice that when Rp becomes extremely large, the Rp² term in the denominator becomes dominant (i.e., much larger than anything else) and so only terms in the numerator that contain Rp² can survive being divided by the denominator without vanishing. Thus as R→∞, Z'→R+jX. Common sense also tells us that when Rp=0, the resulting impedance is a dead short, i.e., Z'=0+j0.
The effect of a resistance Rp in parallel with an impedance Z=R+jX can also be considered as the effect of a conductance Gp=1/Rp in parallel with an admittance Y=G+jB, in which case:
Y' = G + Gp +jB.
Since the susceptance B=-X/(R²+X²) is completely defined by the original impedance, it remains constant regardless of the value of Rp. Therefore we may also write:
B = B' = -X' /( R'² + X'² )
i.e., the new susceptance is the same as the old susceptance. This equation may be rewritten:
 R'² + X'² +X'/B = 0 6.2
which is the equation of a circle.
Compare this with the general equation of a circleof radius r with its centre at Ro,Xo (with the variables changed to X' and R' to suit the current problem):
R'² + X'² -2R'Ro -2X'Xo + Ro² + Xo² - r² = 0
Note that in expression (6.2) there is no term equivalent to -2R'Ro, and so Ro must be zero; which means that the centre of the circle lies on the X axis. This reduces the circle equation to:
R'² + X'² -2X'Xo + Xo² - r² = 0
There are no terms equivalent to Xo²-r², therefore Xo²=r², therefore Xo=±r. The circle just grazes the point R=0, X=0.
Finally, we may equate -2X'Xo=X'/B, i.e., Xo=-1/(2B). Therefore the centre of the circle lies at Ro=0, Xo=±1/(2B) (there being two alternatives depending on the sign of X in the original impedance), and the radius is 1/(2B).
The conclusion is that when a resistance Rp is placed in parallel with an impedance Z=R+jX, the resultant impedance Z' moves on a circle, which has its centre at 0-j/(2B), and a radius of 1/(2B), where B=-X/(R²+X²). Z' is constrained to lie between the original impedance R+jX and 0+j0, and since R is always positive in conventionally defined impedance problems, the curve is only part of a circle (an arc) and so is called an arc of constant susceptance. In the diagram above, the curve is shown for positive X (and hence negative B). If X is negative, the curve has the same shape but is reflected about the R axis. Note that if X is greater than R, reducing the parallel resistance can actually cause the resistive component of the resultant impedance (R') to increase to a maximum of |1/(2B)| when X'=±j/(2B). The effect of Rp however, is always to reduce the load phase angle, i.e., it has the effect of swamping the reactance, but it can never bring the phase angle to zero except at 0+j0.
To give an idea of the rate at which Z changes as the parallel resistance is reduced, the table on the right shows computed values of R' and X' for various values of parallel resistance, Rp, connected across an initial impedance of 100+j100 Ω. Equation (6.1) was used for the calculation. Notice, in this instance, that the parallel resistance has little effect on the resistive component of the resultant impedance, R', provided that it is about 10 or more times greater than the initial resistive component R. In general, when the initial reactance X is large in comparison to R, parallel resistance has a stronger effect on X' than it does on R', but this is no longer true when X is small.

 Rp R' X' ∞ 100 100 32768 99.998 99.392 16384 99.993 98.787 8192 99.971 97.588 4096 99.886 95.236 2048 99.567 90.709 1024 98.429 82.346 512 94.799 68.170 256 85.373 47.929 128 67.734 26.433 64 45.794 11.101 32 27.071 3.734 16 14.734 1.091 8 7.681 0.295 4 3.920 0.077 2 1.980 0.020 1 0.995 0.005 0 0 0

 7. Parallel Reactance: In the case of an impedance shunted by a pure reactance, the general expression for impedances in parallel reduces to:
 Z' = RXp²R² + (X+Xp)² +j XXp(X+Xp) +XpR²R² + (X+Xp)²

7.1

where Z' is the impedance that results when an initial impedance Z=R+jX is modified by the connection of a parallel reactance Xp. Notice that when the magnitude of Xp (i.e., its value regardless of sign) is infinite, then only terms in the numerator containing Xp² can survive being divided by the denominator, and the expression reduces to Z'=R+jX. Also, when Xp=0, i.e., when the initial impedance is shunted by a very small inductance or a very large capacitance, then Z'→0+j0.
The effect of a reactance Xp in parallel with an impedance Z=R+jX can also be considered as the effect of a susceptance Bp=-1/Xp in paralell with an admittance Y=G+jB, in which case:
Y' = G +j(B + Bp)
Since the conductance G=R/(R²+X²) is defined by the original impedance, it remains constant regardless of the value of Xp. Therefore we may also write:
G = G' = R' /( R'² + X'² )
i.e., the new conductance is the same as the old conductance. This equation may be rewritten:
 R'² + X'² -R'/G = 0 7.2
which is the equation of a circle.
Compare this with the general equation of a circleof radius r with its centre at Ro,Xo:
R'² + X'² -2R'Ro -2X'Xo + Ro² + Xo² - r² = 0
Equation (7.2) has no terms equivalent to -2X'Xo, and so Xo=0. The centre of the circle lies on the R axis. The circle equation is therefore reduced to:
R'² + X'² -2R'Ro + Ro² - r² = 0
Equation (7.2) has no terms equivalent to Ro²-r², therefore Ro=r (strictly, Ro=±r, but here we will confine our analysis to positive resistances). The circle just grazes zero.
Finally we may equate -2R'Ro=-R'/G, i.e., Ro=1/(2G). The centre of the circle lies at
Ro=1/(2G), Xo=0, and its radius is 1/(2G).
Thus we conclude that the effect of adding reactance in parallel with an impedance is to move the point Z' around a circular path. The circle so described has its centre on the R axis at
[1/(2G)]+j0, just touches the origin of the graph (0+j0), and is known as a circle of constant conductance. The direction of rotation in the Z-plane is clockwise when a capacitor is connected across Z and anticlockwise when an inductor is connected across Z. The effect of a parallel reactance is least when the reactance is at its highest value, and increases at the reactance is reduced. Consequently, if a variable capacitor is used, increasing its capacitance (reducing Xc) moves the resultant Z in a clockwise direction; and if a variable inductor is used, reducing its inductance (reducing XL) moves the point Z in an anticlockwise direction. Note that the point 0+j0 is only approached if the parallel capacitance becomes infinite (Xc→0), or the parallel inductance goes to zero (XL→0), both situations corresponding to an effective short-circuit.

 8. Transformers:
 In [AC Theory, Section 41], it was shown that the effect of a tightly coupled conventional transformer, to a good first approximation at least, is to carry out an impedance transformation according to the square of the turns ratio. Z' = ( Np / Ns )² Z Thus a transformer scales the load impedance by a factor (Np/Ns)², where Np is the number of turns in the primary winding, and Ns is the number of turns in the secondary winding. This operation moves the resultant impedance Z' along a line starting at 0+j0 and passing through the original impedance Z. The transformer modifies the magnitude of the impedance but leaves the phase angle φ unchanged, thus the resultant impedance is constrained to lie on a line of constant phase. This type of transformation can be achieved using both conventional and transmission-line transformers; but transmission-line transformers are to be preferred on the grounds of efficiency, bandwidth, and freedom from core-saturation effects. Any transformer included in a matching unit will also normally carry out a balanced to unbalanced load transformation if required, i.e., the transformer is also the balun.      Dr Jerry Sevick, W2FMI (see references  - ) has described highly efficient wideband toroidal cored baluns and ununs, with impedance ratios of 1:1, 4:1, 9:1 and 16:1 (i.e., turns ratios of 1:1, 2:1, 3:1 and 4:1), and slightly less efficient intermediate ratios. When driving a centre-fed wire antenna, a 1:1 balun is normally used, but other ratios can be convenient for shifting awkward impedances into different parts of the Z-plane.

 9. Impedance Matching Strategies: From the above, it can be deduced that the process of matching an impedance to (say) 50Ω without introducing unnecessary losses, is a matter of manipulating the transmitter (generator) load Z' on to the point 50+j0 by moving it along lines of constant resistance and around circles of constant conductance. A switchable transformer gives the added possibility of moving Z' along a line of constant phase. All of the possible operations not involving resistance are summarised in the diagram below: Thus we are free to move anywhere in the Z-plane, but constrained in terms of the paths that can be taken. In order to work-out which operations should be used, we can consider the problem backwards by observing the attributes of the target impedance, i.e., 50+j0 Ω: The first point to note is that the target impedance, 50+j0 Ω, lies on the 50Ω constant resistance line. An initial impedance Z that does not lie on this line can always be brought on to it by moving it around a circle of constant conductance, i.e., by placing a reactance in parallel with it. An intermediate impedance that lies on this line can always be brought to 50+j0 by placing a reactance in series with it. Therefore impedance matching can always (in principle) be carried out in a two-step operation.      The constant conductance circle on which 50+j0 Ω lies is known as the 20mS constant conductance circle (i.e., 20 milli-Siemens or 1/50 Siemens). Its radius 1/2G=25Ω, and its centre lies at the point 25+j0 Ω. It crosses the resistance axis at 0 and at 1/G=50Ω.      If an initial impedance has a resistive component of less than 50Ω, it can always be manipulated onto the 20mS constant conductance circle by placing a reactance in series with it. An intermediate impedance that lies on the 20mS circle can then always be brought to 50+j0 by placing a reactance in parallel with it.

 10. Z-Plane Regions: The diagram below shows six regions of the Z-plane, identified according to their relationship to the target impedance 50+j0. It also shows some of the matching networks that give the shortest possible route to Z=50+j0. The encircled numbers indicate the operations that must be performed, the order in which to perform them, and their effects. It is assumed that parallel reactances are initially set to their highest possible values (effective open circuit), and series reactances are initially set to their lowest possible values (effective short circuit).   TX to Ae Z matching - Part 2 >>