Quick Summary:  This Article describes a device and procedure for quantifying several characteristics of crystal sets.  They are: (1) Insertion power loss, (2) Selectivity, (3) RF input impedance match and (4) Audio output resistance. 

First, an acknowledgement:  This article was inspired by a paper written on 9/15/99 by Charlie Lauter at: Lautron@aol.com .  It can be accessed at:  http://home.t-online.de/home/gollum/testing.htm .  He led the way with a good procedure for sensitivity and selectivity measurement, but I wanted a more general approach.  Here is mine:
 

This article is divided into six sections. The first describes the IPL (Insertion Power Loss) measurement method.  The second gives a theoretical derivation.  The third shows a method for the measurement of selectivity.  The fourth shows how to measure the input impedance match of a CSUT (Crystal Set Under Test).  The fifth shows a method for measuring the output resistance of a Crystal Set.  The sixth gives some comments and suggestions on how to improve crystal set performance.

A quick definition:  The IPL of a crystal set may be loosely defined as 10 times the log of the ratio of the audio power delivered to the output load divided by the maximum RF sideband power available from the antenna.

Here comes a more rigorous definition of IPL:  The function of a crystal set is to convert (demodulate) the modulated RF signal sideband power received by the antenna-ground system and deliver as much of that power as possible to the output load as audio output power.  Understand that all of the signal information modulated on a carrier and picked up by an antenna-ground system resides in the power carried in the sidebands of that signal.  No signal information is contained in the RF carrier.   The Insertion Loss Method assumes that a voltage source with a specific internal impedance is connected through a “device under test” (DUT) to a load resistor.  We can say that the DUT is “inserted” between the source and the load.

• First, consider what happens when an Ideal Lossless Crystal Set (ILCS) is inserted as the DUT, tuned to the source signal and adjusted for maximum output.  It is connected between the signal source and a Load Resistor (Rl) representing the average impedance of the headphones to be used later.  This ILCS presents a perfect impedance match to the signal source and also to the output load.  It has no internal power losses.  The ILCS will convert all of the Maximum Available Sideband Power  (MASP) in the modulated signal source into useful audio power in the output load.  Power loss is zero when the ILCS is inserted as the DUT.
• Second, consider what happens when a real world Crystal Set is inserted as the DUT.  It probably will not present a perfect impedance match to the signal source nor perfectly match the output audio load, thus incurring mismatch losses.  It will have some internal power losses.  Its output audio power will be less than that of the ILCS.  The IPL of the CSUT is:  IPL = 10*log ((Output power of CSUT)/(Output power of ILCS)) dB.

Now, a brief detour to explain the concept of MAP (Maximum Available Power) and a more detailed look at Insertion Power Loss (IPL) as used in this Article.

Maximum Available Power (MAP)
Assume that any electrical source of power can be represented as a voltage source (Vs) that has an inaccessible internal impedance Zs = Rs + jXs.  See Fig. 1.  Assume that the reactance component (Xs) of this impedance is tuned out.  The crystal set tuner should do this by generating a series reactance whose value is the negative of Xs.  There is a maximum amount of power that Vs, with its internal series resistance Rs, can deliver to any load.  The value of the load (Rl) for maximum power transfer is Rs itself.  This is called an impedance matched condition.  Any other value for Rl will absorb less power from the source than a value of Rs.
 

Here is how to calculate MAP from the Vs and Rs combination.  As mentioned before, the maximum power output occurs if Rl = Rs.  This means that the total load on Vs is the series combination Rs + Rl = 2*Rs.  Since power in a resistor can be calculated as (V^2)/R, the total power dissipated in the two resistors is (Vs^2)/(2*Rs).  Since one half of the power is dissipated in Rs (and lost) and one half in Rl, the maximum power deliverable to Rl is: (Vs^2)/(2*Rs)/2 = (Vs^2)/(4*Rs).  We will use this relationship later on.  Note that Vs^2 means Vs squared and 4*Rs means 4 times Rs. Vs is in RMS volts.  If Vs is given  in peak or peak-to-peak units, a correction factor must be applied.

Definition of IPL when the input signal is an RF carrier, modulated by a sine wave.
Input Power:  Audio information that is amplitude modulated on an RF carrier is contained solely in what are called sidebands.  Sidebands are better called side frequencies if the audio modulation waveform is a single sine wave, as will be the case here.  In sine wave AM, two side frequencies are generated in the modulator.  One is at a frequency above the carrier and one is below it.  They are each spaced away from the carrier by an amount equal to the modulation frequency.  These two side frequencies carry all the information that is in the signal.  The RF carrier carries none.  When we receive a signal on our crystal set it is this sideband power that we want of capture and convert to audio power in our headphones.  The carrier acts only as a “carrier” for the sidebands and generates the DC diode current and DC voltage across the DC resistance component of the load. 

Output power and IPL:  Assume that an RF source with a MASP of Pa Watts is connected to a CSUT and that the CSUT feeds a load resistor.  The source has an internal RF impedance of Za Ohms and the load has an impedance of Rl.  The SCUT is tuned and adjusted to deliver maximum audio power to the load, with the desired selectivity.  Define the output power as Po.  Now imagine the replacement of the CSUT with an ILCS.  It provides a perfect impedance match to the source and perfectly matches the load.  Its output power will equal Pa because there are no losses.  This ideal crystal set will function as a device to convert all of the MASP into audio power.  The ratio of the output power of the CSUT to that of the ICS set is Po/Pa.  This ratio, expressed in dB is the IPL of the CSUT.  IPL = 10*log (Po/Pa) dB.  The load resistor should have a value equal to the average impedance of the headphones to be later used with the CSUT. (See Article #2 on how to measure headphone impedance.)

Section 1.  IPL Measurement Method.
The test equipment required is:

1. An RF signal generator (SG) covering 530 -1700 kHz and capable of linear amplitude modulation up to 50%.  The generator can be a modern function generator or a conventional RF signal generator, provided that the RF waveform has a reasonably low harmonic content.  It should have a 50 Ohm output resistance.
2. A scope with a flat response to at least 1.7 MHz and an accurate calibrated vertical sensitivity of 0.002 V per division or better.  Input resistance is assumed to be 1 Megohm.  Input capacitance (including that of the connection cable) is assumed to be about 175 pF.
3. A special attenuator set up and impedance adjuster unit called AMCS.

It is assumed that by tuning the CSUT for maximum output volume, that the best conjugate impedance match possible is presented to the antenna.  In simpler terms, tuning for maximum volume adjusts the resistance component of Zi to as close to 25 Ohms as possible and the reactive component of Zi to as close a value as possible to the negative of the reactance of La and Ca in series.  This set of circumstances transfers the most signal power possible from the antenna to the CSUT.

 The test procedure that follows involves applying a modulated RF Voltage (Ea) through a dummy antenna to the input of the CSUT and then measuring the Audio Output Power (Po) delivered to the output load.  The IPL of the CSUT is calculated as: IPL = 10*log (Po/(MAP in the sidebands of Ea)).

Measurement of IPL
 

We will use use a special attenuator box between the SG and the CSUT and call it the AMCS.  Refer to the schematic in Fig. 2.  The AMCS has one 3 dB and one 20 dB attenuator that are used in measuring selectivity.  It has an additional 10 dB attenuator in the event some extra attenuation is needed.  The 20 dB attenuator can also be used to determine the voltage Ea at test point P1 when it is so low that it is hard to read.  The series 45.0 and two parallel 11.1 Ohm resistors form a “minimum-loss impedance transforming attenuator”.  Its input design resistance is 50 Ohms.  Its attenuation is set so that the ratio of the voltage at test point Pi to that at P1 is 10:1.  The source resistance feeding the Dummy Antenna and Crystal Set series combination is 5.25 Ohms.  Two 11.1 Ohm resistors are used in place of one of 5.55 Ohm resistor because resistors under 10 Ohms may be hard to find.  Also lead inductance is minimized.  If the 45.0 and 11.1 Ohm resistors are held to within +/- 4%, the attenuation accuracy will be within +/- 0.33 dB. of nominal.  Resistor accuracy tolerances for the other attenuators, to hold a +/- 0.33 dB accuracy are:  3 dB-10%, 10 dB-4% and 20 dB- 2.5%.

The load on the CSUT must be a resistor (R1) of value equal to the average impedance of the headphones used with the crystal set.  One can determine the impedance of the headphones by building and using the FLVORA described in Article # 2 of this series.  Alternatively, it may be estimated as 5 or 6 times the DC resistance of the phones.

To measure the IPL of a crystal set, connect the SG**** to the AMCS and set it to a test frequency of, say, 1.0 MHz.  Turn on the sine wave modulation function and adjust the frequency to 1000 Hz**** and the modulation percentage to 50%****.  (50% modulation exists when Ea_pp is three times Ea_vv.)  Connect the AMCS to the antenna and ground terminals of the CSUT.  Connect the scope to Rl and set it to a sensitivity of 2 mV/div.  Set the SG to a high RF output and tune the CSUT to maximize the 1000 Hz trace on the scope****.  Reduce the SG output as necessary to keep the scope trace on scale****.  Reset the SG to deliver a 4 mV p-p trace on the scope.  Connect the scope to point P1 and measure and record Ea_pp and Ea_vv at Point P1.
 

Here are the labelling conventions that will be used.  Voltages on the input side of the CSUT always start with Ea.  Voltages on the output side start with Eo.  The underscore is a separator from the description suffix  that follows.  fo = carrier frequency,  fmod = modulation frequency,  pp = peak-to-peak,  vv = valley-to-valley,  car = carrier, dc = direct current,  sf = side-frequency, carpp = carrier peak-to-peak,  1sfpp = one side-frequency peak-to-peak, 1sf = one side-frequency,  2sf = two side-frequencies.
 

The IPL of any crystal set depends upon the output power level at which it is operating.  At very low output levels (signal barely readable with sensitive headphones), the IPL increases about 6 dB for every 6 dB reduction in input power.  This results in a 12 dB reduction in output power.  When this happens, the diode detector is said to be operating in its “square law region”.  Because of this effect, I suggest that the IPL be measured at several audio output power levels when characterizing a crystal set, maybe -80 and –110 dBw. 

Section 2.  Derivation of IPL.
Figure 3 shows of the envelope of an AM carrier of frequency fo, modulated at 50% by a sine wave of frequency fmod.  This modulation produces two side frequencies separated from the carrier by fmod.  One is above fo and one is below it.  If no side frequencies were present, Ea_pp would equal Ea_vv and the modulation envelope would be straight lines. With some modulation is present, one half of the total envelope fluctuation is caused by one side-frequency and one half by the other. Two side frequencies, each of amplitude Ea_sfpp, when added to a carrier of amplitude Ea_carpp, will cause the modulation envelope to have a maximum value of Ea_pp = Ea_carpp+2*(Ea_1sfpp).  The minimum value of the envelope will be Ea_vv = Ea_carpp-(2*(Ea_1sfpp)). Define: D = (Ea_pp)–(Ea_vv) = 4*(Ea_1sfpp).  Rearranging terms, we get:  Ea_1sfpp = D/4.  We can calculate the MAP of one side-frequency as: MAP_1sf = ((Ea_1sfpp/(2*sqrt2))^2)/(4*Ra).  The first “2” changes the value of Ea_1sfpp to a peak value.  The “sqrt2” changes the resultant peak value to RMS.  The equation, restated, is MAP_1sf = ((Ea_1sfpp)^2) / (32* Ra).  The total power in the two side frequencies is twice that in one side-frequency and is:  MAP_2sf = ((Ea_ 1sfpp)^2)/(16*Ra).   Now, substituting Ea_1sfpp = D/4, we get:  MAP_2sf = (D^2)/(256*Ra).

The output waveform shown in Fig.3 is a sine wave Eo_pp, having a DC value of Eo_dc.  The audio power it supplies to the output load Rl is: Po = ((Eo_pp/(2*sqrt2))^2)/Rl.  The “2” and the “sqrt2” are needed as before to change Eo_pp from a peak to peak to an RMS value.  Simplifying, Po = ((Eo_pp)^2)/8*Rl.  Since IPL = 10*log (Po/MAP_2sf), we can state the Final Result we've all been waiting for, and it is:

                             INSERTION POWER LOSS = IPL = 10*log (32*((Eo_pp/(D))^2)*Ra/Rl).

The MAP of the RF carrier only from the AMCS to the CSUT is: ((Ea_pp + Ea_vv)^2)/(3200) Watts.

Section 3.  Measurement of Selectivity:
Here is a method for measuring selectivity using the instrumentation used for measuring IPL.  It is adapted from Terman’s “Radio Engineer’s Handbook:  Measure the frequency difference between two points that lie 3 dB down on the selectivity curve.  I’ll call this value S_3.  Measure the frequency difference between two points that lie 20 dB down.   I’ll call this S_20.  The input is measured at test point P1. Depending upon the input signal level chosen for this measurement, the detector may not be operated in the linear part of its operating region, but partly into its square law region.  This non linearity will cause an erroneous result if the measurements are made using a constant input signal level and then measuring the output at each of the four frequencies.  The correct method is to measure the input required at test point P1 to attain the specified fixed output level at each of the four frequencies.  The non linearity will now be the same for all measurements and cancel out.  The Shape Factor (SF), of the selectivity curve of a CSUT, at a particular RF frequency and output audio power is defined as SF = ((S_20)/(S_3)). The lower the number, the better.
Things to remember:  The selectivity of a CSUT varies, depending on coupling, tap settings and frequency of measurement.  It is suggested that measurements be taken at 550, 1000 and 1650 kHz and any other ones where you think there might be a large variation from the average.  With fixed coupling settings, the SF of a CSUT can change if the input signal power is changed.  This effect can be minimized if the correct an audio transformer is used with a correct parallel RC in series with the cold lead of the primary of the transformer.  See Article #1.

Section 4.  Measurement of Input Impedance Match
Impedance Match (IM) refers to how closely the input impedance of a device equals the conjugate of the impedance of the source driving it.  We will define the IM of a CSUT only at the frequency to which it is tuned.  It's assumed that the input impedance is resistive at this frequency.  Impedance match may be defined in terms of "Voltage Reflection Coefficient" (S11) or Voltage Standing Wave Ratio (VSWR).  Either can be calculated from the voltages appearing at test points P1 and P2.  Turn off the modulation of the SG.  Define: RF voltage at P1=EP1_pp and voltage at P2 = EP2_pp.  S11 = 20*log abs(1 - 2*(EP2_pp/EP1_pp)).  VSWR =  (1 + abs(1 - 2*(EP2_pp)/(EP1_pp)))/(1 - abs(1 - 2*(EP2_pp)/(EP1_pp)))  These calculations define how closely the input impedance of the crystal set matches that of the IEEE standard dummy antenna.

Section 5.  Measurement of the Output Resistance (Ro) of a Diode Detector.
The addition of a variable resistor and an ohmmeter are needed to measure the output resistance of the CSUT.  Connect the SG, AMCS and scope as before.  Set the fo of the SG to 1 MHz and the AM modultaion to about 50% at 1 kHz.  Connect the variable resistor to the output of the CSUT and set it to the nominal audio load resistance for which the SCUT is designed.  Call this value RL.  Pick a moderate input power, say one that delivers an audio output power (Po) of -75 dBW to RL.  An output power of -75 dBW is indicated when the 1 kHz p-p output voltage Eo_pp is:  sqrt(RL*(31.6*(10^-9))).  Increase the load resistor to a value 1.3*RL and call the resulting demodulated output voltage Eohi_pp.  Reduce the resistor to 0.7*RL and call the new output voltage Eolo_pp.  Ro = 1.3*RL*((Eohi_pp - Eolo_pp)/((13/7)*Eolo_pp - Eohi_pp)).   Ro varies with change of input power.  At low input power levels, Ro, measured at the diode detector output (before any step-down from an audio transformer), will equal about 0.026*n/Is.  At high input power levels, in the region of peak detection, Ro will approach twice the antenna-loaded RF tank resistance. 

Section 6.  Comments.

1. Remember that output transformer loss is included in the measurement of IPL.  The usual audio transformer loss is in the range of 0.5-2 dB, but some are higher.  It's a good idea to to check the loss of the one being used.  A method is given in Article #5.  Don't forget to reduce the calculated IPL by the 0.9 correction factor if you are using the 100k resistor in series with the scope.  The MAP of the RF carrier only from the AMCS to the CSUT is: ((Ea_pp + Ea_vv)^2)/(3200) Watts.
2. It's possible for two different CSUT to have the same IPL at moderate input signal powers, but differ when receiving weak stations or strong ones. Very low input signal performance is enhanced (better DX) if the RF tank resonant resistance and transformed audio load can be made a high value.  This enables the optimum diode to be one of a lower Saturation Current.  The result is less IPL at low signal levels.  See Article #1. Very high input signal performance is enhanced (louder maximum volume) if diode reverse leakage is kept low.  This point is often overlooked.  Diodes vary greatly in reverse conduction current.  There are two kinds of reverse current effects:  One is a gradually increasing reverse leakage current that loads the circuits more and more if the input signal increases, maybe by tuning to a stronger station.  It acts as sort of an automatic volume control and is rather rare.  Unfortunately, this effect reduces the maximum volume one can get from the crystal set.  The other is normal reverse current that increases rapidly above a certain input signal power and causes audio distortion as well as reduced volume.  This effect can be observed when performing the IPL tests.  For instance, in my single tuned loop set, several Agilent 5082-2835 in parallel, while very good with low signals, distort when the input carrier power at 50 % modulation gets above about -35 dBW.  Several Agilent 5082-2800 or HSMS-2800 work fairly well at low signal levels but do not distort at the highest signal level I can supply.  This improvement comes about because the HSMS-2800 has much less back leakage current than does the HSMS-2820 or 5082-2835 at high reverse voltages.  This effect is more noticeable if the diode load resistance is above the optimum value than if below it.
3. If you use an audio transformer, don't forget to replace the R in the parallel RC with a pot and adjust it for the least audio distortion. Actually, I keep a pot in there all the time because the optimum value is usually zero for weak signals and about 1/2 the loaded RF source resistance driving the diode for strong signals.
4. For a given amount of output audio power, the output voltage is proportional to the square root of the output load resistance.  This may cause a problem for those who use 300 Ohm Sound Powered Headphones (SPHP) and those who may want to make measurements at low output power levels.  With the suggested starting output of 0.002 V p-p, the output power to a 1200 Ohm load (SPHP elements in series) is -94 dBW.  It would be -88 dBW @ 0.002 V p-p if the SPHP elements were wired in parallel. 
5. To take readings at a lower power level, there are several options to consider:
• Use a more sensitive scope.
• Use a low noise 10X gain audio amplifier.  An an improvement on this would be a single tuned band-pass amplifier tuned to 1,000 Hz.  It will filter out some of the noise and hum that will probably be present.
• Temporarily, for the tests, use an output audio transformer that transforms to a higher output resistance, along with its corresponding load resistor.  Going from a 300 Ohm to a 12,000 Ohm output resistance will boost the output voltage by sqrt (12,000/3,00) = 6.3 times.  I use two A.E.S. P-T157 transformers connected as shown in the first schematic in Article #5 as a variable-impedance-ratio transformer to boost the audio signal voltage.  I also use it to experimentally determine if the load on the diode equals the output resistance of the diode.  The switch position that gives the most output voltage is the one that provides the best match: (4, 16, 63 times ratio, or near the mean of two of the adjacent values). 
6. Here are some test results with my single tuned crystal set that uses a 14 " square loop wound with #12 ga. solid wire for the resonator.  The average parallel shunt loss resistance of the tank is 700k Ohms over the frequency range of 550-1650 kHz..  I use three Agilent 5082-2835 diodes (Is = 38 nA) in parallel for the detector and an audio transformer to convert the 700k Ohm (low signal) AC output resistance of the diode detector down to a 12,000 Ohm load resistor.  I have not yet set up to measure a loop set directly, but have coupled in an external antenna connection to a tap on the tank 6 turns from ground.  This, of course, loads the tank and results in a lower tank resistance than 700k Ohms.  The input impedance match is very good  The measured IPL at 1.0 MHz using the external antenna-ground connection is 9.65 dB at an input carrier power of -84 dBW, giving an  audio output power of -102.9 dBW.  The noise and hash on the scope prevented the measurement of selectivity.  Measurements were then made at a carrier input power of -69.4 dBW.  The output audio power became -82.9 dBW,  IPL =  4.5 dB, -3 dB RF bandwidth = 30 kHz and SF = 9.0.  Tapping the antenna 2 turns from ground increased the -3 dB selectivity to 8 kHz, kept the SF at 9.0 and increased the IPL by about 4.9 dB.  Note: The IPL figures use the 0.9 dB correction for the 100k resistor feeding the scope cable and also include the estimated transformer loss of 0.4 dB.  A SPICE simulation of this set-up with no loss in the tank gives, for the 6-turns-from-ground tap condition, an IPL of of 6.1 + 0.4 (for the output transformer) = 6.5 dB instead of the 9.6 dB and 1.7 + 0.4 (for the transformer) = 2.1 dB instead of the 4.4 dB.   This suggests a tank loss of about 2.7 dB.
7. The lower the IPL crystal set, the more noticeable will some of the effects noted above become..  The use of a parallel RC in the transformer primary for reducing distortion when receiving strong signals is important if the audio load resistance is higher than the output resistance of the CSUT.  If the audio load resistance is lower than the output resistance of the SCUT, it becomes less important.  This effect shows up in simple Xtal Sets that do not use an audio transformer.  Here, the headphone impedance is usually lower than the output resistance of the Xtal Set.  Also, the headphones' DC resistance, as a fraction of its AC impedance, is generally 2 or more times larger than the corresponding fraction looking into the primary of a headphone-loaded transformer.  This goes part way towards equalizing the AC and DC impedance of the diode output load.
8. Here is an interesting point of information:  The exact frequency to which the CSUT is tuned is a function of the input level.  Reason?  For small signals, the voltage across the diode is rather small, it is reverse biased for about 1/2 the RF cycle, and the average junction capacitance is close to the zero bias capacitance.  When a large signal is present, the diode tends toward peak detection and is reverse biased for more than 1/2 the RF cycle.  The average back voltage during this period is higher than when small signals are applied.  Since the junction capacitance reduces when reverse bias increased, the average bias over one RF cycle is less than it is for small signals.  Thus, when the signal level applied to a CSUT is increased, the frequency to which it is tuned also increases.  All semiconductor diodes exhibit some of this varactor-type behaviour.
9. If the receiving antenna has a different internal resistance than the 25 Ohms used in the AMCS dummy antenna, the calculated values of S11 and VSWR and IPL will be in error.  I may develop a simple way to measure the input resistance of a CSUT and will add it to this article if I do.