Ri is Ru ?
(published in Electron #10, 2006)
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 start-off with the basic rule about maximum energy transfer. In figure 1 we find a diagram with a generator (Ug), an internal generator resistance (Ri) and a power 'consumer', symbolized by the external load resistance (Rb).
We like to know when maximum available generator power will be transferred to the load.
i = Ug / (Ri + Rb)
This will generate power in the load resistor to the amount of:
Pb = i2 * Rb = Ug2 * Rb / (Ri + Rb)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 radio-amateurs 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.
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 L-C 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 L-C 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 L-C 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 feed-back 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 Ri < (Rb / 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 radio-communication, a transmitter and / or HF amplifier usually are driving the antenna, that in many systems is to be regarded as an L-C 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 low-Q antenna because this may lower feed-line 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 not-characteristically terminated transmission-line, 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.
As before we may regard the transceiver as a generator with an 'internal voltage source', Ug with an 'internal resistance', Ri, that is externally loaded with a resistor Rb, (the antenna) as in figure 3.
The 'generator', Ug and 'internal resistor', Ri 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 Rb, 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, Ug and Ri, so we have to perform two measurements to get hold of. In figure 4 we may find the measurement set-up.
As a first step, the transmitter is tuned to a specific output power, that is measured as a voltage V1 across load resistor Rb1. The latter is 50 Ohm as prescribed by the manufacturer and should be a firm and reactance-free dummy load at the measuring frequency.
Voltage V1 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 dummy-load in front of the detector voltmeter.
In a second step an additional (much bigger) resistance is switched in parallel to Rb1. The value of this resistor is selected to present a small, but significant voltage change at the meter, now showing V2. Total load resistor however should not deviate to much from 50 Ohm. A good value for this resistor appears to be 220 Ohm, non-reactive of course.
The parallel circuit of Rb1 and 220 Ohm is called Rb2, 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 V2 is measured with the same accurate voltmeter.
The 'internal resistance' (output impedance) of the transmitter may now be calculated to:
Ri = Rb1.Rb2 (1 – V1/V2) / (Rb2.(V1/V2) – Rb1)
This formula is showing the importance of the accurate voltage measurements since the accuracy is depending on the ratio of the two voltage measurements.
Based on the above set-up, 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 set-point 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 feed-back 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 feed-back, 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 %).
efficiency is about equal to the efficiency of a class AB amplifier, that
indeed is present in the output stage of the TS
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 Ri = Rb. 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 'by-product' of the overall design.
Bob J. van Donselaar,