Index

R_{i} is R_{u} ?
(published in Electron #10, 2006) Introduction
One of the basic rules in electronics is a
generator to supply maximum power when the internal generator resistance is
equal to the external load resistance. In practice the assumption is easily
made the generator (power amplifier, transmitter) to exhibit an internal
resistance equal to the optimal load of the generator as prescribed by the
manufacturer. When designing a 'power generator' however there is 'more
between heaven and earth' that will determine optimal generator load of which
the internal resistance (output impedance) usually is not the most important
factor. Maximum power transfer Let's startoff with the basic rule about
maximum energy transfer. In figure 1 we find a diagram with a generator (U_{g}),
an internal generator resistance (R_{i}) and a power 'consumer',
symbolized by the external load resistance (R_{b}). We like to know when maximum available
generator power will be transferred to the load.
In figure
i = U_{g}
/ (R_{i} + R_{b}) This will
generate power in the load resistor to the amount of: P_{b}
= i^{2} * R_{b} = U_{g}^{2} * R_{b} /
(R_{i }+ R_{b})^{2} To get an
idea about various load conditions, we will vary the load in relation to the
internal resistance and will obtain the graph of figure 2.
Figure 2: Relative power in the load Figure 2 is showing maximum available power
will be generated in the load resistance when this is equal to the internal
resistance of the generator. Since this is a basic rule that we are being
taught early in our (professional) life, an automatic mechanism is generated
that for every situation where power is delivered to a load, the aim is to
always go for a situation where the external load is equal to the internal 'resistance'. For radioamateurs this 'automatic assumption'
may translate into the idea the output resistance of our transmitter is equal
to 50 Ohm since this is the resistance as prescribed by the manufacturer to
have the transmitter deliver its specified maximum power. In this article we will discuss this
assumption at various 'generator types' where 'internal resistance' usually
is a complex entity and is more often designated as the 'output impedance' of
the system. An example in DC
An example at low frequency
For high fidelity sound reproduction it is
important the loudspeaker to always follow exactly the movements as dictated by
the amplifier. In a first order approach, a loudspeaker system is behaving
like a parallel LC circuit with the mass of the loudspeaker diaphragm and
associated air volume as the capacitor and the voice coil suspension as the
inductor. This parallel LC circuit is exhibiting a
natural resonance frequency, usual somewhere in the lower audio register.
When kicked from the neutral position, the speaker will have a tendency to
keep on swinging for a few periods before all energy has been spend. This
swinging movement is specific to the loudspeaker and usually has no
connection to the music it is supposed to reproduce. Therefore this speaker
resonance should be damped rigorously and the speaker cabinet manufacturer
and amplifier designer are working together to accomplish this. From an electronics point of view we are
familiar with LC circuits, that may be damped by connecting a low value
resistance across. At high fidelity audio amplifiers this is accomplished by
ensuring the output impedance is as low as possible, e.g. by applying
feedback from the output to the input. This will make the amplifier look
like an firm 'voltage source'. When the output is not following the input
accurately, a difference signal is enhancing drive to the output stage to
make this difference as small as possible. This mechanism will make the
output impedance of the amplifier very low and will prevent any free swinging
of the speaker diaphragm. At high quality audio amplifiers the output
damping factor is an important part of specification and is telling how much
lower the amplifier output impedance is relative to the prescribed speaker
impedance. Damping factors of 20 dB and more are no exception, meaning the R_{i}
< (R_{b} / 10). Therefore it is bad practice to connect
loudspeaker cabinets to the amplifier using low diameter conductors, thereby
spoiling the high damping factor that came as a significant part of the
amplifier costs. An example at HF frequencies
In radiocommunication, a transmitter and /
or HF amplifier usually are driving the antenna, that in many systems is to
be regarded as an LC circuit. At low bandwidth applications high antenna Q
may be an advantage because this may enhance total system efficiency and also
provide some harmonic filtering. At high bandwidth systems we may prefer a
lowQ antenna because this may lower feedline loss over a wider frequency
range. Both considerations are pointing to different aspects of the antenna
and will have a different impact to the optimal transmitter design.
Measuring transmitter output impedance
To obtain an impression of the output
impedance of a real life transmitter, I have performed a number of
measurements to a TS440S transceiver by Kenwood. Since this transceiver was
also to be used after this test series, measuring output impedance was to be
regarded with some prudence. Many transceivers are specified to deliver
maximum HF output power when connecting into a 50 Ohm load. The output stage
is designed in such a way that at the specified load the output voltage and
current are below maximum value for the active devices, to ensure some margin
at small deviations from this optimal load resistance. When the transceiver is connected to a
notcharacteristically terminated transmissionline, some of the output voltage will be reflected back to the
transceiver. At SWR =2, the reflected voltage will be 1/3 of the input
voltage and this may add at the output terminals to 1,3 x the voltage in a
correctly terminated system. To keep the active output devices within the safe
operating area (SOAR), the transceiver manufacturer is specifying a maximum
allowable mismatch usually at SWR < 1,5. On top of this the output circuit
is safe guarded against adverse output conditions by means of an output
measuring system to shut down drive. Because of these output load
specifications, some care has to be taken when measuring output impedance and
load conditions therefore should remain as close as possible to the specified
50 Ohm resistance.
Measuring setup
As before we may regard the transceiver as a
generator with an 'internal voltage source', U_{g} with an 'internal
resistance', R_{i}, that is externally loaded with a resistor R_{b},
(the antenna) as in figure 3.
The 'generator', U_{g} and 'internal
resistor', R_{i} are model values that will be not be present in this
way in our receiver. These model parameters though will be reasonably simple
to measure at the transceiver and to allocate meaningful real life system parameters to. Load
resistor R_{b}, in practice will be the antenna radiation resistance
but will be exchanged in the model for a 50 Ohm dummy load, as prescribed by
the manufacturer. Looking back from the load to the
transceiver, we find two unknown factors, U_{g} and R_{i}, so
we have to perform two measurements to get hold of. In figure 4 we may find
the measurement setup.
As a first step, the transmitter is tuned to
a specific output power, that is measured as a voltage V_{1} across
load resistor R_{b1}. The latter is 50 Ohm as prescribed by the
manufacturer and should be a firm and reactancefree dummy load at the
measuring frequency. Voltage V_{1} should be measured with
some accuracy, preferably using a (digital) HF rms voltmeter. When not
available, also a (digital) DC voltmeter will do with a HF diode detector in
front. For this measurement I have been using the 20 dB attenuator output at
the dummyload in front of the detector voltmeter.
In a second step an additional (much bigger)
resistance is switched in parallel to R_{b1}. The value of this
resistor is selected to present a small, but significant voltage change at
the meter, now showing V_{2}. Total load resistor however should not
deviate to much from 50 Ohm. A good value for this resistor appears to be 220
Ohm, nonreactive of course. The parallel circuit of R_{b1} and
220 Ohm is called R_{b2,} is representing a value of 40,75 Ohm and
will generate SWR = 1,23 which is low enough to be fully compliant to the
load specification for the TS440. Next to the dummy load, the 220 Ohm resistor
will also be able to handle some power since this resistor will dissipate
18,5 Watt at a total input power of 100 W. Voltage V_{2} is measured
with the same accurate voltmeter. The 'internal resistance' (output impedance)
of the transmitter may now be calculated to: R_{i} = R_{b1}.R_{b2} (1 – V_{1}/V_{2}) / (R_{b2}.(V_{1}/V_{2}) – R_{b1}) This formula is showing the importance of the
accurate voltage measurements since the accuracy is depending on the ratio of
the two voltage measurements. Measuring results
Based on the above setup, a
number of measurements have been performed at different frequencies and at
two output power settings. Results may be found in figure 5.
To obtain figure 
output impedance is nowhere near 50 Ohm, 
output impedance is not a fixed value, 
a complicated relation exist between output impedance, output power
and frequency. The measurements have been repeated at an
output power of 50 Watt because I suspected the output power at 100 Watt
could be limited by the output safeguarding circuits. If this suspicion was
valid, the measured output impedance would appear to be lower. When looking
at figure 5, we indeed find a lowered output impedance at some frequencies,
but a raised impedance at other frequencies. This is an ambiguous message, so
a second series of measurements have been performed at a single frequency
where the difference between the measurements at 50 Watt and 100 Watt was
small, i.e. at 14,175 MHz. Results may be found in figure 6.
In figure 6 we find output impedance to
diminish at raising output power. At very low output power impedance is going
up to over 180 Ohm. At this impedance level, the transceiver may be even
regarded as a current source. A few remarks of caution
The output impedance measurements have been
performed at a specific transceiver, so results may not be transferred to
other types and manufacturers. Secondly, measurement accuracy is depending
very much on the accuracy of the components and instruments. For high
accuracy, components and instruments should be calibrated at each new
frequency (and power) setting. Thirdly the measurements have been performed
by applying a (small) load difference. Therefore the setpoint of the active
elements in the output stage may have been changed, also changing the
measured output impedance (second order effect though). At the fourth place the measurements are of a
'static' nature. Since the transceiver is equipped with feedback mechanisms
to protect the active elements, a dynamic measurement may yield somewhat
different results. Reviewing HF measurements
The
transmitter is delivering power to a load resistance. This is not a lossless
process and part of total input power to the load will not be transferred to
the load but will be dissipated internally as heat. The ratio of (DC) input
power to (HF) output power is called the efficiency and this efficiency
obviously is less than 1. In our
model for the transmitter we may find a generator and an internal resistance
(the output impedance), that also will dissipate power that is not
transferred to the load. Some similarity may therefore exist between our
model and the efficiency of our transmitter although some caution is needed.
The output impedance of any type of amplifier is to a large extend determined
by overall feedback, with output efficiency hardly be affected by this
mechanism. At the
measurements to the TS440 transceiver we find output impedance at full power
to be around 25 Ohm at output power at 100 Watt when connected into 50 Ohm.
In our model this would mean about 50 Watt of power is lost in the output
impedance, setting the transceiver at an output efficiency of: h = 100 /(100 + 50) = 0.667 (= 66.7
%). This
efficiency is about equal to the efficiency of a class AB amplifier, that
indeed is present in the output stage of the TS
Finally This discussion has been started to
illustrate that at practical power sources and amplifiers, other design
parameters prevail above the transfer of maximum available power as in R_{i}
= R_{b}. Vice versa there hardly is any association between the
prescribe or optimal load resistance to a (power) amplifier and the output
impedance of the latter, because of the same prevailing parameters and design
considerations. At designing power amplifiers, one of the more important aims
usually is to optimize efficiency; power not delivered to the load is
transformed into heat that is to be carried off at high cost and constructive
effort. Output impedance therefore usually is not a design consideration but
a mere 'byproduct' of the overall design.
Bob J. van Donselaar, 
