G3YNH info: AC theory part 4.

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Ch 1 Contents

Part 3

Part 5

1. An introduction to AC electrical theory: Part 4.
Contents:

1-28. Output impedance and maximum power transfer.
1-29. The potential divider.
1-29a. Output impedance of a potential divider.
1-30. Measuring source resistance.

1-31. Error analysis.
1-32. Antenna system Q.
1-33. The basic impedance transformer.
1-34. Auto transformers.
1-34a. Continuously variable auto transformer.
1-35. Prototype impedance matching network.

1-28. Output impedance and maximum power transfer:
So far we have considered generators to be sources of constant RMS voltage. In reality, in the absence of a control system to keep it constant, the output voltage of a generator will droop as the output current is increased. This means that the generator has an internal impedance, which is somehow distributed throughout its wiring and component parts, but which will be seen from outside as though there is a single impedance in series with an otherwise perfect generator. This impedance is known as the source impedance or the generator's output impedance, and must often be taken into account when carrying out circuit analysis. In particular, it is necessary to include the source impedance explicitly when determining the characteristics of the parallel resonator bandpass filter; but there are various connotations relating to power transmission in general which must be addressed. The basic matter is that of the effect that the load impedance has on the amount of power delivered to the load, and is encapsulated in a set of relationships known as the maximum power-transfer theorem.
For the special case of a generator with a purely resistive output impedance and a purely resistive load, we can obtain the maximum power-transfer condition using a graphical method. The circuit to be considered is shown below, where Rg is the generator output resistance, and R is the load. V is the off-load generator voltage, i.e., it is the voltage which will be seen at the generator terminals when the load R is disconnected. It should also be obvious by inspection that no power is delivered when R=0 (short-circuit), and also that no power is delivered when R is disconnected (i.e., when R®¥). Hence we expect a peak in power output at some intermediate value of R, and we can obtain this value in relation to Rg by determining the relationship between P and R and plotting it as a graph.

In the circuit on the right, the power delivered to the load is:
P = I² R
Where:
I = V / (R + Rg)
Hence:

P = V² R / (R + Rg)²

1-28.1

This function is plotted below, for constant V, and shows that maximum power output occurs when R=Rg.

This result is, of course, well known, but it is by no means the whole story, and its interpretation is subject to various common misconceptions. We can settle all of these issues by deriving the complete maximum power transfer condition (see box below). This requires the use of calculus, which will not be explained here, but those unfamiliar with the technique may still avail themselves of the result.

The maximum power transfer theorem:
In the circuit shown on the right, the power P delivered to the load Z is:
P = |I|² R
where |I| = |V| / |(Z + Zg)|
Hence:
P = |V|² R / |(Z + Zg)|²
= |V|² R / |(R+Rg +j[X+Xg])|²

P = |V|² R / [(R + Rg)² + (X + Xg)²]

There are two maximum power transfer conditions to be obtained here, one being the value of load reactance, and the other being the value of load resistance. For changes in either of these variables, there will be a peak in the graph of power versus the variable, and the peak will of course occur at the point where the gradient of the curve is zero. Hence, for the reactance condition, maximum power transfer occurs when ¶P/¶X=0, and for the resistance condition, maximum power transmission occurs where ¶P/¶R=0 (where ¶ is known as "partial d" or "curly d" and indicates a partial differential; i.e., differentiation of one variable with respect to another is carried out with all other variables held constant). In order to carry out these differentiations on the expression above, we can use the quotient rule:

If y = N/D then dy/dx = (DdN/dx - NdD/dx)/D²

Hence if we let N = |V|² R
and D = (R+Rg)²+(X+Xg)² = R²+Rg²+2RRg+X²+Xg²+2XXg
then:
¶N/¶X=0, ¶D/¶X=2X+2Xg, ¶N/¶R=|V|², and ¶D/¶R=2R+2Rg.
Hence:
¶P/¶X = [0 - |V|²R(2X+2Xg)] / D²
= -2|V|²R(X+Xg) / D²
therefore:

¶P/¶X=0 when X=-Xg

(maximum power transfer occurs when the power factor is 1)
and
¶P/¶R = { |V|²[(R+Rg)²+(X+Xg)²] - |V|²R(2R+2Rg) }/ D²
= |V|²[ (X+Xg)² + R² + Rg² + 2RRg -2R² -2RRg ] / D²
= |V|²[ Rg² + (X+Xg)² -R² ] / D²
therefore:
¶P/¶R=0 when Rg²+(X+Xg)²-R²=0 , i.e.,

¶P/¶R=0 when R=Ö[Rg²+(X+Xg)²]

Notice that this latter maximum power transfer condition is a magnitude: it is the same as;
R = |Rg +j(X+Xg)|
i.e., maximum power transfer occurs when the load resistance is equal to the magnitude of the impedance formed by the source resistance and the total reactance. This also means that if the source impedance is purely resistive, then maximum power transfer occurs when the magnitude of the load impedance is equal to the source resistance. Observe also that when the unity power-factor condition X=-Xg is satisfied, the (X+Xg) term disappears and maximum power transfer occurs when R=Rg. Thus the overall maximum power transfer condition occurs when R=Rg and X=-Xg, i.e, when

Z = Zg*

The condition obtained when the load impedance is the complex conjugate of the source impedance is known as a conjugate match.

We may address the most common misconception regarding impedance matching by stating that, although unity power-factor (X=-Xg) is always desirable, is it is not necessary and not always desirable that the load resistance should be equal to the source resistance. The reason can be understood by considering the poor generator, which must dissipate power in its internal resistance, and will therefore get hot.

If we assume that power-factor correction will normally be carried out, then there is no need to consider the reactances in the system, and we can analyse the power dissipated in the generator using the case where both the source impedance and the load are purely resistive. Thus:
Pg = I² Rg
where the current is, as defined earlier: I=V/(R+Rg). Hence:

Pg = V² Rg / (R+Rg)²

1-28.2

We will plot this function shortly; but when doing so it will be interesting to use the comparison between the load power and the power wasted in the generator as a measure of the power transmission efficiency. We can define efficiency as:
Transmission efficiency = Power delivered / Total power generated
and here we will give it the symbol h (Greek lower case 'eta'). Thus:
h = P / (P + Pg)
Now, substituting the definitions of P (1-28.1) and Pg (1-28.2) into this expression we get:
h = V²R/(R +Rg)² / { [V²R/(R +Rg)²] + [V²Rg/(R +Rg)²] }
i.e.,

h = R / (R + Rg)

Shown plotted below for comparison are: P, the power delivered to the load; Pg, the power dissipated as heat in the generator; P+Pg, the total power generated; and h, the ratio of power delivered to power generated.

The tabulated results below show the various power levels as a proportion of the maximum deliverable power Pmax, and are applicable to any power-factor corrected generator-load system.

Load R / Rg	Total Power (P+Pg) / Pmax	Power Loss Pg / Pmax	Load Power P / Pmax	Load power / dB	Efficiency R / (R+Rg)
0	.................... 4.00	.................... 4.00	0.00	-¥	0.00
1/16	................... 3.76	.................. 3.54	. 0.22	-6.55	. 0.06
1/8	.................. 3.56	................ 3.16	.. 0.40	-4.03	. 0.11
1/4	................ 3.20	............. 2.56	... 0.64	-1.94	.. 0.20
1/2	............. 2.67	......... 1.78	.... 0.89	-0.51	... 0.33
1	.......... 2.00	..... 1.00	..... 1.00	0.00	..... 0.50
2	...... 1.33	.. 0.44	.... 0.89	-0.51	....... 0.67
4	.... 0.80	. 0.16	... 0.64	-1.94	........ 0.80
8	.. 0.44	0.05	.. 0.40	-4.03	......... 0.89
16	. 0.26	0.01	. 0.22	-6.55	......... 0.94

Notice in the graph above that as the load resistance R is increased and becomes greater than source resistance Rg, the power delivered to the load tails off gently. The reason for this behaviour is that, as the current drawn from the generator reduces, the output voltage increases; and so the system possesses a self-regulating property when lightly loaded. When the load is twice the source resistance, the power delivered is still 89% of the maximum possible, a droop in output of only 0.51dB.
The major advantage of light loading however is seen in the transfer efficiency. When a conjugate match is achieved, the efficiency is only 50%, but it rises to 67% (2/3) when R=2Rg, and 80% (4/5) when R=4Rg. This means that light loading, when compared to conjugate matching, gives a reduction in generator dissipation and power input for a given power output. In the field of radio, of course, the generator is a radio transmitter; and if the transmitter is designed for light loading it can have smaller heat-sinks and reduced battery or mains power consumption in comparison to a transmitter designed for conjugate matching. Consequently, the figure often referred to as the "output impedance" of a radio transmitter (often 50W) is usually nothing of the sort, it is instead (and should be called) the preferred load-resistance, or alternatively the design load resistance. The preferred load resistance of a broadband transistor power amplifier is usually higher than the output impedance, and attempting to provide such an amplifier with a conjugate match will result in excessive internal dissipation, overheating, and possibly catastrophic failure. Fortunately, most modern amplifiers are provided with protection circuitry to prevent over-dissipation, and this circuitry gives the transmitter a loading characteristic which makes it appear that the source resistance is higher that it really is. This loading characteristic will be different from the power transfer curve derived above because it is caused by the action of non-linear circuit elements (level detectors etc.), and so the load resistance which corresponds to the middle of the permitted operation window is known as the pseudo output-impedance (or, if you like, the pseudo source-resistance). The diagram below shows what the power transfer curve might look like with the operation window centred on twice the source resistance (unprotected transfer-function shown dotted).

Notice that the protection circuitry also operates when the load resistance is higher than the preferred value. This is not usually necessary for the protection push-pull transistor power amplifiers (the most common type of output stage in current practice); but it helps to ensure that any harmonic suppression filter after the amplifier will function correctly, and it occurs because the load impedance is traditionally detected using a bridge circuit (often called a reflectometer or SWR bridge, but really an impedance bridge) balanced for a particular value of resistance.

An interesting discussion of the effect of load impedance on power amplifier efficiency and the conditions which provoke transistor failure is given by Bob Pearson, G4FHU [14].

If the protection circuitry is correctly designed and adjusted, the pseudo output-impedance should be the same as the preferred load-resistance. When determining the effect of source impedance on the Q of antenna systems and bandpass filters however, it is the true output-impedance, not the preferred load-resistance which must be used. Unfortunately, this quantity is often impossible to obtain from the manufacturer's data; but, as we shall see shortly, it can be measured with the aid of two dummy load resistors of different value.

1-29. The potential divider:
The potential divider is the simplest three-terminal electrical network. We have made some use of its properties already, but there comes a point when it is useful to characterise it formally. Here we will do so for the general case, which is that of defining the voltage at the intersection of two impedances. Referring to the diagram:

Vout = I Z1
where, since Vin= I(Z1+Z2) :
I = Vin / (Z1 + Z2)
hence

Vout = Vin Z1 / (Z1 + Z2)

1-29.1

and if we multiply the right-hand side by Z2/Z2:

Vout = Vin (Z1 // Z2) / Z2

1-29.1a

Note that Z2 is the sum of the source impedance and any additional impedance placed in series with the generator. Vin is the off-load generator voltage.

If the impedances are pure resistances, the formula above reverts to:

Vout = Vin R1 / (R1 + R2)
1-29.2

where R1 is the resistance across which Vout is said to appear.
Alternatively, multiplying by R2/R2:

Vout = Vin (R1 // R2) / R2
1-29.2a

1-29a. Output impedance of a potential divider.
The output impedance of a network is defined as that impedance which, when placed in series with a hypothetical perfect generator, accounts for the drop in output voltage which occurs when a load is connected. Shown below is a representation of a potential divider network loaded with an impedance ZL. If the load is removed, the output voltage is Vo, but when the load is connected, the output drops to a new voltage Vo' (the single inverted comma is pronounced "prime"). This situation is modeled on the right as a perfect generator with an output Vo in series with an impedance Zo, the latter being the output impedance we wish to define.

Using the definitions given in the diagram:
Zo = ( Vo - Vo' ) / IL
where:
IL = Vo' / ZL
Hence, combining these two equations:
Zo = ZL ( Vo - Vo' ) / Vo'
     = ZL [ ( Vo / Vo' ) - 1 ]         . . . . . . 1-29.3
From equation (1-29.1a) given above:
Vo = V (Z1 // Z2) / Z2
and by considering ZL as part of the potential divider itself:
Vo' = V (Z1 // ZL // Z2) / Z2
Hence
Vo / Vo' = (Z1 // Z2) / (Z1 // ZL // Z2)
             = [ (1/Z1) + (1/ZL) + (1/Z2) ] / [ (1/Z1) + (1/Z2) ]
             = 1 + (1/ZL) / [ (1/Z1) + (1/Z2) ]
Vo / Vo' = 1 + (Z1 // Z2) / ZL
Substituting this into (1-29.3) gives:
Vo = ZL { 1 + [ (Z1 // Z2) / ZL ] - 1 }
i.e.:

Zo = Z1 // Z2

The output impedance of a potential divider is the parallel combination of the component impedances.
Note that the output impedance of the main generator is part of Z2. If however, as is often the case, this output impedance is small in comparison to the total Z2, then it can be neglected.

1-30. Measuring source resistance:
In the test setup shown below; the output voltage of a generator (radio transmitter, etc.) is measured with two different load resistances, all other variables being kept constant, and the circuit being constructed in such a way as to minimise stray capacitance and inductance (i.e., using very short wires). It is assumed that the source impedance is purely resistive, this being reasonable in the case of a transistor RF amplifier, but very unreasonable in the case of a tuned valve (tube) RF amplifier. In order to avoid interference from any protection circuitry, the test should be carried out at a low power level (<10% of maximum output). The voltmeter should be capable of measurement at the generator frequency and should have a high input resistance. An oscilloscope with a high-impedance probe is suitable, but ordinary multimeters do not work at radio frequencies. Only the voltage ratio needs to be determined accurately, the absolute voltages are immaterial.

Let the output voltages be V1 when R1 is connected, and V2 when R2 is connected. The source and load resistances form a potential divider. Hence (using equation 1-29.2):
V1 = V R1 / (Rg + R1)
and
V2 = V R2 / (Rg + R2)
Rearranging both of these expressions to get V on its own and then equating them gives:

V = V1(Rg + R1)/R1 = V2(Rg + R2)/R2
R2V1(Rg + R1) = R1V2(Rg + R2)
Rg (R2V1 - R1V2) = R1R2 (V2 - V1)

Rg = R1R2(V2 - V1) / (R2V1 - R1V2)

If (say) V1 is factored out of the numerator and denominator, a form is obtained which makes it clear that only the voltage ratio is needed:

Rg = R1 R2( [V2/V1] - 1) / (R2 - R1V2/V1)

1-30.1

A respectable difference between the two load resistors is necessary in order to minimise the effect of measurement errors, but too large a deviation from the preferred load resistance is likely to provoke a transistor power-amplifier's protection circuitry. For a transmitter designed to operate into a 50W load; 25 and 100W dummy-load resistors correspond to the upper and lower 2:1 SWR points and should give an easily discernible output voltage difference. A 25W resistor can be had by connecting two 50W dummy load resistors in parallel with a coaxial T-piece. 100W coaxial resistors are less readily available, but an old-fashioned 75W load will do instead. In the calculation, the actual resistance of the load measured with an accurate resistance meter should be used (rather than the nominal value stamped on the resistor).

Example: The output voltage of a Kenwood TS430S 100W HF transmitter was measured with two different dummy loads. The measurement frequency was 1.9MHz, and the test power level was very approximately 1W. One load was a 75W nominal coaxial resistor measuring 75.1±0.7W, the other was the combination of this resistor and a 50W nominal coaxial resistor in parallel with it, the combination measuring 29.6±0.3W. The voltage ratio was measured using an oscilloscope with a 10MW ´10 probe. The resistors and the probe were attached directly to the antenna socket using coaxial T-pieces (no cables). The measurement was made by attaching and removing the 50W resistor from the T-piece with the transmitter running and noting the change in the peak to peak excursion of the output waveform. Using the following designations: R2=75.1W, R1=29.6W, the voltage ratio V2/V1 was 1.364 ±0.04. Using equation (1-30.1), the source resistance Rg was calculated to be 23.3W. An error analysis (see next section), gave an estimated standard deviation of 3.4W; i.e., Rg=23.3±3.4W. Note incidentally, that this determination assumes that the output impedance does not change with power output level. Given that power transistors are non-linear devices, this may not be the case.

1-31. Error analysis:
While it would be inappropriate here to delve too deeply into the subject of scientific data analysis; the reader should nevertheless be aware that all physical measurements are meaningless unless they have some kind of error-window or confidence interval associated with them. This is not a serious problem when taking a reading with say, a multimeter, because (assuming that the instrument has been calibrated), the manual will say what the measurement accuracy is. A digital multimeter, for example, might have a quoted accuracy of ±0.8% ±1digit (i.e., ±1 in the last decimal place) for its resistance ranges, so if we obtain a resistance reading from this instrument of (say) 75.1W, the actual measurement will have a confidence interval of ±0.6 ±0.1, i.e., the reading should be recorded as 75.1±0.7W. Scientists and engineers normally equate error boundaries stated in this way with the estimated standard deviation (ESD) of the measurement; where, on the assumption that errors are scattered randomly according to a 'normal' or Gaussian distribution, a standard deviation represents a region where we have a 68% confidence that the true result will lie. The standard deviation is usually given the symbol s (Greek lower case 'sigma'), and so if we obtain a measurement x±s, we have 68% confidence that the true answer lies between x-s and x+s. From the properties of the Gaussian error distribution also, we have a 95.5% confidence that the true answer lies between x-2s and x+2s, and a 99.7% confidence that the true answer lies between x-3s and x+3s [15]. The use of standard deviations rather than 'brick wall' tolerances reflects the reality that there is always a finite probability that the true result will lie outside the stated error range. We can only ever have absolute confidence that the magnitude of the true answer lies somewhere between zero and infinity, but we expect only 3 measurements in every 1000 to fall outside x±3s.
     It is always advisable to try to write down an ESD for every measurement made. This is a reasonably straightforward matter where direct measurements are involved, but a difficulty arises in situations where several measurements are made and then put into a formula in order to obtain the required result. The problem is that of working out how much influence the deviation of a particular variable has on the overall result, and how to add the various deviations together in order to arrive at the overall ESD. It is therefore fortuitous that we have been engaged in the study of vectors, because it turns out that this is a problem of vector addition and magnitudes.
     If two or more measurements are made in such a way that the outcome of one has no influence on the outcome of any of the others, the measurement errors are said to be uncorrelated. An example of uncorrelated errors is that of readings taken from two separate instruments, where an error or inaccuracy in the reading of one instrument is not related to any error or inaccuracy in the reading of the other. On the other hand, the errors in two measurements made using the same instrument may be correlated, in the sense that if the instrument always reads too high or too low, it will introduce errors in the same direction in both cases. If measurement errors are correlated, then it means that there is some systematic (design, interpretation, or calibration) defect in the measuring process; but if we believe that the measurements have been made to the best of our abilities with the equipment available, then it is usually sensible to assume that any measurement errors are uncorrelated.
     Now, if the errors in two or more measurements are uncorrelated, this means that a deviation from the true value in one measured quantity can occur without influencing the deviations in any of the other quantities. If we determine a quantity by applying a formula to a set of measurements, each measurement will contribute a random error to the result, but there is just as much chance that the error due to one measurement will partly cancel the error due to another as that there is that a pair of error contributions will both increase or decrease the result. Therefore it will be unduly pessimistic to add the uncertainty contributions of the individual measurements directly. Instead, we should allow for the independence of the uncertainty contributions by regarding each one as a vector pointing in a direction which is at right angles (orthogonal) to all of the others. In effect, by virtue of its randomness, each uncertainty contribution exists in its own dimension, and we may identify its magnitude as its length in that dimension. It follows that the overall uncertainty is the length (i.e., the magnitude) of the vector which results from the addition of a set of orthogonal uncertainty vectors. This situation is represented in the diagram below, where U1, U2, and U3 are the uncertainty contributions to the determined value of an unknown, and U is the overall uncertainty in the result. We can easily find U by successive application of Pythagoras' theorem, i.e.:

Let the magnitude of the vector sum of U1 and U2 be U12:
U12 = Ö( U1² + U2² )
Then U is the magnitude of the vector sum of U12 and U3:
U = Ö( U12² + U3² )
but U12² = U1² + U2²
Hence:
U = Ö( U1² + U2² + U3² )

This process can be extended to find the magnitude of a vector in an arbitrary number of dimensions (we can't make perspective drawings in more than three dimensions, but there is no restriction on the number of dimensions that a vector can have). Hence:
U = Ö( U1² + U2² + U3² + . . . . + Un² )
Now note that this formula says: "to find the overall uncertainty; calculate the sum of the squares of the uncertainty contributions and take the square root." The uncertainty contributions are not the same as the uncertainties in the measurements made.
Imagine that an unknown quantity x is given by a formula f, which is a mathematical function involving measurable quantities (variables) m1, m2, m3, etc. We can express this situation by writing:
x = f(m1, m2, m3, ...)
and we can determine x by plugging m1, m2, m3, etc. into the formula. We can also determine the uncertainty contribution due to any one of the variables by changing it and noting the change which occurs in x. The obvious amount by which to change the variable is its standard deviation, hence:
x+sx1 = f(m1+s1, m2, m3, ...)
Here we have assumed that a positive change in m1 will cause a positive change in x. This may not be the case, but since we intend to add the contributions from changes in each of the variables as orthogonal vectors, it makes no difference either way. Now, restoring m1 to its original value we determine the uncertainty contribution due to m2.
x+sx2 = f(m1, m2+s2, m3, ...)
and so on. If we work through all of the variables in this way and determine their error contributions, we can obtain an estimate of the standard deviation of x by summing the squares of the contributions and taking the square root:
s = Ö( sx1² + sx2² + sx3² + . . . . + sxn² )
Note that there are a number of assumptions inherent in this procedure: firstly, as discussed before, that the uncertainties are uncorrelated; and secondly that we have assumed that the function f is linear for changes in any of the variables. The latter condition is normally true to a good approximation for small changes, and the effect of any non-linearity is mitigated by the fact that the object of the exercise is to obtain an estimate.

Example:
The output resistance Rg of an RF amplifier was determined by loading the output with two different resistances and noting the change in the output voltage with all other conditions held constant. The applicable formula is equation (1-30.1):
Rg = R1 R2 (NV - 1) / (R2 - R1 NV)
Where NV is the ratio of the output voltages:
NV = V2/V1
The voltage measurements were made using an oscilloscope, and it was considered that each measurement had an uncertainty of about 2%. It was also considered that these uncertainties were uncorrelated because they were incurred by different operations; one operation being to set the transmitter carrier level and oscilloscope Y-shift until the waveform just touched the top and bottom of the measuring graticule with the higher value resistor connected, the other being to read the height on the graticule with the lower value resistor connected. The overall uncertainty of the voltage ratio measurement was therefore taken to be the square root of the sum of the squares of the two voltage measurements; i.e., Ö(2²+2²)=2.8%, which was rounded to 3% in view of the approximate nature of the estimate. The actual voltage ratio was 1.364, and 3% of 1.364 is 0.04. Hence:
NV = 1.364±0.04
The resistances were measured using a multimeter known to read correctly within ±0.1W against a standard resistance of 100.0W. The stated accuracy of the instrument was ±0.8% ±1 digit. The measured resistances were R1=29.6W and R2=75.1W. Hence:
R1=29.6±0.34W
R2=75.1±0.7W
The output impedance Rg was calculated from the formula (1-30.1) using a spreadsheet program and determined to be 23.3W. The output impedance was also calculated with each of the measured values individually incremented and decremented by an amount equal to its estimated standard deviation, and the resulting deviation in Rg was noted. The spreadsheet is shown below:

Output impedance of TS430s. Spreadsheet calculation (Open document spreadsheet (ods) file. Best viewed using Open Office).

Note that the formula is somewhat non-linear in its behaviour because the deviations caused by incrementing and decrementing a variable are not exactly equal and opposite. The correct way to allow for this effect is to take the average of the deviation magnitude (RMS) for each case. Therefore, the estimated standard deviation in Rg is:
s = Ö( 0.579² + 0.253² + 3.359² ) = 3.418
Hence:
Rg = 23.3 ±3.4 W
Notice that the major contributor to the uncertainty in Rg in this case is the uncertainty in NV. We cannot ignore the effect of the resistance uncertainties however, because if we repeat the experiment with more closely spaced values for R1 and R2 , we will find that their contributions to the uncertainty increase dramatically.

1-31a. Analytical approach to error analysis:
While the error analysis technique just described is perfectly respectable; those who write computer programs will generally prefer an analytical approach. The derivation of an error function from a formula requires the use of calculus. Those who are unfamiliar with calculus may either proceed to section 1-32 or refer to Appendix 1 (mathematical techniques).

The analytical form of an error function is obtained from the observation that an error in a variable is transmitted through a formula according to the rate of change of the formula with respect to the variable. Thus the error contribution from a variable is the partial derivative of the formula with respect to the variable multiplied by the deviation in the variable. Hence if
x = f(m1, m2, m3, ...)
and the ESDs of the measured quantities are s1, s2, s3, etc.; the contribution which the variable m1 makes to the ESD of x is given by:
sx1 = ( ¶f/¶m1) s1
and so on (strictly we should take the modulus of the derivative because standard deviations are by definition positive, but it does not matter in this case because orthogonal addition involves squaring of the error contributions). Hence the analytical form of the error function is:
s = Ö{ [(¶f/¶m1)s1]² + [(¶f/¶m2)s2]² + [(¶f/¶m3)s3]² + .... }

Example:
The output impedance of a generator is obtained from the formula:
Rg = R1 R2 (NV - 1) / (R2 - R1 NV)
Differentiation of this function requires the use of the quotient rule:
If y = N/D then dy/dx = (DdN/dx - NdD/dx)/D²
where, in this case, the numerator is:
N = R1R2(NV - 1) = R1R2NV - R1R2
and the denominator is:
D = R2 - R1NV
Differentiating the numerator with respect to each of the variables gives:
¶N/¶R1 = R2(NV - 1) ,      ¶N/¶R2 = R1(NV - 1) ,     ¶N/¶Nv = R1R2
and differentiating the denominator with respect to each of the variables gives:
¶D/¶R1 = -NV ,      ¶D/¶R2 = 1 ,      ¶D/¶Nv = -R1

Using these results we obtain:
¶Rg/¶R1 = [ D(¶N/¶R1) - N(¶D/¶R1) ] / D²
             = [ (R2 - R1NV) R2(NV - 1) - R1R2(NV - 1)(-NV) ] / (R2 - R1NV)²
             = [ (R2 - R1NV) R2(NV - 1) + R1R2(NV - 1)NV ] / (R2 - R1NV)²
             = [ R2²NV - R2² - R1R2NV² + R1R2NV + R1R2NV² - R1R2NV ] / (R2 - R1NV)²
¶Rg/¶R1 = R2²(NV - 1) / (R2 - R1NV)²

¶Rg/¶R2 = [ D(¶N/¶R2) - N(¶D/¶R2) ] / D²
             = [ (R2 - R1NV)R1(NV - 1) - R1R2(NV - 1) ] / (R2 - R1NV)²
             = [ R1R2NV - R1R2 - R1²NV² + R1²NV - R1R2NV + R1R2 ] / (R2 - R1NV)²
¶Rg/¶R2 = R1²NV(1 - NV ) / (R2 - R1NV)²

¶Rg/¶NV = [ D(¶N/¶NV) - N(¶D/¶NV) ] / D²
              = [ (R2 - R1NV)R1R2 - R1R2(NV - 1)(-R1) ] / (R2 - R1NV)²
              = [ R1R2² - R1²R2NV + R1²R2NV - R1²R2 ] / (R2 - R1NV)²
¶Rg/¶NV = R1R2(R2 - R1) / (R2 - R1NV)²

The error function in this case is:
s = Ö{ [(¶Rg/¶R1)sR1]² + [(¶Rg/¶R2)sR2]² + [(¶Rg/¶NV)sNv]² }
The derivatives all share a common denominator D², and so on writing the expression in full, a factor (1/D²)² can be removed from the square root bracket. Hence:

s = [1/(R2 - R1NV)²] Ö{ [R2²(NV-1)sR1]² + [R1²NV(1-NV )sR2]² + [R1R2(R2-R1)sNv]² }

In the previous section, we determined Rg = 23.3W from the following measurements:
R1=29.6±0.34W , R2=75.1±0.7W , NV = 1.364±0.04
These give:
D² = (R2 - R1NV)² = 1205.8673
¶Rg/¶R1 = R2²(NV - 1) / D² = 2052.9636 / 1205.8673 = 1.7025
¶Rg/¶R2 = R1²NV(1 - NV ) / D² = -435.0100 / 1205.8673 = -0.3607
¶Rg/¶NV = R1R2(R2 - R1) / D² = 101144.68 / 1205.8673 = 83.8771
s = Ö{ [(¶Rg/¶R1) ´ sR1]² + [(¶Rg/¶R2) ´ sR2]² + [(¶Rg/¶NV) ´ sNv]² }
= Ö{ [1.7025 ´ 0.34]² + [0.3607 ´ 0.7]² + [(83.8771 ´ 0.04]² }
= Ö{ 0.5789² + 0.2525² + 3.3551² }
Note that the error contributions in the expression above are very close to the averages of the deviations calculated by the incremental (spreadsheet) method used previously. Finally we have:
s = 3.414
Rg = 23.3 ±3.4W

For a spreadsheet version for this calculation (which can be used as a template) see Rg_errfunc.ods .

1-32. Antenna system Q:
In previous sections we showed that conjugate matching is not necessarily a good idea, and that radio transmitter manufacturers do not necessarily design power amplifiers to work into a conjugate load. Besides the obvious advantages in terms of output regulation and efficiency however; there is a further reason to load a radio transmitter lightly in cases where the input impedance of an antenna system changes rapidly with frequency or is subject to variable environmental factors. Recall from section 1-28 that the true maximum power transfer condition occurs when:
R=Ö[Rg²+(X+Xg)²]
Consequently, if the antenna system is subject to disturbances which can cause a reactive component to appear after matching has been carried out, then the best average maximum power output will be obtained when the load resistance is somewhat higher than the source resistance. Possible causes of transient residual reactance are many and various, including: changing physical environment (of mobile and portable transmitters), wind, rain, component heating, birds, etc., and (as is easily forgotten) modulation.

In section 1-7, we discussed an electrically-short inductively-loaded vertical antenna system. The equivalent circuit of this antenna is shown on the right; with one extra resistance added, that being the (true) source resistance Rg. With this additional piece of information, it becomes possible to calculate the Q, and hence the bandwidth, of this system.
If we take the same example component values as were used before, we have:
ZL = RL + jXL = 7.5 +j3000
Za = Ra + Rr +jXa = 2.5 -j3000

giving an input resistance of 10W for a whip length of about 0.07l. The whole system is of course, a series resonator, and we can define the circuit Q as:
Q0 = |XL|/Rtotal = |Xa|/Rtotal
Since bandwidth is proportional to f0, problems of excessive Q are likely to occur at low operating frequencies, so let us see what happens if this antenna is built to operate on (say) 1.9MHz, with the generator source resistance adjusted to be 5W. This will make the total series resistance 15W, and with XL=3000W, the Q will be 3000/15=200. The -3dB bandwidth of the antenna will therefore be f0/Q=9.5KHz. This is wide enough to accommodate a communications SSB signal (2.7KHz bandwidth) but there is very little latitude for incidental detuning, and the antenna will exhibit a small variation of input impedance depending on the modulation frequency. Light loading of the generator will help to offset these problems, because it will create a situation where transient detuning forces the generator-load system closer to its maximum power transfer point (although detuning won't increase the amount of power delivered, light loading will give better regulation than a conjugate match).
Note incidentally, that the antenna discussed above is not physically small when designed for operation in the 160m band. The wavelength at 1.9MHz is c/f =157.8m; and so a 0.07l rod will be 11m long, and consequently far too large for mobile use. To make a mobile antenna, the rod must be shortened; and this will reduce the antenna capacitance (and hence increase the reactance), and sadly for efficiency, will cause the radiation resistance to fall. The larger antenna reactance will necessitate a larger loading reactance, and although this will bring more resistance with it, the increase in reactance will be greater than the increase in total resistance and the Q will rise. A point can be reached where serious curtailment of the modulation bandwidth occurs, although, for this system, it it not predictable using lumped-component theory. The coil can be regarded as a lumped component provided that the whip is long enough to ensure that most of the radiation occurs from the whip rather than from the coil. If that condition applies, then the Q of the antenna system can never be larger than the Q of the loading coil because the total series resistance will always be that of the coil plus a little extra. The maximum tolerable Q (causing some, but not serious, audio degradation) occurs when the antenna system bandwidth is the same as the audio bandwidth, and for SSB on 1.9MHz this figure is 1900/2.7=704. It is extremely difficult to make a lumped inductor with a Q of greater than about 400, so on 160m the Q limit can be avoided by controlling the length of the coil. If, on the other hand, the whip is of length comparable to or shorter than the coil, then most of the radiation occurs from the coil. In that case, the lumped component description fails completely, and the system is best described as a quarter-wave transmission-line resonator. In the transmission-line regime, the Q and hence the voltage magnification can become enormous, and the useable input power is limited by the tendency for the air around the top of the coil to ionise and become electrically conductive. Coils operated at or slightly below the quarter-wave transmission line resonance frequency are used for artificial lightning experiments, in which context they are known as 'Tesla coils'. In particular, the voltage-magnifier coil connected in series with the output from a step-up transformer is known as the 'Extra Coil'. The transmission-line properties of coils will be discussed in chapter 3.

While on the subject of MF and HF mobile antennas; when forced to use a very short whip, it is possible to increase the antenna capacitance artificially (and hence reduce the reactance) by adding a capacitance hat to the antenna (some prongs sticking-out sideways symmetrically; or, if there's a risk that you might poke someone's eye out, an aluminium disk ). Reducing the antenna reactance in this way reduces the amount of loading inductance required, and hence allows the coil to be wound with thicker wire for a given size (less resistance). Placing the hat at the top of the antenna moreover, increases the current in the vertical section, and actually increases the radiation resistance slightly (every little helps).

Capacitance hat

1-33. The basic impedance transformer:
In the previous section it was implied that the generator output impedance could be adjusted, but we have yet to offer any methods for doing so. In later chapters, it will become apparent that there are a large number of options in this respect, but for the present purpose it will be sufficient to have just one: the transformer. We will look at transformers and other magnetic devices in detail in chapter 3; but here we will avail ourselves of the properties of the straightforward (but unfortunately mythical) perfect or ideal transformer. In truth, well-designed transformers can have power-transfer efficiencies of more than 98% within a certain band of frequencies, and so the myth of the ideal transformer is not so far from reality. Here we will assume that a tightly-coupled transformer is perfect when operating within its pass-band, on the understanding that we will need a more advanced analysis later in order to determine what the pass-band is.
A transformer loaded with an impedance Z is represented in the diagram below:

Here Np is the number of turns in the primary (generator side) winding, and Ns is the number of turns in the secondary (load side) winding. The dots next to the windings indicate either the start or the finish (it doesn't matter how this is designated, as long as it is done consistently), and it it assumed that both coils are wound in the same sense (clockwise or anticlockwise when looking at a particular end of the coil). The dotted line between the coils indicates that the transformer is wound on a magnetic core, the purpose of which (in this instance) is to produce a very tight magnetic coupling between the windings. If all of the magnetic field from the primary winding is enclosed by the core and transmitted to the secondary winding (i.e., if there is no magnetic leakage), and if the coils and the core have no heating losses, then all of the power delivered by the generator (neglecting reflected power) is transferred to the load. Also, if the inductive reactance of the windings is much larger than the magnitudes of the impedances seen on either side, and the capacitance of both windings is very small, then the secondary voltage will be in phase with the primary voltage, and the secondary current will be in anti-phase with the primary current (i.e., as a current appears to flow into the primary, a current appears to flow out of the secondary). If the number of turns in the secondary winding is greater than the number of turns in the primary, then Vs will be larger than Vp, and vice versa; and the voltage transformation will be in proportion to the turns ratio, i.e.,
Vs = Vp Ns / Np . . . . (1-33.1)
It follows that if the power produced by the generator is transferred to the load without loss, then the VI product will be conserved; which means that if the voltage is stepped up, then the current will be stepped down to keep VI constant (and vice versa). This implies that the transformer performs on the current the inverse of the transformation it performs on the voltage, i.e. (interpreting the currents in the sense of the arrows in the diagram above):
Is = Ip Np / Ns . . . . (1-33.2)
Now, by definition, the impedance looking into the transformer primary is:
Z' = Vp / Ip
which gives, using (1-33.1) and (1-33.2) as substitutions:
Z' = Vs (Np/Ns) / [ Is (Ns/Np) ]
and since Z = Vs/Is :

Z' = Z (Np/Ns)²

1-33.3

Thus, to a reasonably good approximation; a tightly-coupled transformer properly designed for its purpose scales an impedance according to the square of the turns ratio.
Now let us consider the problem in reverse, and see what a transformer does to the output impedance of a generator. Here we will call the apparent source impedance as seen from the secondary side of the transformer Zg', with Zg as the actual generator output impedance. The relationship between Zg' and Zg is perhaps guessable; but to derive it mathematically requires a trick, which is that of defining an equivalent circuit with all of the source resistance moved to the secondary side of the transformer. A suitable approach to the derivation is then to write expressions for the voltage V across the load using both the original and the equivalent circuits and then equate the two expressions.

For the left-hand circuit, let us define Z' as the load impedance seen by the generator, its relationship to to the load Z being given by equation (1-33.3) above:
Z' = Z (Np/Ns)²
The voltage V' is then the output of a potential divider formed by Zg in series with Z', i.e.:
V' = Vg Z' / (Zg + Z')
and V' is related to the load voltage V by the turns ratio, ie.:
V = V' Ns / Np
Hence:
V = (Ns/Np) Vg Z' / (Zg + Z')
V = (Ns/Np) Vg [1 + (Z' / Zg) ] . . . . (1-33.4)
For the right hand circuit, V is the output of a potential divider formed by Zg' and Z :
V = Vg' Z / (Zg' + Z)
V = Vg' [1 + (Z / Zg') ]
where:
Vg' = (Ns/Np) Vg
Hence:
V = (Ns/Np) Vg [1 + (Z / Zg') ]
Equating this to expression (1-33.4) gives:
1 + (Z' / Zg) = 1 + (Z / Zg')
i.e.,
Zg' = Zg Z / Z'
but, by rearrangement of equation (1-33.3), Z / Z' = (Ns/Np)², hence:

Zg' = Zg (Ns/Np)²

1-33.5

Thus a tightly-coupled output transformer scales the source impedance according to the square of the turns ratio, a generator with a low output impedance being converted into a generator with a high output impedance by means of a step-up (Ns>Np) transformer and vice versa.

The broadband output transformer of a fairly typical 100W short-wave radio transmitter (the Kenwood TS430s) is shown on the right. The transformer core is a block of ferrite with two hollow channels passing through it (known colloquially as a "pig nose"). The primary winding consists of two short lengths of copper or brass tubing passing through the core and connected together at one end by a strip of copper-laminate board. The secondary winding is a length of PTFE-coated multi-strand silver-plated copper wire threaded through the copper tubes (the reason for the choice of materials will become apparent in chapter 2). To make a complete turn around the core, a conductor must pass through one hole and back through the other. As shown diagrammatically, the copper tubes form a centre-tapped single turn, with the DC power supply (B+) connected to the centre tap, and the other ends connected to the collectors of the RF power transistors (a matched pair of 2SC2290s). Only three secondary turns are shown in the diagram for clarity, whereas the transformer in the photograph has four turns and so increases the amplifier output impedance by a factor of 16.

There is something more to the use of an output transformer than impedance transformation however, the principal issue being that the transmitter discussed above uses a 13.8V power supply and yet must deliver 100W into a 50W load. The required output power and target load impedance defines the output voltage as V=Ö(PR)=Ö(100´50)=70.7V RMS, i.e., 70.7´2Ö2=200V peak-to-peak (p-p). A simplified version of the power amplifier circuit is shown below, and we may deduce the minimum allowable transformer step-up ratio by examining it.

This is a so-called push-pull amplifier circuit, in which one transistor provides the positive half-cycle of the output waveform, and the other transistor provides the negative half-cycle. When a bipolar transistor is turned hard on, its collector voltage does not go to zero, but stops at some saturation voltage, which is usually around 1V. Also, it is not a good idea to drive the transistors close to saturation because this will lead to considerable distortion of the output waveform. Therefore we must assume that the output stage can produce positive and negative half cycles of no more than about 12.5V across half of the primary winding, i.e., 25V per transistor across the whole winding , hence 50V p-p. To obtain 200V p-p (70.7V RMS) therefore, a voltage step-up ratio of 1:4 is required. The fact that this transformation increases the source impedance by a factor of 16 is a secondary consideration; and is of no great concern unless the source impedance begins to approach the design load resistance, the latter situation being associated with low transfer efficiency and poor load regulation as discussed earlier. It follows, that to keep the output impedance as low as possible, a step-up ratio just sufficient to provide the required output voltage is optimal. The actual output impedance (Rg') of the TS430S transmitter (measured at the antenna socket, see the example at the end of section 1-30) is about 23W (measured 23.3±3.4W at 1.9MHz) for a design load resistance of 50W. The output impedance of the power amplifier (Rg) is therefore approximately 23/16=1.4W.
Dye and Granberg in their book "Radio Frequency Transistors" [16] give an approximate formula for calculating the output impedance of a transistor power amplifier below 100MHz as:
Rg = (Vcc - Vsat)² / Pout
where Vcc is the supply voltage, and Pout is the maximum power available from the amplifier. If we assume a saturation voltage Vsat of about 1V, this gives:
Rg = 12.8² / 100 = 1.64W.
Multiplying this by 16 gives Rg'=26.2W, which is within 1s of the measurement without taking any of the circuitry between the power amplifier and the antenna socket into account.
Notice incidentally, that the power amplifier is shown as feeding into a low-pass filter (LPF) before connection to the antenna system. Such a filter is always necessary with a broad-band transistor power amplifier, because such amplifiers produce relatively high levels of harmonics. The push-pull configuration actually cancels even harmonics, but there are still high levels of odd harmonics (3rd, 5th, 7th etc.) which must be removed (in engineering, the first harmonic is the same as the fundamental). In section 1-28, we noted that the power amplifier protection circuitry operates when the load impedance is too high, as well as when it is too low. This is not necessary for the protection of the amplifier, but the LPF may not provide the required degree of harmonic attenuation when incorrectly terminated; and so the protection circuitry helps to keep spurious emissions within acceptable limits if the load impedance is too high.

1-34. Auto transformers:
'Auto-transformer' (self-transformer) is just another name for a tapped inductor. The transformation rules for tightly-coupled auto-transformers are identical to those for tightly-coupled transformers with separate windings. The significant functional difference between the two types of transformer is that an auto-transformer does not provide DC isolation between source and load. A more subtle difference is that a transformer with separate windings, by judicious use of electrostatic shielding, can be made in such a way that the coupling between the primary and secondary windings is almost entirely magnetic. An auto-transformer will always exhibit some stray capacitive and resistive (potential divider) coupling, and so if its inductance is part of a filter circuit, the filter may exhibit poor attenuation of signals outside its passband.
The step-up and step-down auto-transformer configurations are shown below. Also shown is the somewhat redundant 1:1 auto-transformer, otherwise known as an inductor; the point in including it being to draw attention to the inductive reactance which every transformer places in parallel with its load.

Step-down

step-up

1:1

It should be obvious by inspection of the '1:1' auto-transformer circuit, that the voltage - current relationship for the load seen by the generator is given by:
V/I = jXL // Z
If the coil has losses moreover, we can represent these as a resistance (RL say) in series with the coil:
V/I = ( RL + jXL ) // Z
We can also transform the impedance of the coil into its parallel form (see section 1-16), in which case the load on the generator becomes:
V/I = RLp // jXLp // Z
The implication is that, unless the magnitude if the inductive reactance is very much larger than the magnitude of the load impedance, the transformer will not preserve the load phase relationship. If the load is reactive, the parallel loss component will also alter the load phase relationship slightly.
In the previous section, we introduced the idea that an impedance located on one side of a transformer can be transferred to the other side in an equivalent circuit by the act of multiplying it by the turns-ratio squared. So we might represent the inductance of a transformer as a separate inductance L in parallel with the primary side of an ideal transformer (of otherwise infinite inductance), or we might represent it as an inductance L' in parallel with the secondary side. The transformation rule (1-33.3) tells us that:
j2pf L = j(Ns/Np)² 2pf L'
i.e.,
L = (Ns/Np)² L'
This is a remarkable result because, not only does it give us the basis for constructing equivalent circuits to serve as models for real transformers, it also tells us something about inductors. The expression can only be true if the inductance of the coil is proportional to the square of the number of turns in it. We can see why by considering the two 1:N auto-transformer equivalent circuits shown below:

In the left-hand circuit, the inductance of the transformer is referred to the primary side, and for reasons of convention is given the symbol AL. In the right-hand circuit, the inductance is referred to the secondary side and is given the symbol L. From the foregoing discussion, we can immediately write the relationship between L and AL:

L = N² AL

We can also interpret L as the inductance of the whole coil, and AL as the inductance of one turn of the coil.
AL is known as the inductance factor, and depends on the physical dimensions of the coil and the nature of any magnetic core material. It may be interpreted either as the inductance of a one-turn coil, or as the inductance of an auto-transformer referred across a one-turn tap. AL has the units of inductance (Henrys), but is more informatively given units of inductance / turn² ("Henrys per turn-squared").

1-34a. Continuously variable auto-transformer:
One of the drawbacks of ferrite or iron-cored transformers as impedance matching devices is that the the transformation ratio can only be altered in a stepwise fashion, by changing windings or tappings one turn at a time (or half a turn if the core has two holes). If the turns in the coils are few, as tends to be the case in radio-frequency applications, then the steps available can be very coarse indeed. It is however possible to make a continuously variable inductor or auto-transformer by rotating a coil about its axis and tapping into it with a rolling contact, the coil end-connections being made by slipping contacts (known, for historical reasons, as "brushes"). Such a device is known colloquially as a "roller coaster", and an example is shown in the photograph below:

This is the motor-driven variable impedance transformer from a 1957 vintage Collins 180L-3A automatic HF antenna tuner. The tuner is designed to match end-fed wire (Marconi) antennas of 14 to 40 metres in length over a frequency range of 2 to 25MHz, and is for use with transmitters with an output of up to 150W and a preferred load impedance of 52W. An interesting feature of the transformer is that it achieves a continuous transition from step-down to step-up by having an overwind (see diagram right), i.e., the brush contact at one end of the coil goes to a centre-tap, and the end of the coil is left unconnected. The coil has 28 turns, and the input tap is at 14 turns, so a maximum impedance step-up of approximately 4:1 is obtainable.

The disadvantage of the Collins transformer is that the coil does not have a magnetic core. The stray magnetic fields will therefore induce currents (eddy currents) in the surrounding metalwork and give rise to resistive losses. The open magnetic circuit also implies that the impedance transformation obtained will not be exactly proportional to the square of the turns ratio, and due to the absence of a magnetic core the inductance might appear on first consideration to be rather low. The inductance for the whole coil, estimated using Wheeler's Formula is about 20mH, giving only about 5mH when referred to the primary side. This will give rise to significant phase shift at lower frequencies, the inductive reactance seen by the transmitter at 2MHz being something around 2p´2´10

´5´10

=63W. It transpires however, that the choice of primary reactance about equal to the target input impedance at the lowest operating frequency is sensible, because in addition to the impedance transformer, the antenna tuner also has power-factor correction components. In the process of adjusting a reactance in series with the antenna to achieve a resistive input impedance, any phase shift due to the transformer is automatically taken into account. Consequently, it is possible to keep the inductance small, which helps in the avoidance of self-resonance problems at the high-end of the operating frequency range.

1-35. Prototype impedance matching network:
An antenna matching system based loosely on the Collins 180L-3 is depicted in the diagram below. The only major difference is that the transition from step-down to step-up is accomplished by means of a change-over relay. This increases the transformation range in comparison to the overwind method, but also increases the complexity of the control system.

This is the prototype of all antenna tuners in the sense that it approaches the impedance matching problem in the simplest possible way. The object of the exercise in every case is to transform the impedance in its two dimensions: magnitude and phase, and the most direct approach is to do so using one device which only affects the magnitude and one device which only affects the phase. The magnitude-correcting engine is the variable auto-transformer, and the phase-correcting engine is a series reactance; relays being provided to insert a series coil in the event that the antenna is capacitive, or a series capacitor in the event that the antenna is inductive.
Such a matching unit can, of course, be controlled manually, by the expedient of providing it with control knobs and switches instead of motors and relays. This approach replaces the automatic control system with a human being, but makes no allowance for the fact that humans in general have little aptitude for the task. Here we monitor the load magnitude and phase using bridge circuits, which are described in detail in chapter 6. The bridges produce error signals, which tell their respective control systems which way to go in event that the error exceeds a certain preset threshold. Not shown on the diagram, but necessary to make the system work, are limit switches, two for each variable device. These tell the control system when a motorised device has hit one of its end-stops: so that the change-over relay can be switched and the motor direction reversed in the case of the impedance transformer; so that the switch-over from coil to capacitor or vice versa can be made in the case of the series reactance network; and as protection against motor burn-out in the event that the load is outside the matching range. The control systems for magnitude and phase are shown as being completely separate; which they are except in respect of common signals, such as the request for a tuning carrier or an instruction to reduce power, which they might send to the transmitter on detecting a matching error. The independence of the two systems is possible because the two chosen matching criteria are independent, i.e., the two matching processes can proceed simultaneously without altering the outcome. The system can even adjust itself when presented with a speech SSB signal, but will reach a solution fastest when the error signals are continuously available. One fortuitous property of this matching system, and of all of the matching systems which we will consider in later chapters, is that it corrects for the defects of its own components. In this case, when the phase control system adds series inductance (for example), the increasing resistance of the coil will increase the impedance magnitude seen at the input, but the magnitude control system will simply back-off to compensate. Similarly, the inductance of the impedance transformer will cause a positive phase shift, but phase control system will back-off in the capacitive direction to compensate.
While the simple magnitude-phase matching system is entirely practical however, it has never been particularly popular. The reason is that it is difficult to design an efficient and resonance-free variable broadband transformer. The required transformations can just as well be obtained using only variable capacitors and inductors, and we will explore this subject in detail in chapter 5.

Part 5