Antenna Matching Networks
Terry Rogers
WA4BVY
Ver 1.1, 22 Oct 2009 as extended 2010
(Includes T-Network analysis)
Note that to follow the links back and forth in the MSoft Word version of this document, the web
tool bar must be enabled.
Summary
Investigating two and three element networks intended to adjust antenna impedance to the
transceiver nominal value of 50 ohms with no reactance, we find that:





Minimal operating Q is desirable for minimal losses and maximum bandwidth
Antenna impedance, if other than nominal, will itself set a minimum network operating Q
The lowest operating Q (excluding transformers) is had with an L network compared with
Pi or T
Pi or T networks can be used if an inductor adjustable with small increments is not
available (see Engineering Considerations).
A T Network Matchbox can be designed at lower cost than either L or Pi networks
Circuit Q and Losses
XL
Xc
Rload
Rs
Fig 1
Consider the resonator above consisting of an inductor with reactance +jXL and capacitor with
reactance –jXc1. In parallel is the load represented by Rload. Rs is the inductor‟s parasitic
resistance which if an air core coil is determined by the effective2 skin resistance of the wire in the
inductor. Typically, capacitors have equivalent loss resistance ten times smaller than that of
1
If the „j‟ in +jXL is strange to you, see Appendix A: Complex Numbers
The skin resistance in the coil is higher than what one would calculate from the standard wire
skin resistance formula because the magnetic field from adjacent turns reduces the current in the
wire close to that turn (proximity effect). There is less effective “skin” to conduct current and thus
even higher resistance. Also, the closer the turns, the higher the packing factor, the higher the
coil pitch, the greater the effect.
2
Page -1-
inductors because the conducting surface area is much larger. Therefore, capacitor loss is not
considered. The ratio of circulating resonator reactive power to resistive or dissipative power is
defined as the Q or quality factor. Neglecting for the moment Rload which s external to the
resonator, the current in the inductor is the same as in Rs, the ratio of reactive to dissipative
power is the same as the ratio of reactance to resistance. This defines the coil and therefore
resonator Qul or unloaded Q (Q not considering Rload, of course  ).
Qul 
XL
Rs
Eq 1
Coil Q Definition with Series Coil Resistance Equivalent
In the L Network section, it is shown that the equivalent parallel resistance, a resistor in parallel
with Rload, that would have the same effect as Rs is the parallel equivalent resistance Rp:
Rp 
Rs2  X L2
Rs
Eq 2
Parallel Resistance Equivalent of Coil Resistance
Divide numerator and denominator by 1/Rs 2:
Rs2 / Rs2  X l2 / Rs2
Rp 
 Rs 1  Q 2
2
Rs / Rs


Eq 3
Parallel Resistance as function of Q
Substitute Eq 1 in Eq 3 to introduce Q, the quality factor of the coil at the frequency of interest 3. A
further substitution for Rs from Eq 1 into Eq 3 leads to:
1 Q2
 XL
Q
if Q  1 then R p ~ QX L
Rp 
Eq 4
Parallel Resistance as function of Q and Reactance
The last form of Equation 4 is another definition for coil Q if the losses are represented by a
parallel resistance.
3
The skin effect causes a resistance increase proportional to
frequency whereas XL is
proportional to frequency so the Q of a coil actually goes up with frequency per any particular coil,
at least at first. If you want a particular value of XL which is likely, then tapping or cutting the coil
requires the number of turns also vary with the square root of the frequency since inductance is
proportional to N2, the square of the turns count. That is, the same style or tapped coil will have
about the same Q at all frequencies until at the upper end of the frequency limit, dielectric losses
from the supports begin to be important and the Q for constant X L begins to go down. Air cores
with thin polystyrene supports are best with (high alumina) porcelain or steatite second and third.
Polystyrene is 100 times lower loss than standard steatite but there is a low loss version.
Porcelain has a loss factor as low as polystyrene (and better than cross linked) but the dielectric
constant is higher which effectively makes it slightly less desirable.
Page -2-
Q
Rp
XL
Eq 5
Coil Q Definition with Parallel Coil Resistance Equivalent
Since Rload is in parallel, Equation 5 can be used for loaded or operating Q as well. To be
accurate, we would use Rp in parallel with Rload as the operating Q.
The RF power dissipated in the coil is:
Pdis  E
2
Rp
Eq 6
Coil Dissipation
The RF power dissipated in the load is:
Pload  E
2
Rload
Eq 7
Load Power
The loss ratio or ratio of useful power in the load with no coil dissipation (Q→∞) to that with
dissipation is:
E2
Loss Ratio 
Rp
Pload  Pdis
P
R
 1  dis  1  2
 1  load
Pload
Pload
Rp
E
Rload
Eq 8
One can then substitute Eq 5 as is for Rp (unloaded Q or Qul) and modified for load Q or Qload
assuming Qul >> Qload.
Loss Ratio  1 
Qload X L
QUL X L
Eq 9
And of course XL cancels:
 Q 
Loss dB ~ 10 log1  load 
QUL 

Eq 10
Resonator Signal Loss when QUL>>Qload
Which means that with the same resonator…typically determined by the coil…the network loss
is lowest when the loaded Q is the lowest possible compared to unloaded Q. Low operating
Q is a reasonable goal for matching networks because suppressing harmonics is not the function
of the network. For an entire network with several resonators, Eq 10 is considered for each
Page -3-
resonator4. However, for L, Pi and T networks, there is only one resonator. Consider the
following situation for one resonator.
QUL
20
20
Qload 3
10
Loss 0.7dB 3dB
Table 1: Same Coil/Resonator with Different Loaded Q
Operating or loaded Q is also defined as (for a single resonator):
Qload 
fo
f
Eq 11
Q and Bandwidth
Where
fo
Δf
center frequency
bandwidth
So we get the lowest losses with the widest bandwidth, both desirable goals, with the lowest
operating Q possible for a given resonator/coil Q UL.
L Network
A full network analysis is shown rather than just present the component value equations to
demonstrate the degrees of freedom, if any, how network Q and bandwidth is determined…and
losses…and the nature of the problems encountered by both network designers and operators.
The latter is fully explored in the section Engineering Considerations after the basic network
analysis is complete. At all times, remember that we are designing impedance matching rather
than harmonic suppression networks.
The method for analysis and design in this section and latter is Ladder Network Analysis and
more specifically the backward propagating impedance method. See Appendix B: Network
Analysis (brief). The reason to use this rather than Modern Filter Theory as in newer text books
and even the ARRL Handbook (about page 12.5) is that:



The math and concept is easier
We get impedances directly which is the goal of an impedance matching filter
Passband is secondary (in Modern Filter Theory, passband shape is the goal)
In this method, you can get the net transfer function (output passband shape) for a low loss
network by taking advantage of the fact that what is reflected is not transmitted and vice versa.
If complex numbers are not familiar, see Appendix A: Complex Numbers.
4
Which leads to a situation for transforming networks where there is an optimal number of
resonators (sections) for a given bandwidth where the loss is lowest.
Page -4-
1
2
XL
ITX
Rgen
Xc
Rload
Rload < Zo = Rgen
Tx
Fig 2
L Network
In the L network above, Xc, capacitive reactance is always on the high impedance side. If the
transmitter impedance is 50 ohms, Rload would need to be less. If Rload were greater, say 200
0hms, then we just urn the network around putting the capacitor on the R load side. The numbers
are node numbers where all the elements are connected together and therefore at the same
voltage.
Reactances are used for a frequency independent solution and simpler notation that can latter be
scaled to frequency.
The impedance, by inspection and rules for series connected components at node 1 due to the
L+load branch is:
Z L branch  RLoad  jX L
Eq 12
To add in the effects of Xc, we need to convert Eq 11 and Xc to an admittance since parallel
admittances add just like series impedances.
YL branch 
R
 jX L
1
 Load
2
RLoad  jX L
RLoad  X L2
Eq 13
Admittance of Inductor + Rload branch
The real or resistive part is
 j
XL
2
Rload
 X L2
Rload
and the imaginary or reactive part is
R  X L2
2
load
. The capacitive reactance, Xc, will be made such that it cancels the
inductive reactance above. That is:
j
X
1
 j 2 L 2 0
Xc
Rload  X L
Page -5-
solving for Xc:
2
RLoad
 X L2
Xc 
XL
Eq 14
Capacitive Reactance Formula for L Network
See Eq 17 for XL
Note that if XL >> Rload , then Xc is just the same magnitude as XL as you would expect. For the
real part as an impedance, invert the real part of the admittance from above.
Rinput
2
Rload
 X L2

Rload
Eq 15
Input Impedance for L Network on High Z End
Consulting figure 2, Rinput should be set equal to Rgen for best power transfer and 1:1 VSWR. This
is the same equation that is copied over with appropriate nomenclature changes in Eq 2 above.
Equations 1,4 and 5 also apply here interpreting R load in the same way as Rs. According to
Equations 1 and 5, the Q is determined for you once the input and output resistance is known.
Combining equations 1 and 5 with resistances relabeled as R load and Rinput, we get:
Qoperating 
Rinput
Rload
Eq 16
Operating Q for L Network with Rinput > Rload
Latter I will show that setting operating Q of a Pi-Network to a very low value causes it to
degenerate to an L network.
Solving 15 for XL:
2
X L  RinputRload  Rload
Eq 17
Inductive Reactance for L Network with Rinput > Rload
You can also substitute Eq 17 into Eq 14 for a simpler form for X c:
Xc 
RinputRload
XL
Eq 18
Capacitive Reactance for L Network
See the associated L Network design Excel Spread Sheet. This is an older program from 2006
not updated other than to put in ham radio frequencies and component values. The Pi-Network
design sheet is much better and defaults to an L network in the case where output Q is set to
near zero so I did not upgrade the L Network program.
Page -6-
The example has:



75 meters: center frequency 3.75Mhz
50 ohm transmitter
200+j0 antenna load
No reactance in the antenna load is not typical but that situation is covered below. Note that the
network is turned around because the antenna impedance is higher than nominal. The spread
sheet is fixed in this configuration. Using Eq 17:
X L  200 * 50  50 2  86.60Ohms
Using Eq 18:
Xc 
200 * 50
 115.47Ohms
86.60
Converting these to component values at 3.75Mhz, e get:
L = 3.68µH
C = 366pF
Using Equation 16:
Qoperating 
200
2
50
A satisfactorily low value with good bandwidth of about +/- 0.94 Mhz to the 3dB points which are
VSWR=5.6:1 and reflection S11=0.707. Obviously you want to have a very wide bandwidth
because it‟s unlikely you would want to operate beyond 2:1 let alone 3:1 VSWR which is well
inside the bandwidth. Of course you could readjust per frequency. Actually, the bandwidth is
wider because the formula in Eq 16 assumed Q>>1 which it is not. Therefore, it is calculating the
minimum operating Q to expect.
The spread sheet calculates the frequency dependent reactance of the components from (in this
case) from 2 to 7Mhz (or whatever one copies into the frequency column…you must be familiar
with Excel to use it). In then does the same backward ladder network impedance calculation we
did above but numerically for each frequency and finally calculates the reflection coefficient and
from that the VSWR. This calculation is accurate (neglecting coil losses) as it uses no
approximations.
Page -7-
L Matching Network
1.000
0.900
0.800
0.700
|S11|
0.600
Abs(S11)
0.500
S11 at load
0.400
0.300
s11=0.5
is 3:1 VSWR.
s11 < 1 or
100%
0.200
0.100
0.000
0.0
2.0
4.0
6.0
8.0
MHz
Figure 3
Example Reflection Voltage (S11).
The reflection was also calculated at the 200Ω load and is a constant 0.6 or 4:1 VSWR. We are
seen to get no reflected power right on 3.75Mhz. On the high side, we don‟t get reflection
S11=0.7 until 5.7Mhz or 1.95Mhz over the center frequency. On the low side, from inspection, at
very low frequency approximating DC, the VSWR would only be 4:1, 200Ω/50Ω, since the
capacitor and inductor would be ineffective and it would simply be a 200Ω load connected to the
transmitter. Therefore, we never do get to S11=0.707 or 3dB down response so the Q
approximation does not work for bandwidth (it does for loss calculations) since the network is
actually a low pass filter unless the impedance ratios are extreme…you are trying to load a short
wet noodle. This is typical of matching networks which in any event are not designed to be
bandpass networks.
Page -8-
VSWR @ Network Input
3.00
2.80
2.60
2.40
VSWR
2.20
2.00
1.80
1.60
1.40
1.20
1.00
3.5
3.6
3.7
3.8
3.9
4.0
Mhz
Figure 4
Example L Net VSWR
The VSWR chart covers only the 80 meter band with a maximum of 1.2:1 at the band edges. If
the antenna were just 200+j0Ω across this matching network could have one adjustment for the
entire band. What if the antenna were, at 3.75Mhz, 200Ω resistive in parallel with capacitive
100pF? (164-j77Ω series equivalent) Then reduce C from 366pF to 266pF so that 100pF is
supplied by the antenna itself! The result would be the same at 3.75Mhz. The antenna would not
be 100pF parallel equivalent across the band so the network would need to be readjusted on
different frequencies. In general, an antenna will produce a situation where you do not get as
much bandwidth as with a resistive load even with the same VSWR at the center point as the
resistor.
What if the antenna was 200Ω in parallel with 200pF? Then reduce the L network capacitor to
166pF. If 300pF? (66-j94Ω series) Then the capacitor is 66pF but what if the antenna is 200Ω in
parallel with 400pF? (43.9-j83Ω) Then you could use 366 - 400 = -34pF? This is an inductor of
52µH to ground and then the rest of the network would work as is. However, 52µH is awfully big.
There is another way as you can see. 43.9Ω is almost 50Ω so just cancel the –j83 with the
inductor and forgo the capacitor. Better yet, add a little inductor which will both increase the
parallel resistance to 50Ω and go a little +j. Then use the capacitor on the other side! That is,
just reverse the network. You can always find, at least in theory, an L network that will match the
load providing there is a resistive component.
Here is an example matching 1500Ω to 50Ω also on 3.75Mhz, a tube final amplifier to
transmission line. I choose it because it is the impedance set for an example of Pi-Network in the
ARRL Handbook which I also address in my Pi-Net section below. Using Eq 16, 17 and 18, we
get:
Page -9-
Component
Impedance
Value @
3.75Mhz
Rinput (Rgen)
Rload
Qoperating
L
C
1500Ω
50Ω
5.48
269.3Ω
11.43µH
278.5Ω
152.4pF
Table 2
Tube PA to 50Ω L Network Matching
L Matching Network
1.200
1.000
|S11|
0.800
Abs(S11)
0.600
S11 at load
0.400
s11=0.5
is 3:1 VSWR.
s11 < 1 or
100%
0.200
0.000
0.0
2.0
4.0
6.0
8.0
MHz
Fig 5
L-Net Reflection at Tube Rgen end 1500Ω
If you compare Fig 5 with Fig 3, you can see that it is much narrower (both same scale). The
calculated Q of the former is 2 while that above is 5.485. This Q predicts we will find S11=0.707
(half power reflected) at 384khz high but we find it at 650khz, 4.4Mhz. This time, there S a ½
power point on the low side at about 3Mhz. Regardless, it is much narrower than the previous
example which the VSWR chart below makes more than apparent.
5
All the time remembering at low Qs the BW prediction is not accurate but the loss prediction is.
Page -10-
VSWR @ Network Input
3.00
2.80
2.60
2.40
VSWR
2.20
2.00
1.80
1.60
1.40
1.20
1.00
3.5
3.6
3.7
3.8
3.9
4.0
Mhz
Fig 6
L-Net VSWR at Tube Rgen end 1500Ω
Looking at Equation 16 and knowing that the L network offers the minimum Q of any LC matching
network, one wonders if there is a way to widen the bandwidth if the impedance ratio is large as
in the above example. Consider an RF transformer with wide bandwidth6. If in the 50 to 1500Ω
step up, we used first a 4:1 transformer, then the Q of the L network would be 2.74 instead of
5.48. If there were two 4:1 transformers in series, then the Q (calculated) would be 1.37. This is
the reason that FET inputs for HF receivers which have fairly high input impedance often use
toroidal transformers or even sequential transformers 7. It is also possible to use cascaded L
network section to get better bandwidth but it is costly for transmitters and there is an optimal
number of sections. Cascaded L networks approximate a transmission line matching section
since a transmission line impedance is
L
C
where L and C are inductance and capacitance per
length. If the L network sections are the same ratio, they have the same „surge‟ impedance.
6
The bandwidth of an RF transformer is limited by stray inductance and inter-winding
capacitance. More of either means a lower cut-off frequency.
7
High impedance ratio transformers often have narrower bandwidth than low impedance ratio
Page -11-
Pi Network
1
2
XL
ITX
Rgen
Xc1
Xc2
Rload
Rload < Zo = Rgen
Fig 7
Pi-Network
The Pi-network above, having one more reactive element than the L network, a capacitor
(reactance Xc2), has one more degree of freedom and can match high to low impedances either
way since it has symmetric topology. It has just 33% more parts for analysis, not counting the
transmitter, than the L network but the analysis is several times as lengthy. A few steps
considered just algebra, are omitted.
Xc2 and Rload are connected in parallel so we add their admittances.
Y2 
1
Rload
j
X  jRload
1
 c2
X c2
Rl X c 2
Eq 19
Admittance at node 2 to ground
1/Y2 to ground is Z2 to ground.
2
RL X c 2
Rload X c 2 ( X c 2  jRload ) Rload X c22  jRload
X c2
Z2 


2
2
2
2
X c 2  jRload
X c 2  Rload
X c 2  Rload
Eq 20
Impedance at node 2 to ground
Now add in the inductor in series with node 2 impedance to ground to get L branch impedance to
ground at node 1.
Z1Lbranch 
 

2
2
2
Rload X c22  jRload
X c2
Rload X c22  j X L X c22  Rload
 Rload
X c2

jX

load
2
2
2
2
X c 2  Rload
X c 2  Rload
Eq 21
L Branch impedance to ground at node 1
Take reciprocal to find admittance at node one due to L branch: Y Lbranch=1/ZLbranch.
Page -12-

2
X c22  Rload

2
2
Rload X c22  j X L X c22  Rload
 Rload
X c2
Y1branch
X

2
c2

 
 



2
2
2
 Rload
Rload X c22  j X L X c22  Rload
 Rload
X c2
 

2
2
2
Rload
X c42  X L X c22  Rload
 Rload
X c2


2
Eq 21
L Branch Admittance to ground at node 1
Assume that the –j part (red) will be cancelled by a +j admittance or capacitor to ground, that is
C1. Then the real part (green) of the admittance remains. Taking the reciprocal of the real part
gives the input impedance to the Pi-network once tuned by C1 and with Rload at the output.
Z1real
 

2
2
R 2 X 4  X L X c22  Rload
 Rload
X c2
 load c 2
2
2
2
Rload X c 2 X c 2  Rload



2
Eq 22
Pi-Network Input Impedance
Believe it or not, without some additional definitions, Eq 22 is in its simplest form! Xc1 is
determined by another equation we did not write yet (the red imaginary part) for a total of two
network equations. Rload and Zreal are known but there are three undetermined variables, XL, Xc1
and Xc2. We have one more additional degree of freedom and cannot solve for the component
8
reactances. Either we fix one of the components or the ratio of two components. Fixing the ratio
of a reactance (or sum of reactances) to a resistance (or sum of resistances) is the better choice
since it determines operating Q and therefore both the bandwidth and, if component unloaded Qs
are known, network loss.
As a convenience and for simplicity, I define Q as being:
Q
Rload
X c2
Eq 23
Pi-Network Q (output Q or Q2)
Also Defines Xc2
Note that this can be solved for Xc2. In the ARRL handbook about pg 18.6, is Elmer Winfield‟s
(W5FD) New…Formulas for…Pi…Networks. He defines three different Qs which are not
independent (only one is a free choice).
Q1 
R gen
Q2 
Rload
X c2
X c1
Qo ( perating)  Q1  Q2
Equations 24
W5FD‟s Pi-Network “Q‟s”
8
If Xc1 or Xc2 are fixed at zero, we have the L network again
Page -13-
My Q in equation 23 is seen to be W5FD‟s Q2. His procedure defines Qo arbitrarily or rather to
meet harmonic suppression goals but since that is not a goal in my procedure, we will just choose
a low value, perhaps 1 or less, for my Q (Q2) to reduce losses and widen bandwidth. A typical
choice for Qo is 10..12 in W5FD‟s method when used as a tube amplifier matching network but
this is actually a practical choice for heat and losses since for harmonic suppression, you would
want it as high as possible.
The procedure for a solution is to expand (multiply terms through) equation 22 and then divide
numerator and denominator by Xc24 and then to replace every Rload/Xc2 with Q. Then you collect
terms This is really painful to watch so I give you the final result:
Rinput  Rgen


2
Rload
 X L2 1  Q 2  2QX L Rload

Rload
Eq 25
Input Impedance of a tuned Pi-network
Xc2 is gone and with all but XL defined, you can solve for XL. If the Q>>1 (we won‟t use that
condition for matching networks), 1+Q2 ~ Q2 and with that substitution, the numerator is perfect
square and you get:
Rinput 
 X L Q  Rload 2
Rload
Eq 26
Input Impedance for Pi-network with High Q2
In that case:
XL 
Rload  Rload Rinput
Q
Eq 27
Inductive Reactance for High Output Q2
Eq 27 is not near correct for low output Q.
You can solve Eq 25 for XL, not assuming a high output Q, by multiplying both sides by R load and
then subtracting RgenRload from both sides. You then get a quadratic equation in XL:


X L2 1  Q 2  2QX L Rload  Rload Rload  Rinput   0
Eq 28
Solve for XL
Which you then can use the standard formula:

 Rload 
R
 
Rload  Q  Q 2  1  Q 2  input
Rload  




XL 
2
1 Q


Eq 29
Inductive Reactance for Pi-network
Page -14-
Sharp observers wonder if the inductor goes imaginary or turns into a capacitor when R load>Rinput
and the answer is that you don‟t let that happen. You turn the network around and reverse the
names so that Eq 29 always produces a real answer.9 Alternately, take the absolute value so you
get:


R
 Rload

Rload  Q  Q 2  1  Q 2 input
Rload 



XL 
1 Q2


Eq 30
Alternate Pi-Network Inductive Reactance Formula with no „imaginary‟ results
That‟s just the mathematical way of saying the same thing.  Note this equation looks similar to
W5FD‟s equation for calculating Q1. You then use Q1 (and operating Qo) to calculate XL. It‟s the
same result in disguise as we shall see working the same example.
Returning to Eq 21, choose the imaginary part (red…plus the denominator) and set this equal to
C1‟s admittance which is 1/Xc1. You get:
X c1
R

X

2
X c22  A 2
2
 Rload
A
load
2
c2


where

2
2
A  X load X c22  Rload
 Rload
X c2
Eq 31
Input Capacitor Reactance for Pi-Net
see below
Remember that Xc2 was defined in Eq 23 as:
X c2 
Rload
Q
Eq 23b
Xc2 Output Capacitor for Pi-Net
Equation 31 is a bit ugly and can, like Equation 30, be stated in terms of Q by substituting
Equation 23b for Xc2. Eventually, you get:
X c1 


2
Rload
 X L2 1  Q 2  2 X L RloadQ
X L 1  Q 2  RloadQ


Eq 31b
Input Capacitor Reactance for Pi-Net
Compare o equations 17 and 18 to see how much complexity is added with just one additional
part! Typically, larger networks are designed using modern filter theory and either normalized
tables of components or computers. Computer design can either use tables or an optimization
metric (like low loss or bandwidth) and methods like Newton-Raphson or half value convergence
and a generic analysis method. As early as 1974, I wrote software to start with table driven
design and then move to microstrip with Newton-Raphson since the latter network can be
analyzed but not synthesized in closed form.
9
and we use the + only and not +/- after the first numerator Q.
Page -15-
The first example is that from the ARRL Handbook about page 18.6. The values can be
calculated from equations 30, 31b and 23b above or use the related spreadsheet. Since this
example calculated an output Q2 of 1.62 after choosing operating Qo of 12, we choose Q=1.62.
Component
Impedance
Rinput (Rgen)
1500Ω
Rload
50Ω
Qoperating
12.00
Value @ 3.75Mhz
Q1 (input)
Q2 (output, our Q)
C1
L
C2
10.38
1.62
144.52Ω
293.7pF
166.54Ω
7.026µH
30.86Ω
1375.1pF
Table 2
Tube PA to 50Ω Pi Network Matching
If you check the L network values in table 2, you will see both capacitors in this network are larger
but the inductor is smaller. As the Q increases beyond 12 (operating), capacitors will get still
bigger but the inductor still smaller. Again, the total of the capacitive reactance is close to but not
equal to the inductive reactance.
Pi Network
1.000
0.900
0.800
0.700
|S11|
0.600
Abs(S11)
Abs(S11) at load
0.500
0.400
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
0.300
0.200
0.100
0.000
0.000
1.000
2.000
3.000
4.000
5.000
6.000
7.000
8.000
Mhz
Fig 8
Pi-Network Reflection at 1500Ω input
The characteristic impedance for the reflection calculation is 1500Ω, the image impedance of the
50Ω load presented to the vacuum tube power amplifier as per the example. The red line is the
reflection from 50Ω on a 1500Ω system if the Pi-Net were not there. The power to the load would
Page -16-
be much less. Compare the bandwidth to figure 5. It‟s much less and losses with the same type
of components are much higher.
Abs(Zin)
1600.00
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
1400.00
Ohms (absolute)
1200.00
1000.00
800.00
600.00
400.00
200.00
0.00
2.000
2.500
3.000
3.500
4.000
4.500
5.000
Mhz
Fig 9
Input Impedance (absolute) at Pi-Network Input
The Pi_Net spread sheet also calculates the absolute input impedance of the network with 50Ω
load. Absolute means both reactance and resistance are included and voltage phase angle
ignored (current is controlled by the vacuum tube). However, near the peak most is resistive. To
the first approximation, output voltage at the tube plate is:
Eout  g m Z input
Where Eo is tube output voltage (not on the output 50Ω resistor but at the network node 1), g m is
tube transconductance (how much output current you get per unit of input „grid‟ voltage…or
equivalent for tube or transistor) and Z input is the network image input impedance, 1500Ω at the
peak. Note Rgen, the tube output resistance, is assume to be infinity, non-existent10. It‟s working
into a 1500Ω load but it‟s own output impedance is much higher. A typical number for gm would
be 7000µMho for a low frequency gain of 10.5 11. If the impedance is halved, the gain and output
voltage is halved. When the voltage drops to .707 of center value, the output power drops to
.7072 = 0.5, it‟s halved. That happens at 3.62Mhz and 3.945Mhz according to the spreadsheet.
This is 0.325Mhz 3dB BW or 3.75/0.325=11.54 which is another way of calculating operating Q.
This is about the value used in the W5FD method of calculation.
10
It was actually bout 5000Ω
3000Ω would be better for this tube at 600 volts but that is immaterial here. Tube input Z is
very high so despite a voltage gain of just 10, the power gain at 3.75Mhz would be ~35dB!
..class AB1. Gain at 2 meters was only 10dB 
11
Page -17-
If you measure the BW another way, by inspecting S11 for a reflected voltage of 0.707 which is a
return loss of 3dB meaning half goes forward to the load and half comes back, you get 0.665Mhz
or a Q operating of 5.6, just about half. The difference is that the previous, assuming the device
output or Rgen is infinity is the singly loaded BW but the latter which implicitly assumes all the
reflected power is absorbed12, is the doubly loaded BW or implicitly Q. There are many engineers
that get really confused about this point.
VSWR
5.00
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
4.50
4.00
3.50
3.00
2.50
2.00
1.50
1.00
3.500
3.550
3.600
3.650
3.700
3.750
3.800
3.850
3.900
3.950
4.000
Mhz
Fig 10
VSWR at Pi_Net input assuming Z o = 1500Ω
Compare Fig 10 with that of the L network in Fig 6. Obviously, we don‟t have the same BW for
VSWR, same as S11 from which this is derived.
If I now enter 0.001 for Q2 (Q or output Q in the equations above) in the spreadsheet, I get:
C2
L
C1
0.8 pF
11.430 uH
152.4 pF
which are the values from the L network calculation (except for very small C2 value which you
can discard). The lowest Q Pi-network IS the L network. The input impedance at he same scale
is shown below.
12
else we would have to include the reflection as useable power because it would be re-reflected
meaning you could not calculated BW this way at all…meaning the generator impedance is a
match for the input impedance whatever it is at a frequency. That only happens with signal
generators, not transmitters.
Page -18-
Abs(Zin)
1600.00
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
1400.00
Ohms (absolute)
1200.00
1000.00
800.00
600.00
400.00
200.00
0.00
2.000
2.500
3.000
3.500
4.000
4.500
5.000
Mhz
Fig 10
Pi_Net Zin if Q2 (output Q) set to 0.001 (zero). Same as L Net
If I enter Q2=.001 and Rload=200Ω while Zin=50Ω, I get:
C2
L
C1
367.6 pF
3.678 uH
0.8 pF
which are the values from the same example for he L-network. The program switches the
definition of “output” Q around to always mean the low impedance end so C1 at the 50Ω input
disappears so we still get the right L-network configuration. You must use the equations above
the same way. For a Q output of 1 and the same resistances/impedances, you get:
C2
L
C1
561.4 pF
3.868 uH
848.8 pF
Note again that the capacitors have increased in size AND you need two of them! You can play
with the input parameters in the spreadsheet. As expected, the VSWR BW decreased compared
to the L network. See Fig 4.
Page -19-
VSWR
3.00
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
2.80
2.60
2.40
2.20
2.00
1.80
1.60
1.40
1.20
1.00
3.500
3.550
3.600
3.650
3.700
3.750
3.800
3.850
3.900
3.950
4.000
Mhz
Fig 11
Pi-Net VSWR 200Ω load to 50Ω system, Qout = 1
Compare Fig 4
T Network
1
2
3
Xc2
Xc1
XL
ITX
Rgen
Rload
Fig 12
T Network
The method of analysis and degrees of freedom are the same as for the Pi-network except that it
is in some ways the mirror network. To obtain a very low Q at one end, you need a very large
capacitor instead of a vanishingly small one. If higher operating Q is tolerable, then caps can be
smaller than the Pi-net if at the same time inductors are larger. Output Q is:
Page -20-
Qoutput 
X c2
Rload
Eq 32
T Network Q
Most of the differences are in the engineering of the design, practical limitations rather than
theory and it shares many of the limitations of the Pi-network for matching applications and ne
advantage, smaller capacitors. However, it is a high pass network instead of a low pass network
like the Pi-network. That is, it will have poor harmonic suppression in comparison but this is not
the prime design goal regardless.
Analysis is similarly complex to the Pi-network and like that above, a few steps considered just
algebra, are omitted.
Zin, input impedance for the Xc2 branch at node 2:
Z 2 xc  Rload  jX c 2
Eq 33
Input impedance at node 2 for branch Xc2
The equivalent admittance is:
Y2 xc 
Rload
R  jX c 2 Rload  jX c 2
1
 load
 2
 jX c 2 Rload  jX c 2 Rload
 X c22
Eq 34
Admittance at node 2 for branch Xc2
Now add the inductor‟s admittance, -j/XL:
Y2 XLC  Y2 xc 
j
R  jX c 2
j
 load


2
2
XL
Rload  X c 2
XL


2
X L Rload  jX c 2   j Rload
 X c22

2
X L Rload
 X c22
X L Rload


 j X X  R
X R  X 
L
L
c2
2
load
2
load
 X c22

2
c2
Eq 35
Admittance at node 2 for branch Xc2 || XL
One cannot just take the real part of Y2XLC since Xc1 will not be added in shunt as with the
Pi-network. One must find the series equivalent so that Xc1 will compensate the reactive part in
series leaving the real component. Thus the T network analysis is even more complicated than
the Pi-network.



2
X L Rload
 X c22
Z 2 XLC  1

Y2 XLC X R  j X X  R 2  X 2
L load
L
c2
load
c2
Page -21-

Z 2 XLC 

X L Rload






2
2
X L Rload
 X c22
X L Rload  j X L X c 2  Rload
 X c22

2
2
 j X L X c 2  Rload
 X c22 X L Rload  j X L X c 2  Rload
 X c22




2
2
2
2
X load
Rload Rload
 X c22  jX L Rload
 X c22 X L X c 2  Rload
 X c22

2
2
X L2 Rload
 X L X c 2  Rload
 X c22




2
Eq 36
Impedance at node for branch Xc2 || XL
…and there are just three reactive parts but only two are involved above!13. If you add Xc1 in
series, Eq. 36 gives the T-network input impedance.
The real part may now be extracted and this is indeed the real component of the T-network input
impedance even at node 1 since Xc1 change only the reactive part.

2
X L2 Rload Rload
 X c22
Z1real  Z 2 real 
2
L
2
load
X R

 X L X c2  R
2
load

 X c22

2
Eq 37
Real Component of T-Net Input Impedance
As before, substitute output Q for Xc2 as defined in equation 32,
Z1real 

3
X L2 Rload
1 Q2
2
L
2
load
X R


 X L QRload  R
2
load
X c 2  QRload .
1  Q 
2
2
Eq 38
Real Component of T-Net Input Impedance in terms of output Q
Obviously, in equation 38, some of Rload could cancel out. That s accomplished correctly in a
painful expansion shown next.
Z1real


X L2 Rload 1  Q 2
 2
2
2
X L 1  Q 2  2 Rload X L Q 1  Q 2  Rload
Q 2 2  Q 2  Rload






Eq 39
Real Component of T-Net Input Impedance Expanded
If you remember high school algebra, you can see a perfect square coming in the lower right of
the denominator but that is handled latter. All in equation 39 is known except X L. To solve for
that, set Z1real equal to Rgen, the desired input or matched impedance, usually 50Ω. The result can
be viewed as a polynomial in XL, a quadratic.






2
X L2 1  Q 2 Rgen  Rload   X L 2Rgen RloadQ 1  Q 2  Rgen Rload
1  Q2

2
0
Eq 40
13
I made several errors further simplifying it the first time but fortunately it‟s easy to detect that an
error is present if not where it is located. Contrary to the mess created with a generic closed
solution, calculating impedances of a ladder network numerically is easy since each step
collapses to just a single complex number at each step. Thus once the closed solution to
anything (just Z2 here) is finished, a numerical example can be run to check it…providing two
compensating errors are not made that work with unique numbers. Thus always choose strange
numbers, fractional Q for example.
Page -22-
Quadratic in XL
One can now find XL using the quadratic formula which after simplification is:
XL 




Rload RgenQ  Rgen Rload 1  Q 2  Rgen

Rgen  Rload
Eq 41
Value for Inductive Reactance for T-Network
And note that Rgen is the same as desired input resistance, Rin. Also, the determinant above must
be positive which leads to a restriction on Q. Specifically that:
Q
R gen
Rload
1
Eq 42
T-Network Minimum Output Q
If the minimum Q is used, then the value of the other capacitive reactance, Xc1, is zero. This can
be implemented with … a wire. 
I find through experimentation that only the minus sign in equation 41 will give the correct answer
for a T-network that has a capacitor in branch one. That is, Xc1 is a capacitor and not an inductor.
If the positive sign, + , is used, the reactance at node 2 is still capacitive reactive so another
capacitor cannot cancel it. You must use an inductor. However, this is quite legitimate and that
type of T-network with two inductors and one capacitor does match the two impedances. I will
stick with the traditional two capacitor T-network so we will need to use the minus sign. Both Eq
41, now that we are using only the minus sign, and Eq 42 also mean that:
Rgen  Rload
Eq 43
Input Output Resistance Requirement
Xc2 has actually been previously defined in Eq 32 which can be rewritten:
X c 2  QoutputRload
Eq 32b
Xc2
So what if the antenna has an impedance higher than the nominal transmitter output, R gen?
Simply reverse the network and make the transmitter output impedance R load and the antenna
Rgen.
Now that XL has been found, the imaginary part of Eq 36 can be used to determine X c1.
Z 2imag  X c1 


2
2
X L Rload
 X c22 Rload
 X c22  X L X c 2

2
2
X L2 Rload
 Rload
 X c22  X L X c 2
Eq 44
Input Capacitive Reactance for T-Network
Page -23-

2

Eq 44 will simplify only if a substitution for Xc2 to QRload is made but I won‟t do that this time. Also,
one can define an input Q as Xc1/Rgen and also make Q operating=Qinput+Qoutput as before.
The first T-Network example is that from the ARRL Handbook about page 18.6, the same as used
for the Pi-Net example above. The values can be calculated from equations 41, 32b and 44
above or use the related spreadsheet. An output Q of 10.4 gives an operating Q of 12 as before.
Component
Impedance
Rinput (Rgen)
1500Ω
Rload
50Ω
Qoperating
12.00
Value @ 3.75Mhz
Q1 (input)
Q2 (output, our Q)
C1
L
C2
1.62
10.4
2436.6Ω
17.4pF
453.91Ω
19.265µH
520Ω
81.6pF
Table 3
Tube PA to 50Ω T Network Matching
Comparing to the Pi-network values in Table 2, you can see that the T-network is not a very good
choice for matching vacuum tubes to transmission lines. The values are just large enough to be
practical on the 80 meter band but totally impractical for 10 meters (C1 would be
2.3pF…adjustable!). Also, the input and output Qs have swapped around compared to that of the
Pi-network.
Pi Network
1.000
0.900
0.800
0.700
|S11|
0.600
Abs(S11)
Abs(S11) at load
0.500
0.400
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
0.300
0.200
0.100
0.000
0.000
1.000
2.000
3.000
4.000
5.000
6.000
Mhz
Figure 13
T Network Reflection at 1500Ω Input
Page -24-
7.000
8.000
The reflection in Fig 13 is very much like that of the Pi-Network in fig 8 except that it is skewed
the other way being ultimately a high pass network rather than a low pass network like the Pinetwork. This is yet another reason not to use a T network for a tube output matching network as
it would pass many of the higher harmonics.
Abs(Zin)
6000.00
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
5000.00
Ohms (absolute)
4000.00
3000.00
2000.00
1000.00
0.00
2.000
2.500
3.000
3.500
4.000
4.500
5.000
Mhz
Fig 14
T Network Input Impedance (Absolute)
With the input and output Qs flipped around, fig 14 is not a duplicate of fig 9 for the Pi-network. L
and C2 become resonant just above the design center frequency of 3.75Mhz at which it is the
required 1500Ω. At L || C2 resonance, the impedance is much higher.
Page -25-
VSWR
5.00
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
4.50
4.00
3.50
3.00
2.50
2.00
1.50
1.00
3.500
3.550
3.600
3.650
3.700
3.750
3.800
3.850
3.900
3.950
4.000
Mhz
Fig 15
VSWR at T_Net input assuming Zo = 1500Ω
However, the in band VSWR is the mirror image of the Pi-net from fig 10 with the difference again
being due to a T net being a high pass instead of a low pass network.
If again I enter .001 for the output Q, I end up with an L network once again with a minor
difference.
C2
L
C1
157.6 pF
11.822 uH
424413.2 pF
Just a wire
The L network values for the system calculated above were 11.43µH and 152.4pF. Actually, it is
the impedances that are mirror images of each other and not the component values in pF and
µH. The reactance values at ~0 output Q are:
Pi-Network
XC2
XL
XC1
T-Network
XC2
XL
XC1
50000.00 ohms
269.31 ohms
278.54 ohms
Cap
removed
Just a wire
269.26 ohms
278.54 ohms
0.10 ohms
The above is for tube matching network impedances of 1500Ω to 50Ω. The series branch in both
cases is 269Ω while the shunt branch is 278Ω. The math works out the same either way with just
Page -26-
a change in sign for the reactive part. Also it was the output cap that is deleted in the Pi-network
to make an L network but the input cap that is shorted to make an L network with the T-network.
Also, the output Q in the Pi-net is zero whereas in the T network, it must satisfy Eq 42 so it is in
fact 5.4 which is also the operating Q for both ways of putting it together. The resulting L
networks are also mirror images of each other.
For the case of a 200Ω load, 50Ω input impedance and minimum output Q which is 1.7, the
values at 3.75Mhz are:
XC2
XL
XC1
C2
L
C1
0.10 ohms
115.47 ohms
86.60 ohms
Just a wire
424413.2 pF
4.901 uH
490.1 pF
VSWR
3.00
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
2.80
2.60
2.40
2.20
2.00
1.80
1.60
1.40
1.20
1.00
3.500
3.550
3.600
3.650
3.700
3.750
3.800
3.850
3.900
Mhz
Fig 16
T-Net VSWR 200Ω load to 50Ω system, Qout = min = 1.7
(input cap shorted out)
Compare above with figs 4 and 11.
Page -27-
3.950
4.000
If the output load suddenly becomes 200+j100, then the VSWR changes to:
VSWR
3.00
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
2.80
2.60
2.40
2.20
2.00
1.80
1.60
1.40
1.20
1.00
3.500
3.550
3.600
3.650
3.700
3.750
3.800
3.850
3.900
Mhz
Fig 17
T-net same but load = 200+j100
Then you just unshort C2 and change it to 424pF to compensate as shown below.
C2
L
C1
424.0 pF
4.901 uH
490.1 pF
Page -28-
3.950
4.000
VSWR
3.00
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
2.80
2.60
2.40
2.20
2.00
1.80
1.60
1.40
1.20
1.00
3.500
3.550
3.600
3.650
3.700
3.750
3.800
3.850
3.900
3.950
4.000
Mhz
Fig 18
T-net oad = 200+j100 but C2 = 424pF
The same compensations can be made just as with the Pi-Network. If the output is capacitive
reactive (e.g. 200-j100), then an output Q greater than the minimum must be chosen until the
required output reactance is more than or equal to that of the load such that the output capacitor
can be adjusted to compensate.
Engineering Considerations
Note; make sure discuss how transformers can make wider bandwidth
Engineering considerations are more than components values and Q. Specifically, one should
consider in order of importance assuming the product will be sold:




Cost
Size
VSWR Range & Power Handling (Tech Specs)
Operator Use & Controls
General Comparison
You might think the L network wins on all accounts but the fact it has just two components
means that both must be continuously variable to achieve a good null..and the customer is
probably not satisfied with 1.1:1 VSWR. Roller inductors are more expensive. MFJ tuners start
at $69.95 for the MFJ-16010 and in fact this one IS an L network with a tapped inductor (not roller
type) BUT it says it‟s not for coaxial lines, just random wire antennas. The 10-80M „travel tuner‟,
MFJ-902 (150W) at $99.95 is their lowest cost „coaxial‟ type and it‟s a T network with tapped
inductors and two variable capacitors which does permit a 1.0:1 VSWR adjustment. However,
Page -29-
the lowest cost roller inductor type is MFJ-969 (300W) at $219.95. Unless you forgo the ability to
always set a complete null at 1.0:1 VSWR, the L network is not the lowest cost14.
Note that toroidal inductors make a design compact but introduce the possibility of harmonic
generation if the current gets too high which can be induced even at low power with a high
impedance ratio15 or operator induced hi Q. Ferrite cores can saturate and become non-linear
which generates harmonics16.
The L network will handle the highest power for a particular class of components 17. It has only
one, unambiguous setting for any load. In the grounded capacitor configuration 18, the capacitor
requires no isolation for the tuning knobs19.
The Pi-network has multiple possible settings for low VSWR20 operator confusion and the related
Q/BW and power handling limitation as discussed. However, you can get a VSWR null using a
cheaper tapped inductor. Isolation is not required for hunt capacitors and the series inductor
would require isolation regardless. It‟s also a low pass network for harmonic suppression21.
Lowest Q and & best BW and power handling is at the smallest low impedance side capacitor
setting. Not knowing which side, minimum initial capacitance both sides and minimal inductance
is best. This is the minimum reactance condition. Therefore, typical operation with lower VSWR
will use the smallest components.
The T-network also has multiple possible settings for low VSWR operator confusion and related
Q/BW/power handling limitations. You can get a VSWR null using a cheaper tapped inductor.
Isolation is required for the tuning capacitor controls if it is a manual adjust unit (not auto-tune). It
does not suppress harmonics but it can reduce low frequency broadcast/IF interference problems
in your receiver. Lowest Q and best BW and power handling is at the largest low impedance side
capacitor setting. Operator manuals often just say (quote from MJ-902 manual): “Be sure to use
the highest possible capacitance for each band.…use the minimum amount of inductance as
possible. Minimum inductance gives the best efficiency, maximum power handling, and widest
bandwidth.” The assumption is probably that you do not know which side is low impedance so
both caps at maximum will do. This is the minimum reactance condition. Therefore, typical
operation with lower VSWR will use the largest possible components. Voltages across the
capacitors are lower than with the Pi-network22.
Auto-tune vs Manual Tune
While the power supply and relays required for auto-tune cost more, the difference is not as great
as supposed. Power can be derived from the transceiver supply if, as likely, it is assumed to be
solid state and have an accessory power take off available. Units custom designed for a
particular transceiver or interface may actually be the same cost as simple tuners as the
transceiver display and controls may be used. Additionally, microprocessor control can avoid
14
But if you accept the limitation, it is!
Equation 16 sets a lower Q limit not just for L networks but also Pi and T networks which of
course must have higher Q yet.
16
The VSWR will ominously jump up at higher power when that happens. You can sometimes
hear people on 75 meters asking what that problem might be!
17
Unless you adjust a Pi or T-net until it is just an L network as shown in the Pi-Net section
18
You can also have a series capacitor and a shunt inductor for an L-network
19
Sometimes manufacturers just use the knob itself as isolation! This can lead to RF burns on
the knob screw and make it „touchy‟ (as the manual often says) …literally…on the high bands.
20
Infinite with a roller inductor
21
Except in the case of toroidal current overload
22
For normal operation… It‟s possible for both Pi-net and T-net to be adjusted for very high
voltages if the load is very high impedance.
15
Page -30-
trouble like high power when un-tuned or adjustment to the China Syndrome setting. It will also
neatly resolve the ambiguity problem. With some additional cost, it can actually read the antenna
impedance at the transceiver connector and double as an antenna bridge (if power can be
reduced for out of band) especially useful if the transceiver can operate on battery at low power.
Specifications
My experience is that even with Radio Amateurs, technical specifications, performance is
important but not the number one consideration or even number two. Although many will
complain about usage problems, few will investigate before purchase so this is not at the top.
However, it‟s important for the return rate since it‟s possible to set some tuners so that they will
self destruct. I will consider cost factors but not in absolute terms. Size reduction will depend on
both the specs and how much money one might be willing to spend. One of the biggest problems
is designing a reasonable cost and size tuner that will handle a wide VSWR range over a wide
frequency range. This is a really difficult challenge with a traditional, low cost manually operated
tuner. Perhaps because of that, most of that class tuner has no tuning range specification at all!
Sometimes a wire length + coax length range is specified instead…but it doesn‟t specify the
height above what kind of ground so this specification is unbounded as well. Also, such a tuner
will progressively match higher VSWRs with the same components at higher frequencies so the
lowest frequency for the VSWR range should be stated. Below see a Smith Chart with 5:1 VSWR
circle. Everything in the red circle should match if the tuner is rated at 5:1 VSWR which seems to
be the general desire. However, many auto tuners operate only inside the blue circle at 3:1
VSWR. As you can see, 5:1 VSWR (the center is 1.0=50 ohms so multiply by 50) is a load of
.2*50=10+j0Ω (no reactance) or 5x50=250+j0Ω but also a infinite number of other combinations
such as 25+j50Ω or 100-j100Ω. Very often a 5:1 VSWR rated tuner will match 10Ω and 250Ω but
not 5:1 with substantial reactance like 100-j100Ω.
The power rating is another problem. The most likely limiting factor is loss in the coil and
temperature limitations. Therefore, the power rating, if thermal, would be different if there were
vent holes which there most often are not. MFJ instructions often talk about arcing implying that
the limitation is voltage rating of capacitor or inductor. They show length of coax and antennas
that are not good combinations that will cause arcing problems saying very high impedances are
not desirable. This would imply switch voltage limitations. In that case, the tuner could handle
more power at low impedance 5:1 VSWR than high impedance 5:1 VSWR. AT low impedances,
heating is more likely to be a problem with high currents. Either way, it‟s likely that the tuner is
not so much VSWR limited as impedance limited at either the high or low end or both. Ergo, the
best VSWR spec is probably none at all!
The MFJ 941E Versa Tuner II manual states, “Trying to load an 80 meter (or higher frequency)
antenna on 160 meters can be a disaster for both your signal and the tuner.” This happens to
be the condition that will mostly likely develop into a high voltage or high current, hi-Q situation
depending on the coax length. I‟ve seen one with the switch melted after trying just such an
operation.
Military gear using short or improvised antennas will often have integrated tuners such that power
is limited according to the condition of impedance and frequency and not a VSWR limit.
Page -31-
Fig 19
5:1 and 3:1 VSWR Circles
Tuner Ratings
Page -32-
China Syndrome
Taking a look at a Pi-Network and realizing that all components are variable without restriction
unless computer controlled, see the current path marked in red.
1
2
XL
ITX
Rgen
Xc1
Xc2
Rload
Rload < Zo = Rgen
Fig 20
Self Destruct Current Path
The problem is if Rload is quite high for any of the networks, a series short to ground can
develop. Suppose that one was tuning a 40M dipole but on 80M about 3.75Mhz. Then if the
radiation resistance at the other end went very low but you had an odd multiple of quarter
wavelengths of coax, then the input impedance might soar to say 2000Ω. Under these
conditions, if you tuned:
C1
L
C2
does not matter
3.8µH (value used nominally on 80M for 3:1 VSWR)
470pF (just short of the value used with 150Ω, 3:1 VSWR load)
then you would get the following input impedance with a Pi-network (or same with T network
where C2 doesn‟t matter, C1 = 470pF but inductor the same):
Page -33-
Abs(Zin)
100.00
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
90.00
80.00
Ohms (absolute)
70.00
60.00
50.00
40.00
30.00
20.00
10.00
0.00
2.000
2.500
3.000
3.500
4.000
4.500
5.000
Mhz
Fig 21
Short to Ground
Impedance not VSWR
You get essentially a short to ground with output currents probably more than 10 times normal if
you keyed at full power. The VSWR is at about 9:1 so you wouldn‟t stay there for long BUT in the
meantime, ferrite core inductors would possibly saturate on peaks making the impedance
perhaps even lower with a T network so hat you could not tell which way to adjust as all directions
of the caps seem to make it worse. In the T-network, you don‟t even need to have the antenna
be in such a strange state. You can be tuning an 80M antenna on 80M! Just open up C2 to
minimum capacitance and you have just sent the output impedance sky rocketing as in the same
condition above. Then you can adjust C1 and L to ground (T network) to a very good short. If it
doesn‟t work on 80M, it sure will on 40M. You need to go all the way to 10M before something
similar will happen with the Pi-network.
A high impedance, high voltage China Syndrome is possible with the T network. If the output
impedance is very high, maybe nothing connected, then it‟s possible to adjust C1 to a very small
value and actually match up to the impedance of the coil on higher frequencies (toroid coils will
have Qs of like 50 to 100 and not the several 100s of air cores) so that all the transmitter power is
used to heat the matching network itself at very high voltages while the VSWR is 1:1! . I have
seen this happen such that the inside of the T network box was turned into a plasma torch. It‟s
usually not possible to do that with typical parts and transmitter output impedances using a Pinetwork.
Page -34-
Component Ratings
The component values will depend on:




Frequency (band) range
VSWR matching rating
Operating Q range
Topology
The component current and voltage ratings will depend on:



Power handling rating
VSWR matching rating
Resistance to misadjustment
Quite often there is nothing to prevent misadjustment.
Component sizes must increase for lower frequencies and either higher Q (Pi-Net, capacitors) or
lower Q (T-net, capacitors)…opposite for inductors. There must be a large value range
adjustment if simultaneously the upper bands are useable or to handle a large VSWR range.
Also, f you do not restrict operation to certain parts of the VSWR circle, the required adjustment
range will be larger.
The L network will require the least component range while auto-tune types which can rapidly
switch parts in and out of the circuit can have an extremely large range of component values
without difficulty regardless.
At 100 watts, the capacitors, switches and other parts need to be rated at 170+ volts to ground for
3:1 VSWR and preferably plate to plate even if in series. For 5:1 VSWR, they need to be rated at
225+ volts if the input/output Q is the minimum in equation 16! (not likely for Pi or T nets)
Caps, coils and switches need to be rated at 2.45+ Amperes for 3:1 VSWR and 3.16+ Amperes if
the operating Q is minimum (not likely). If the operating Q is allowed to go to 10, it should
be rated for possibly as high as 32 Amperes! If the network is also permitted to operate at
VSWR 5:1 at 300 watts, then the current rating sails to 55 Amperes! I don‟t think the switch
contacts are rated at 50 amperes for 28Mhz in most tuners so I don‟t think that most of them can
take the rated power at VSWR 5:1 key down. In ICAS (Intermittent Commercial and Amateur
Service), you can de-rate the current only (not voltage) to 50% of the maximum. Also, that
current would occur only if the parts have infinite Q, not likely. Therefore look for 300Watt
currents of 30A or less with ICAS=15A and 100Watt currents of 17A or less with ICAS=8.5A.
This is about where the components are rated, I believe. Note that Qs of 10 could increase
voltages to over 2000 if that were possible. The capacitors arc before that at from 600 to 1200
volts depending on size. The broadcast type 365pF are rated at 250 volts and would arc at from
500 to 600 volts.
The voltage and current rating will always be least for the L net. Use the top most estimate. That
is, 225 volts for 5:1VSWR for 100W and 390 volts for 300W. The broadcast type variable ganged
cap 365pF type will handle up to 300W at 3:1 VSWR. They should only be used to about 100W
in Pi and T net tuners.
Note the voltage rating is for peak power, not ICAS.
Page -35-
L Network Values & Design
Using the Pi-net spread sheet but with very low output Q (.001), we get an L network. Running
over the range of VSWRs for high and low Z and frequencies (loads are resistive only):
Frequency
Low Z 5:1 VSWR @ 3.5Mhz
Hi Z 5:1 VSWR @ 3.5Mhz
Low Z 3:1 VSWR @ 3.5Mhz
Hi Z 3:1 VSWR @ 3.5Mhz
Low Z 5:1 VSWR @ 29Mhz
Hi Z 5:1 VSWR @ 29Mhz
Low Z 1.5:1 VSWR @3.5Mhz
Hi Z 1.5:1 VSWR @ 3.5Mhz
Low Z 1.5:1 VSWR @ 29Mhz
Hi Z 1.5:1 VSWR @ 29Mhz
C1
1819pF
L
0.91µH
4.55µH
1.07µH
3.22µH
0.11µH
0.55µH
1.08µH
1.61µH
0.13µH
0.19µH
1286pF
220pF
653pF
79pF
C2
368pF
428pF
43.9pF
429pF
52pF
So you need a capacitor that will range from about 40pF to 1819pF for 5:1 80M/10M or 1286pF
for 3:1 VSWR and an inductor that runs from 0.1µH to 4.6µH and it must be roller type
continuously variable (or very close taps and no guaranteed VSWR null). Note that extreme high
capacitance is required for low frequency plus low impedance operation. If the impedance is
higher, even at 5:1 VSWR, you need far less capacitance. This is the reason the manuals say,
“try adding a length of coax” and that MFJ advises against certain lengths of coax in the tuner
manuals. Typically, this large capacitance is just not provided…except in the auto-tuner types
where it is easier. The problem above is that if the capacitor is made larger for 1800pF, then it
won‟t adjust down to 40pF and probably it would need to be either huge or close spaced in which
case the voltage rating won‟t be good enough. You can also see that if it can handle the higher
VSWRs, it can also work with lower such as 1.5:1. The inductor needs to roam over about a 7:1
turns ratio.
FYI, on 160M, for 3:1 Hi Z VSWR, you need 790pF and 5.9uH.
XL
Rgen
Rload
Xc
Fig 22
L network Design
Page -36-
Fig 23
Homebrew L Matching Network
Fig 16 is a simple 100W (300W moderate VSWR) homebrew design. The capacitor is a triple
ganged 3x365pf=1095pF type and the inductor is a 12 µH, 46mm diameter roller type on steatite
of which only 60% would ever be used. The beginning of the coil is to the right where the pitch is
halved from 0.5 on the left to 0.33 for higher Q at higher frequencies. The right hand, wide
spaced part is .75µH. An insulating coupling is required because the axel is used to contact one
end of the coil. The tap is a spring loaded roller. This unit is crude not having the capacitor
switch and the cap is permanently wired on one side meaning one needs to reverse coaxes to go
from high to low Z. This network will adjust to 2.5:1 low Z VSWR and all other combinations in
the table. It‟s also useful for hi-Z 160 meter antennas to 5:1 VSWR that need 670pF and 8.4uH.
However, most likely the 160M antenna will be low Z unless one has Nx80 feet of coax where
N=1,3,5… The coil calculates to Q of 250 in the typical range at 3.75Mhz and 400 to 500 at
29Mhz using the coarse end. That dissipates about 100mW to 700mW from a 100W transmitter
at 5:1 VSWR for a 1° to 2°C rise. Probably the contacts on the coil and cap dissipate as much as
the coil. The limiting factor is the capacitor voltage rating which makes this a 100W design, 300W
at less than 3:1 VSWR or low Z.
Page -37-
Pi-Network Values & Design
For the same conditions as the L-net, the values are shown in the table below for output Q=1 and
again for output Q=3.
„Output‟ Q = 1
Frequency
Low Z 5:1 VSWR @ 3.5Mhz
Hi Z 5:1 VSWR @ 3.5Mhz
Low Z 3:1 VSWR @ 3.5Mhz
Hi Z 3:1 VSWR @ 3.5Mhz
Low Z 5:1 VSWR @ 29Mhz
Hi Z 5:1 VSWR @ 29Mhz
Low Z 1.5:1 VSWR @3.5Mhz
Hi Z 1.5:1 VSWR @ 3.5Mhz
Low Z 1.5:1 VSWR @ 29Mhz
Hi Z 1.5:1 VSWR @ 29Mhz
C1
2728pF
910pF
2031pF
910pF
329pF
110pF
1296pF
910pF
156pF
110pF
L
0.91µH
4.55µH
1.23µH
3.68µH
0.11µH
0.55µH
1.82µH
2.75µH
0.22µH
0.33µH
C2
4547pF
546pF
2723pF
678pF
548pF
66pF
1378pF
857pF
166pF
104pF
„Output‟ Q = 3
Frequency
Low Z 5:1 VSWR @ 3.5Mhz
Hi Z 5:1 VSWR @ 3.5Mhz
Low Z 3:1 VSWR @ 3.5Mhz
Hi Z 3:1 VSWR @ 3.5Mhz
Low Z 5:1 VSWR @ 29Mhz
Hi Z 5:1 VSWR @ 29Mhz
Low Z 1.5:1 VSWR @3.5Mhz
Hi Z 1.5:1 VSWR @ 3.5Mhz
Low Z 1.5:1 VSWR @ 29Mhz
Hi Z 1.5:1 VSWR @ 29Mhz
C1
6366pF
2728pF
4893pF
2728pF
768pF
329pF
3421pF
2728pF
413pF
329pF
L
0.46µH
2.27µH
0.64µH
1.91µH
0.06µH23
0.27µH
1.02µH
1.53µH
0.12µH
0.19µH
C2
13641pf
1273pF
8169pF
1634pF
1646pF
154pF
4134pF
2269pF
499pF
274pF
Unfortunately, the tables above are filled with unworkably large capacitor values and two
impossibly small inductance values presuming that a variable capacitor will be used and a single,
large diameter or single permeability ferrite core inductor. Even if the output Q ~0 (becomes L
net), one needs 1800pF at 5:1 VSWR low Z 3.5Mhz. Even an auto-tuner would have a difficult
time with a 13641pF capacitor, essentially an RF bypass capacitor (a .01µF). Essentially, the
higher Q setting Pi-network is unworkable on 80 meters. Even at output Q=1 (which is actually
the 50Ω side for a hi-Z load), a low-Z, high VSWR load t 3.5Mhz doesn‟t work with variable caps.
You need to add a section of coax cable to transform upwards.
T-Network Values & Design
In fact while the Pi-network is always chosen for tube or high impedance output matching to 50Ω
for the reasons discussed above, a 50Ω to antenna matching network is almost always a
T-network if not L network. The majority of commercial designs are T-network for the reason that
variable capacitors are of a value to be physically smaller and more economical or even
realizable. Typically only the capacitors are continuously adjustable with the inductor tapped and
switched. A design tradeoff is that losses are higher as the inductor has more wire or is even a
ferrite type. It has the same ambiguity and potential for China Syndrome as the Pi-network.
However, that has been mediated in one MFJ design by employing a differential capacitor and
23
Too small to be practical
Page -38-
roller inductor. As the input capacitance grows larger, the output grows smaller. With only two
controls, there is no adjustment ambiguity.
For the similar conditions as the L-net, the values are shown in the table. An output Q of 1 is
impractical as a design criterion and in fact is the wrong way for best component values for low
cost (another reason losses are higher). Instead, the calculation is for a maximum variable
capacitance of 250pF24. This is more practical than 1000pF limit in the L-net design because the
caps will not always be on the low voltage side and they must be isolated from the control knob
as both sides are hot. It‟s also the size cap used in the low cost MFJ tuners. The operating Q is
reported and the loss with Qul = 200 if operating Q is high.
Capacitor Values Limited to 250pF maximum
Frequency
C1
Low Z 5:1 VSWR @ 3.5Mhz
112pF
Hi Z 5:1 VSWR @ 3.5Mhz
250pF
Low Z 3:1 VSWR @ 3.5Mhz
146pF
Hi Z 3:1 VSWR @ 3.5Mhz
250pF
Low Z 5:1 VSWR @ 29Mhz
249pF
Hi Z 5:1 VSWR @ 29Mhz
55pF
Low Z 1.5:1 VSWR @3.5Mhz
205pF
Hi Z 1.5:1 VSWR @ 3.5Mhz
250pF
Low Z 1.5:1 VSWR @ 29Mhz
125pF
Hi Z 1.5:1 VSWR @ 29Mhz
110pF
L
5.75µH
6.49µH
5.30µH
5.80µH
0.12µH25
0.66µH
4.72µH
4.90µH
0.22µH
0.35µH
C2
250pF
133pF
250pF
157pF
246pF
245pF
250pF
210pF
128pF
127pF
Operating Q
26 (-0.6dB)
5.0
17 (-0.4dB)
3.6
2.7
2.1
9.9
6.5
2.2
1.6
It should be noted that in the case in the first row, the bandwidth will be quite narrow and the
network will lose 13 watts (!) per 100 watts probably mostly in the coil even if the unloaded Q is
20026. This will overheat a small toroid if there is no forced air cooling27.
Note the dramatic increase in operating Q and losses for low impedance, hi VSWR operation at
lower frequencies of 5:1 compared even to 3:1. Losses are much better for 17Ω compared to
10Ω load28. If operation is extended to 160 meters, losses could become severe (if 10 to 20 watts
per 100 watts is not already bad enough at 80M). Operating manuals for MFJ T-net match boxes
say that such a situation should be avoided advising one to add a length of coax29.
Below is the operating VSWR bandwidth with 5:1 low Z VSWR at 3.75Mhz and with a design
criteria of 250pF maximum.
24
In the spreadsheet, increase output Q until you get the right size capacitors
Consider C1 48pF, L 0.22µH (same as 1.5 VSWR low Z), C2 100pF, Q operating 7.8 for a more
reasonable size inductor and one less tap. It may be best to accept higher Q operating on higher
bands so that the caps do not approach resonance regardless.
26
More likely it will be ~100 as a ferrite toroid and the loss will be 26 watts at 5:1 VSWR low Z!
27
A reason the Yaesu FC-30 integral (and FT-1000 internal) 100 watt tuners for VSWR 3.5:1 max
low and high Z are forced air cooled.
28
An indication of the difficulty of designing transistor PA matching networks since they have
output impedances of just a few ohms.
29
Which will probably lose the power regardless but spread out over an area and undetectable
from the operating position without some tricks.
25
Page -39-
VSWR
3.00
Right click each axis
and set range if
necessary. Click
'Scale' tab. Set
check box to see
complete graph and
zoom after
2.80
2.60
2.40
2.20
2.00
1.80
1.60
1.40
1.20
1.00
3.500
3.550
3.600
3.650
3.700
3.750
3.800
3.850
3.900
3.950
4.000
Mhz
Fig 24
T-Network Operating Bandwidth 5:1 VSWR Low Z
See text
Some Commercial Designs
There are no auto or manual Pi-network designs listed for the reasons discussed in the section
Pi-Network Values & Design. There do not seem to be any designs using ferrite transformers to
reduce the required range of reactances. Ferrite appears as lumped inductance and baluns only.
L Networks
MFJ 16010 Random Wire 200W 160M – 10M
Page -40-
This is a very basic L network much like my homebrew version where one must switch coax
connectors to go from high to low Z. It‟s intended to be a portable, Field Day unit. A compromise
for cost and size is that the inductor is switched in 12 increments rather than being a roller
inductor, for which there is neither room nor money.
This antenna tuner has no specifications other than 200W after matching adjustment and 160
meters through 10 meters usage. There is no VSWR or impedance range specification but then it
is just $69.95.
Ten Tec 238C 3KW 160M – 10M
This is a top of the line manual, modified, L network. Designs for high power tend to be L
networks. Modified means that you can switch in assistant inductors to deal with antenna
extremes like high reactance and low resistance, a situation encountered with short (wavelength)
whip and hair-pin antennas.
From the front panel witch it is obvious that you are controlling whether the capacitor is shunt at
the input (low Z antenna) or output (hi Z antenna). The inductor is a full roller type. An antenna
switch and full VSWR metering is built in. Like most others (except economy types), it includes a
balun and connectors for twin lines.
Page -41-
Basic Switched Configurations
This design solves the giant cap problem noted in the analysis sections above by switching in
additional fixed capacitors via the front panel although the switch is just labeled low Z 1..5 and hi
Z 1..5. A short hair-pin antenna also requires large shunt caps. It even covers the case where a
low Z antenna might have a lot of series equivalent capacitive reactance (short whip) by switching
in a series inductor. In this way, they minimize operating Q while still matching extreme
impedances.
Page -42-
This design is fully specified.








Circuit Type: L network
RF Power Rating: 2000 watts
Frequency Range: 1.8-30 MHz continuous
Input Impedance: 50 ohms nominal
Output Matching Range: At least 10:1 SWR, any phase angle.
Input/Output Connectors: Input and four antenna coax connectors are SO-239, UHF type.
Studs with wing nuts for single wire and balanced feeders.
Capacitor Voltage Rating: 3500 volts, 2400pF max with switched caps (1200 watts at
10:1 VSWR hi-Z, TER)
Inductor: 0.2-18 µH silver-plated roller inductor.
The VSWR spec marked in bold is unusual especially at this power level (1500 watts avg, 3KW
peak) since there are no caveats about high VSWR at low Z which from the L network design
analysis above, is shown to present a problem. For 160M use, Ten Tec includes a 1kV, 1000pF
which can be shunted n the antenna terminals in back to achieve 3400pF! If you scaled the
values in the table in L Network Values & Design , you can see that 3400pF would be just about
enough for low Z 5:1 on 160M or 10Ω. Also the voltage rating is adequate only for 400W at 5:1
VSWR if Hi-Z. On other bands, it will go down to 5Ω which is exceptional. I have used its
predecessor on many occasions and find the design capable of its specifications in general. At
$795, you get what you pay for.
The balun can be seen in the interior photograph but I did not include that part of the schematic.
Page -43-
Manual T Networks
MFJ 902 150W 80M – 10M
This is a basic T-network tuner also billed as a Travel Tuner or Field Day unit. Having three
controls, there is no need to switch coax connectors for high and low Z antennas but it introduces
the tuning ambiguity as mentioned in the analysis section above. Knowing that, the manual has a
prescribed tuning procedure. Unfortunately, I think it does not attain the lowest operating Q
position and in fact will lead to some confusion because you start with the antenna capacitor at
one limit (that might be especially bad on 10 meters). Elsewhere in the small manual, there is a
note that you should, “Be sure to use the highest possible capacitance for each band. This will
provide the smoothest tuning, highest efficiency, and greatest power handling capability.” They
also tell you max capacitance is at setting 0 and maximum inductance at switch position “A.”
Page -44-
The schematic is misleading in that it implies a roller or continuously variable inductor. The
picture above and adjustment procedure make it clear there are instead 12 switched inductor
values.
There are no specifications per se. Items mentioned in the manual text or schematic include:




150W maximum after tuning
600 volt capacitors (corresponds to 144W peak at 5:1 VSWR hi Z, TER)
80M – 10M (warning not to operate n 160M in the manual)
anywhere with any transceiver – use any coax fed or random wire antenna.
The last specification is unlikely to be true at least at 150W. Refer to the table in section TNetwork Values & Design. Very likely, given the size of the unit and toroid inductors, a low Z,
high VSWR antenna (5:1 or worse) is likely to cause excessive coil or switch contact heating or
worse saturate the inductor ferrite and cause harmonics along with catastrophic Currie Point
collapse. That is the temperature at which high permeability cores suddenly lose their
permeability and the inductor becomes a simple piece of wire. At that point, the transmitter would
suddenly be subject to high VSWR even though the antenna tuner was adjusted correctly 30. High
currents could also the peak power VSWR phenomenon when the core saturates on RF peaks
but not otherwise.
Most reviews I could find on line did not mention high temperature but rather failing switches and
capacitors for mechanical reasons. About half use it QRP so it never encounters thermal
problems. One fellow did said (from eHam reviews):
“I tried initially tuning an 80m dipole fed by a ladderline, and it was fine on 80 to 10m. For
comparison, the automatic tuner in my K2 had problems on 17m. On 40m, wires (but not
30
Remember that one can adjust for very high Q, coil currents and flux on some bands because
of the tuning ambiguity
Page -45-
cores) were getting hot after sending 100W for 20s; on SSB heating was minimal. It
seemed though that most of the power was getting through. There was also some
heating on 80m.”
Of course an 80M center fed dipole will be a very high, high VSWR antenna n 40M but the line
length can rotate that to low Z without much changing the VSWR (improve it if anything from the
losses). However, it‟s just $99.95 and so should be considered a fair deal best used with QRP or
moderate VSWR antennas, preferably hi Z if not 50Ω, at 100W.
MFJ 941 300W Versa Tuner 160M – 10M
This is MFJ‟s entry level base station tuner. As such it has a back lighted VSWR meter and
antenna switch.
It also has the tuning ambiguity and their traditional 12 value switched inductor. If you are used to
the Travel Tuner, you will find that the capacitors are backwards being maximum capacity at
position 10 instead of 0. The inductor switch works the same way. Although it‟s the same
T-network design (different maximum capacity values compared to Travel Tuner), the tuning
procedure is different starting at both caps mid range and adjust inductor first. This would be less
confusing, I believe. The inductor is air core and so does not have the possibility of Currie Mode
Failure as does the Travel Tuner.
Page -46-
Again, there are no specifications as such. At various points in the manual it states or implies:






160M to 10M
300 watts (once adjusted)
1000 volt capacitors (400W at 5:1 VSWR hi-Z, TER)
Air core inductor (likely Qul 300 to 500, TER)
Cross Needle VSWR Meter
4:1 Balun
This mannual also contains the sage advice mentioned before about line lengths and antenna
lengths to avoid. It also explicitly includes a China Syndrome warning targeted at 160 meter
operation and short antennas. At $139.95, it‟s a good value product. On line reviews mentioned
that meter fonts are just 6 point (magnifying glass size). There were no comments on heating but
several seem to have used the tuning ambiguity to adjust into high Q, narrow and difficult to
adjust situations where there should be none.
Page -47-
MFJ 986 3KW 160M – 10M
The 986 is a unique T-net design that eliminates tuning ambiguity in a design for higher power. It
has a differential capacitor, a combined input and output capacitor on one knob, such hat ne
increases whilst the other decreases. That also requires the continuously adjustable roller
inductor. You also get a larger VSWR meter and antenna switch.
Once again, there are no specifications per se. At various points in the manual it states or
implies:
Page -48-





160M to 10M
1.5KW (once adjusted)
Air core roller inductor (likely Qul 300 to 500, TER)
Cross Needle VSWR Meter
Current Balun
A note on the latter specification from me. A current source is an extremely high impedance,
current limited device which is difficult to construct as a balanced ferrite transformer. Most of
them are nothing of the sort but I have no knowledge of the quality in this unit. An imbalance
leads to radiation from the twin lead and a change in effective antenna impedance. The
advertising says that it is exactly the other way around such that (it says), “More tuners use a
voltage balun that forces equal voltages on the two antenna halves. It minimizes unbalanced
currents only if the antenna is perfectly balanced -- not the case with practical antennas. The
MFJ-986 uses a true current balun to force equal currents in the two antenna halves -- even if the
antenna is not perfectly balanced! That would certainly be the case if it were a “current balun.”31
There is no specification for VSWR or impedance range. The same recommendations as to
antenna wire and transmission line length are in the manual.
The manual has a good method for adjustment but then includes the warning, “When adjusting
the tuner, use the lowest amount of CAPACITANCE and INDUCTANCE that produces a good
SWR. This will reduce tuner losses and increase the power rating of the tuner.” It is impossible to
conform to the warning since there is only one position with minimum VSWR. That was the
whole point of the differential capacitor design! There should be (and I hear are) places where
the capacitor control is rather broad in which case it says to get it to the lowest number as
possible adjusting also the inductor. I am not sure that comment is valid. Note that the
advertising says, “You get minimum SWR at only one setting a broadband response that ends
constant re-tuning.” At $349.95, it is a good value. There are no negative comments in on line
reviews that I could find.
Auto Tune Networks
There are two common types of auto tuners mechanically. There are the older motor driven type
and the newer relay driven. Relay driven can be surprisingly fast making adjustments every 50
msec or less. The design is intrinsically distributed so it can have more form factors. Solid
dielectric high voltage capacitors are much smaller even collectively than an air variable.
Together with the form factor advantage, it can allow a huge range in capacitance and thus
bandwidth and VSWR range. Typically there is a collection of inductors instead of a single
tapped inductor. For toroids, this is an advantage since automatic winding does not handle toroid
taps well. 8 inductors can be made to have the same effect as a single coil with 256 taps! 8
capacitors can similarly produce 256 values. At least the capacitors must be arranged this way.
Motor driven types take no or little power once tuning finishes. Typically relay type auto tuners
must disconnect in receive if on battery power meaning receive sensitivity is degraded which may
be of no effect on the lower bands. It is possible to install (more expensive) latching relays and
use no power once tuned in relay types. As of yet, there are no FET switched HF “relay” tuners,
even at 100 watts, as there is on VHF that I know of. Silicon consumption would be impressive if
there were.
Switched transmission line types seen at VHF/UHF are not present on HF, even with synthesized
or “lumped” lines that I know of.
There are no HF auto tuners that make use of tapped RF transformers that I know of.
31
The current baluns that I have purchased have the property such that if you touch one terminal
you hear band noise but if you touch the other, you hear essentially nothing. This is not good.
Page -49-
There are no HF tuners that make use of circulators that I know of 32.
Restricted range auto tuners are common in solid state rigs whose RF PA is not adjustable like
the older tube RF PA. Solid state PAs begin to degrade performance significantly at only 1.2:1
VSWR unless they have a large design margin and are then either inefficient or have an on
demand modulated PA power supply.
Yaesu FC-30 Integrated T Net 100W 160M – 6M
This auto-tuner is dedicated for use with Yaesu transceivers covering a wide frequency but
narrow VSWR range.
The FC-30 is the attached and compatibly styled box on the left.
32
It would only be useful to absorb reflected power regardless
Page -50-
In the drawing below, you can see the cooling fan used on transmit.
The circuit board(s) contains a micro-processor which gets its power and commands from the
transceiver via a “CAT” or computer transceiver cable. As automation, there is no user ambiguity
despite being a T network or any danger switches will be thrown at full power. It‟s specified to
work at 3:1 VSWR at any angle (high or low Z or combination of reactance and resistance)
although I measure it to adjust up to 3.3:1. Being strictly VSWR limited, part of the circuit
operating range is thrown away33. It‟s intended use is to flatten the VSWR of otherwise well
tuned antennas so that the RF PA maintains full specifications and output power over the entire
band. Thus an 80 meter dipole, tuned to 3.75Mhz, would work end to end (3.5..4.0 Mhz) using
the tuner. Without it, end frequencies would be about 3:1 and about half the band would not
permit full power and/or intermod and efficiency specs. Likely the power supply would trip at one
end of the band. It‟s also useful for flattening the response of a 20 meter Yagi beam, especially
the compact models or matching RF amp input impedances which can sometimes be 2:1 VSWR.
33
Of necessity, the same circuit would match a wider range of mostly resistive loads.
Page -51-
The above coil unit schematic shows that the shunt coil is not designed as a binary combination.
That would require each coil to be half the size of the next where up to seven at a time would be
shorted out by relay contacts. Instead, there are effectively 9 coils and 8 inductance values34. It‟s
effectively a tapped coil the same as in the MFJ design. That wastes some of the resolution but it
was not necessary regardless as there is one extra degree of freedom in a T-network. It also
distributes the energy in the ferrite cores more evenly (possibly…I didn‟t calculate). It‟s not
known what the maximum inductance might be but it‟s likely to be about 12µH.
The capacitor switching board/unit is shown below. It is two of binary combination capacitors with
each capacitor in the string for each adjustable cap (C1 and C2 in my analysis) being half that of
the one next to it for a total of 256 steps. Note that each cap is made up of two (hopefully
matched) units for a rating of 1000 volts35. When the relays are all closed, all caps are connected
in parallel. The least step is 2.5pF and the maximum is about 2.5 x 256 = 640pF but is probably
closer to 680pF. Consulting the table of values in section T-Network Values & Design, this would
be enough for 3:1 VSWR on 160M to 6M.
34
The shunt coil may not be completely switched out as this is a coax line short to ground! …the
same thing as a pin in the coax.
35
If that rating means for RF as well as DC. The two are often not the same.
Page -52-
The relays at the right take the T-net out of the circuit when power is removed in receive unless a
transceiver option is set or the transceiver detects it is on an AC supply.
Specifications listed in the installation instruction manual are below.










FREQUENCY RANGE: 1.8 ~ 30 MHz, 50 ~ 54 MHz
INPUT IMPEDANCE: 50Ω
MAXIMUM POWER: 100 Watts
MATCHED SWR: 1.5:1 or less (simply the transceiver fold back spec limit. In most
cases the match is better than 1.1:1)
TUNE-UP POWER: 4 W ~ 60 W
TUNE-UP TIME: 5 seconds or less
IMPEDANCE MATCHING RANGE: 1.8 ~ 30 MHz, 50 ~ 54 MHz: 16.5Ω ~ 150Ω
(16.5Ω and 150Ω are both 3:1 VSWR. The spec is really 3:1 VSWR as sold and
not a resistance: TER)
IMPEDANCE MATCHING MEMORIES: 100 channels
INPUT VOLTAGE REQUIREMENT: 13.8V ±15% (supplied from transceiver)
OPERATING TEMPERATURE RANGE: 14° F ~ 122° F (−10 °C ~ +50 °C)
LC value ranges are adequate to match much higher impedances than 150Ω on most of the
bands but the software does not permit it.
The unit also has 100 internal set up memories so that when it is given a frequency by the
transceiver, it‟s able to either switch to the nearest tuned and tested value or interpolate. One
can hear relays clicking if the transceiver frequency dial is turned far enough. If the setting is not
good enough on a particular frequency, RF power will fold back. The user can then press an
auto-tune button on the transceiver which causes the FC-30 to run it‟s adjustment algorithm and
memorize the setting on the current frequency.
Page -53-
At $199.99, the FC-30 is a fair but not exceptional deal for owners of the FT-897. The same
design is an integral part of the FT-2000 with the same limitations. The limitations appear to be a
consequence of adapting an older design to micro-processor control rather than taking advantage
of new possibilities offered by it.
LDG AT-897 Integrated L Net 100W 160M – 6M
The FC-30 above, has a number of unexpected limitations addressed by both Yaesu itself in the
FC-40 external auto tuner36 and LDG‟s AT-897 shown attached below. Both are L networks.
The AT-897 is intended to be a drop in replacement for the FC-30 integral to the Yaesu FT-897
transceiver and as such also has microprocessor control. However, it‟s a switched L network
which greatly reduces the number of parts.
LDG does not provide a schematic, just a picture of the PCB.
There are seven toroidal inductors and either capacitors all in the binary half step set of values for
128 inductance and 256 capacitance values. The total value range is unknown but should be
36
Intended to be mounted at the antenna although this is practical mostly in mobile installations.
Yaesu sells a matching mobile whip and the FC-40 and whip together are a replacement for their
mechanically faulty ATAS-120 auto-tune screwdriver.
Page -54-
comparable to that of the FC-30. The inductor is always series and the capacitors shunt to
ground to form the low pass configuration of an L network. The capacitor branch is switched from
the input to output as required as shown in L Network Values & Design. It does not have a
cooling fan and the claim is that it‟s not needed. Since the L network always arrives at the
minimum possible Q and losses with a given set of component Qs (see L Network), it‟s logical
that the losses and heat would be lower compared to Yaesu FC-30 Integrated T Net 100W 160M
– 6M. Note it has fewer circuit boards (one). However, the capacitors shown installed in the
board do not look like they are capable of 100W, 10:1 VSWR operation even at minimum
possible Qoperating. They do not even look large enough to be temperature stable. VSWR drift and
other reliability problems have been reported in on line reviews for this design.
In the past, I have had trouble isolating microprocessor noise from RF circuitry on a single board
if the grounds are not split (isolated) or at least precautions like component placement and large
numbers of ground coupling through holes are used. I don‟t see any of that in the design above.
However, the microprocessors could be in standby in normal operation.
This unit provides a CAT port extension which the Yaesu FC-30 does not.
Styling leaves something to be desired. AT least it could be painted black.
At $179.99, it is a flawed implementation that nevertheless provides 10:1 VSWR range for
popular portable transceiver expected to encounter high VSWR. Yaesu‟s FC-40 is $299.99 as is
not an integral design (remote mounted). The schematic for the FC-40 is below and is likely
similar to that of the AT-897.
Schematic of switched L auto-tuner (Yaesu FC-40)
Page -55-
MFJ 998 Intellituner L Net 3KW 160M – 10M
This auto tuner is designed for full gallon high power. It s wide frequency and VSWR capable
with a cross needle combination power and VSWR meter as well as a digital and bar graph digital
VSWR display. There are many features to this high end design.
The auto-tuners mentioned above were dedicated to a particular set of transceiver designs
(Yaesu) while this tuner will work with any down to 5 watts (manual tuning below that or outside
amateur bands). The previous tuners used the transceiver‟s VSWR bridge and user interface
while this unit has its own. However, it can operate under transceiver control as well through the
rear panel interface as is compatible with radios that can use Alinco EDX-2, Icom AH-3, Kenwood
AT-300 or Yaesu FC-30 auto-tuners. It‟s CAT compatible.
The matching network design is a switched L having a range of 256 inductance values to 24µH
and 256 capacitance values to 3900pF. There are only 64 capacitance values if connected to the
output (Hi-Z antenna) and the range is then just to 970pF, entirely adequate. This limitation is
made deliberately to avoid tuning a super high Z antenna manually and applying high power, thus
arcing the capacitors and relays.
It does not handle balanced lines directly but it can switch between two antennas. Below is a
block diagram.
Page -56-
Page -57-
Note that the VSWR range is limited on the low impedance side but it still attains 41 low while
matching 32:1 on the high side. This requires 2200 VDC capacitors at 1500 watts. The relays
are rated at 1000 volts. There is a warning not to operate the tuner with the cover removed since
dangerous voltages (due to RF) could be present. Instead, these voltages are brought out to a
wing nut connection for your convenience37.
Note that it does not cover 6 meters as do the previous, 100W auto-tuners covered.
As an acknowledgement of the microprocessor spurious problem that I mentioned before, MFJ
says that the microprocessors go into idle mode even shutting off internal clocks when idle.
There are many, many other features including protection for the RF power amplifier from high
VSWR events.
There are no bad reports about this tuner in on line reviews that I could find. At $649.99 for
1.5KW and a wide frequency and VSWR range, it‟s a good deal.
37
Any operator running a random wire into the shack or bringing open line close to the ground
and using 1500 watts should probably not have a license.
Page -58-
Appendix A: Complex Numbers
See also Wikipedia® Complex Numbers.
Complex numbers of the form A + jB (or A + Bi) where A and B are both numbers (but not
necessarily integers, e.g. 3.14 or -2.78). Complex means that the number has two parts, a real or
ordinary part that is like what might appear in money accounting (A) and an imaginary part (j B).
The two together are the complex number38.
An imaginary number is the product of a real number (B) and the square root of -1 or
written:
Bi
 1 and is
or j B
where both i and j stand for  1 and B is the real number. i stands for imaginary and is used
by mathematicians and physicists while j is used by engineers. The original use for complex
numbers is to provide for a solution or “root” to any polynomial equation. In the general solution
to the quadratic polynomial, ax2+bx+c=0 where a,b,c are real numbers and x is the variable, the
two roots are given by:
x
 b  b 2  4ac
2a
If the determinant, b2-4ac , is negative, then there is no solution unless we use (b 2-4ac)i where i =
 1 (alternately j =  1 ). This situation happens every time in equations describing circuits
with inductors and capacitors if the Q is greater than 1. Therefore, it is absolutely essential to use
complex numbers to work out network impedances.
It can be shown (see Wikipedia® reference at http://en.wikipedia.org/wiki/Complex_number) that a
complex number may also be represented as a vector at some angle from the origin in complex
number space which is essentially a graph with the real part along the abscissa or X part and the
imaginary part along the Y or ordinate (a and b are themselves real numbers).
38
Bernie Madoff used imaginary numbers in accounting but this is not the same thing even
thought it was also complex.
Page -59-
The length of the blue vector to the point a + jb (or a + bi) is called the magnitude and is
calculated:
r  a2  b2
The angle the blue vector makes with the real (X) axis is also called the phase. Impedances can
be represented by such vectors with inductive and capacitive reactance being the imaginary part
and resistance or resistors or the radiation resistance of an antenna being the real part. The idea
of the matching network is to add reactances in such a way that the length of the vector is scaled
to the system impedance, usually 50Ω, and at the input is aligned with the real axis.
Operations on Complex Numbers
The basic four mathematical operations, add, subtract, multiply and divide are shown below.
A,B,C and D are real numbers. I will use the notation A+jB for complex numbers as I did in the
sections before. Note that if j =
 1 , then:
j2 = -1
Addition
(A + jB) + (C + jD) = (A+C) + j(B+D)
That is, you just add the real and imaginary parts independently.
Subtraction
(A + jB) - (C + jD) = (A-C) + j(B-D)
That is, you just subtract the real and imaginary parts independently.
Page -60-
Multiplication
(A + jB) x (C + jD) = (AC – BD) + j(BC + AD)
You can get to this perhaps surprising result by writing the two complex numbers one over the
other and conducting multiplication the old fashioned way. One of the terms will be j2BD.
Remember that j2 = -1 so that j2BD = -BD.
Division
(A + jB) / (C + jD) = (AC + BD) / (C2 + D2) + j(BC – AD) / (C2 + D2)
You arrive at the above expression by first writing down the division thus:
A  jB
C  jD
And then multiplying numerator and denominator by C – jD which is the same as multiplying by
one:
A  jB C  jD

C  jD C  jD
You can then use the method in the multiplication section to get:
A  jB C  jD  AC  BD   j BC  AD 


C  jD C  jD
C 2  D2
Remember that j2 = -1. The two terms C + jD and C – jD are complex conjugates. The product of
complex conjugates is a real number with no imaginary part, which is the point. If you look at the
rules for multiplication above, the product of complex conjugates, making the appropriate
substitutions, would be:
(C2 + D2) + j(DC – DC) = (C2 + D2) + j0
So if one wants to eliminate the imaginary part in the denominator so that the real and imaginary
parts may be separated, then you multiply by the denominator‟s complex conjugate over itself
(identical to one). This trick was used repeatedly in the circuit analysis above.
Phasor Operations
Although not used in the analysis above, here are phasor operations. That is, if each complex
number is represented TBD
Addition
Subtraction
Multiplication
Division
Page -61-
Appendix B: Network Analysis (brief)
See network analysis in Google Books and as referenced in ARRL Handbook esp. Zobel (image
parameter) and Butterworth and/or Darlington (modern filter design). There are obviously many
texts on analog circuit analysis. This is just the briefest outline of one technique used in this note.
There are several common methods for analog circuit analysis. The most versatile are loop
(alternately branch) and nodal analysis respectively named after Kirchoff and Norton. The first is
based on the idea that if you trace around any loop of branches in a circuit diagram, you
eventually return to the same place at the same voltage so the sum of the voltages around any
loop is zero. You write equations that say exactly that until you have the same number of nonredundant equations as components and solve. The second is based on the idea that charge
does not collect (else the voltage goes to enormous values immediately which then moves
charge). That is, the sum of currents entering and exiting any node is zero. You write equations
saying just that, one for each node and solve39. If there are forty branches or nodes, here are
forty simultaneous equations! It‟s easily possible to write computer code that will solve any
specific network with specific component values at a specific frequency (or time) that will work
well for hundreds or even thousands of nodes but not to get a closed form solution in one life
time40.
However, in certain special cases, it is possible to generate a closed form solution without solving
simultaneous equations. These are the ladder networks shown below.
Node 1
Branch 1
Branch 1
Branch 3
Node 2
Branch 2
Node 3
Branch 4
Node 4
Node 4 is the reference or ground node. Each of the branches could be, basically, a resistor,
inductor or capacitor, RLC. One current or voltage source is required as shown. In the case that
the generator is a transmitter, it‟s modeled as either a current source in parallel with 50Ω (a
Norton equivalent) or as a voltage source in series with 50Ω (a Thevenin equivalent). One way of
analyzing the network is to work from left to right replacing each series and shunt element(s) (the
generator, branch 1 and branch 2) with their Thevenin or Norton equivalents (not described here)
39
The method is the same as for thermodynamics in solids, acoustics and coupled vibrations in
mechanical engineering (actually another type of acoustics). You can use circuit analysis SW for
all.
40
There are computer codes that will solve many, linear simultaneous equations for any variable
in closed form but the resulting ratio of polynomials goes on for pages and is not of any practical
use nor understandable for networks having more than a few nodes. Probably there are easier to
understand simplifications and rearrangements (as with the Q substitution herein) but the
computer code will not find them. …not any I have seen anyway.
Page -62-
gradually eliminating all of the components from left to right until you have calculated the output
voltage and current. For example, if the generator was 10 Volts and branch one a 10Ω resistor,
then shorting node 2 to ground produces the maximum current there, 1 ampere. That means the
10 volt source and resistor is equivalent to a one Ampere current source in parallel with a 10Ω
resistor to ground (node 4). It‟s easily possible to find the parallel equivalent of 10Ω or branch
one with whatever branch two might be, say 20Ω. Then you replace the 10 volt source, branch
one and branch two with a 1 Ampere current source and a parallel (to ground) 6.67Ω resistor.
You can then proceed on to the right replacing branch three, the 6.67Ω equivalent resistance and
1 Ampere generator and so on.
This method works just as well if the generator is an RF voltage or current source of some single
frequency and zero phase (since it‟s the reference phase). If the branches are inductors or
capacitors, their „resistance‟ or rather impedance (the complex number version of resistance) is
calculated:
Z l  0  jL
Zc  0 
1
j
 0
jC
C
The real part of the impedance is ideally41 always zero so the imaginary part has it‟s own name,
reactance and is referred to as XL or XC. ω is the radian frequency so that ω = 2πF where F is
the frequency in Hertz.
However, there is an even simpler method which finds the input impedance of the network
working from the output to the input. You also replace components with their equivalent as you
work but there is no generator to find an equivalent. The end result is the input impedance of the
network which can be converted to a reflection voltage and thence to a VSWR with the formulas:
S11 
Z load  Z o
Z load  Z o
VSWR 
1  S11
1  S11
where:
S11
Zload
Zo
normalized reflection voltage (coefficient)
complex network input impedance, R+jX
impedance (resistance) of the nominal transmission line
If there‟s a S11, there is also S12, S21 and S22, the two port S parameters which indicates that
ladder network analysis is a standard, well developed method for RF circuit analysis. It‟s beyond
the scope of this note to describe what they are other than to say they are like the Norton and
Thevenin methods except that you use a 50Ω generator to measure forward and reflected RF
voltages rather than try to find the current through a short or voltage to an open circuit which you
41
Real components have losses so in fact the real part is not zero at all but some relatively small
value compared to X at frequencies for which the component was designed. This is true for all
types of parts except blocking/bypass caps or RF chokes in which case frequently the real
resistance is larger than the reactance and in act is the limiting factor of its usefulness. Bypass
cap equivalent series resistance is often listed as ESR.
Page -63-
rapidly will discover do not exist even in a practical sense at VHF and above frequencies so you
use a transmitter stand-in, the 50Ω signal generator. One of those parameters is quite familiar,
S11 otherwise known as reflected voltage if Zo is known (it is, 50Ω) and therefore reflected power
through Pr = (S11)2 / Zo. (P=E2/R) The VSWR equation is just the definition of the now ancient
VSWR parameter42. S11 can range from -1 (short or zero ohms) to +1 if Z load is infinite or an open
circuit. -1 just means that all the forward voltage comes back, is reflected, but 180° out of phase.
+1 is 100% reflection but in phase. If Zload = Z out, that s it‟s 50Ω, S11, reflected voltage (power) is
zero, of course(!) and VSWR = 0:1.
If you look at the ladder network above, it is the same as in section T Network where branches
one and three are capacitors and branch two is an inductor. Branch four is the antenna
impedance. The way we proceed from output to input is to take advantage of Ohm‟s law and
Thevenin/Norton equivalents. The basic Ohm‟s law in simple form for lumped elements is:
E = IR
Or
I=E/R
Where
E
I
R
voltage
current in a branch in Amperes
branch resistance in Ohms Ω
It works just as well if R iz complex, that is Z or Z = R + jX.
E = IZ
Or
I=E/Z
If then E or I is permitted to be complex. The second form above could be written:
I=EY
Where
Y=1/Z
Y is known as the admittance
This is useful for the same reason as in the explanation of Norton equivalents above, a current in
parallel with a resistance (impedance) which is also that the currents of two parallel impedances
add at their connecting nodes. That is:
Itotal = I1 + I2 = EY1 + EY2 = E (Y1 + Y2)
Where now Y1 + Y2 is the equivalent admittance of two parallel branches. So therefore we can
just proceed from the output, Z load to the input adding impedances in series, converting that to an
admittance and adding parallel (shunt) branch admittances and converting that to a series
42
The original way of finding out if a load was matched to the transmission line was to measure
the voltage at various points along an open wire transmission line (the lab version is called
Letcher Wires) and take the ratio of the highest to the lowest value which is then, sure enough,
voltage standing wave ratio. The waves are standing not that the voltage is not oscillating but
that the wave appears constant along the line measured as an RF voltage. The RF voltage
values are the wave. The method is ancient in that no one uses Letcher Wires any more instead
measuring S11 directly and calculating what VSWR would have been had we used the wires. The
ratio is itself now useless.
Page -64-
equivalent and adding series branches and so on until we get to the input. Then we find the
reflection coefficient and the VSWR.
In the ladder network above, we first add branch 3 and 4:
Zequivalent = Zbranch 3 + Zbranch 4
Branch 4 is, of course, Zload. Then find the admittance of this combination:
Yequivalent = 1/Zequivalent
You can then add in the admittance of branch three which is Ybranch 3 = 1/Z branch 3 .
Ynew equivalent = Yequivalent + Ybranch 3
Then you convert this into the impedance form with Z new equivalent = 1/Ynew equivalent and then add in
Zbranch 1 (and so on if there were more network). You now have the effective Z load to use in the
reflection or S11 equation. If you use the equations for XL and Xc and not the actual numbers at a
frequency, you end up with the closed form input impedance equation for the network with a
given load and then are able to solve for what up to then has been an unknown, the values of XL
and Xc and then the values of the components in Farads and Henries for a particular matching
situation and frequency.
Page -65-