For the benefit of those not on the original mailing list of Tim's
message, I am posting his contribution:
------------------------------------------------------------------------
Gentlemen:
I'd like to make an input the discussion about the tuning of
pi-networks. I built a spreadsheet to analyze all the element voltages,
currents, and dissipations, and power in, power out, and efficiency of a
traditional pi-network (Z1 = C tune, Z2 = L series, Z3 = C load) with
lossy elements (Zi = Ri + j Xi) which transforms an arbitrary fixed load
impedance (ZL = RL + j XL) into the load for an arbitrary voltage source
(open circuit voltage ES and fixed series impedance ZS = RS + j XS), or
the voltage source equivalent for an arbitrary current source. The
values
are calculated for a range of frequencies above and below the nominal
frequency. The range of frequencies is a parameter in the sheet.
I input the component parameters (Rs, L and Cs), output load impedance,
source characteristics, a nominal frequency, and a frequency step size
(in percent of nominal freq). The sheet uses Excel complex arithmetic
functions to calculate the element impedances as a function of the 25
selected frequencies, then it finds the voltages, currents, and
dissipations of each element, the real power into the pi-network, and
the real power into the load for each case.
I'd like to point out that the power in and power out calculated here
have nothing to do with the efficiency of the source which generates the
RF current. A certain amount of RF power is delivered into the pi
network (to the source's load impedance), and most of it (or all of it
in
the case of lossless elements) goes to the termination resistance ZL.
Whether the source itself is 50% efficient or 1% efficient is not
analyzed or considered here.
For this investigation, I used L & C values such that (for lossless
elements)
a load impedance of 50 + j0 ohms was transformed into 1000.000 + j0.000
ohms at 3.5000 MHz, with a Q1 (input L-section Q) value of 10 and a
total Q
of about 12.01. The source was selected to be a 1.50 amp sinusoidal RF
current source shunted by a 10K Norton equivalent resistance. This
resulted
in about 1800 watts power output, which I recognize is not quite a
legitimate
amateur power, but it makes the point.
John Ehler K5JA used this spreadsheet tool to investigate the power out
of the network as a function of frequency for both lossy and lossless
elements, and discovered that, in both cases, the maximum power output
did
NOT
occur at the frequency (3.500 MHz) at which the input impedance was
purely
resistive. The max power output occurred at about 3.5135 MHz, near
where
the resistive component of the load reached a maximum. As frequency was
increased above this point, the resistive part of the load began to
decrease (which reduced the power into the network), and the reactive
part continued to grow in magnitude reducing the share of the source
current passing through the load impedance (further reducing the power
due to the I-squared term in the input power).
I do not have a good explanation of why power peaks slightly off
"resonance." Are the two L-sections that make up a pi-network acting
like coupled resonant circuits, where higher coupling causes a double
peak in the band pass response??? Or since the Q of the output
L-section is low (Q=2.0), is the resonant frequency of that part of the
network dragged away from that given by one over two pi rad elle see???
K5JA made further investigations of the effect of the source impedance
on
the power supplied to the impedance matching network. For the exact
same
pi network component values, when the source impedance is 1000 + j0
ohms,
the output peaks at the design frequency when the elements are lossless,
and
peaks very slightly (about 3.501 MHz) the design frequency when the
elements
are lossy (same Qs as before). The power output the lossless case
peaks at
2250.0 watts, and in the lossy-element case the max power is about
2178.4
watts. In the lossy element case, when the source resistance is set
equal
to
the resistive part of the load (about 970 ohms) at the design frequency,
the
power output peaks at 2245.5 watts at a frequency just slightly below
the
design frequency. It is interesting that "matching" the source to the
load
improves the source output power to within 4.5 watts of the lossless
element case. I almost forgot: he was using a 3.00 amp RF current
source for these calculations. It makes no difference on the
qualitative
results.
It appears from this that the movement of the max power point away from
design frequency may be an artifact of the mismatch between the source
impedance and its load impedance. Comments???
Another thing that is interesting is the significant change in the
source
load
impedance Zin when the element Qs are more like real-world components.
The 1000-ohm pure resistance changed to 969 - j6 ohms when the following
element values were used.
C1 = 454.728 pf R1 = 0.02 ohms Nominal Q = 5000
L2 = 5.40832 uH R2 = 0.3 ohms Nominal Q = 396
C3 = 1830.246 pf R3 = 0.005 ohms Nominal Q =
4969
The input power is reduced, and the output power is further reduced due
to
inefficiencies in the pi network elements. With these values, the
efficiency
of the network is about 96.8% at the nominal design frequency.
The effects shown here are probably not noticeable to the amateur radio
operator tuning his power amplifier, because he tunes for maximum
output, not a resistive plate load. The possibility that his plate load
impedance is slightly reactive is something about which he neither
knows nor cares.
Results details follow below. I hope this table is readable after
transmission via e-mail.
LOSSLESS ELEMENT CASE LOSSY ELEMENT CASE
FREQ Zin Pin=Pout Zin Pin Pout
Eff NOTES
3.4790 958.13+j127.47 1795.05 931.33+j114.26 1753.4 1698.5
.9687
3.5000 1000.00+j0.000 1859.50 969.10-j6.060 1812.2 1754.8
.9683 Nominal freq
3.51335 976.394-j88.81 1823.3 1765.1
.9681 Max power
3.51351 976.395-j89.92 1823.3 1765.1
.9681 Max Rin
3.51365 1008.3630-j90.29
1872.081 Max power
3.51379 1008.3638-j91.23
1872.080 Max Rin
3.521 1006.01-j139.62 1818.3 974.10-j137.02 1819.6 1761.3
.9680
3.535 988.22-j231.80 1840.7 957.62-j223.48 1793.8 1735.9
.9677
3.570 879.13-j427.01 1668.7 857.82-j408.26 1633.9 1580.2
.9671
I look forward to receiving any thoughtful comments you may have
regarding this analysis.
----Tim Bratton K5RA
k5ra@pulse.net
--------------------------------------------------------------------------------------
Posted by John Ehler K5JA
PS: I have done some more analysis since Tim's original post, which I
will be posting in the near future.
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