AM diode detector equations relating diode parameters to the LSLCP and sensitivity

Quick Summary: Several new equations are presented showing various relations between diode detector rectified current, input AC and output DC power, insertion power loss and the 'Linear-to-Square-Law Crossover Point' (LSLCP). The LSLCP is an operating point where the diode detector is operating half way between its linear and square law modes. It can also be thought of as a dividing point between weak and strong signal reception. Bear in mind that the LSLCP is a point (see Fig. 3) on a graph of output DC power vs input RF power of a diode detector system. It is not a point on a graph of DC current Vs voltage of a diode. Article #27 shows actual measurements on a crystal radio set using eleven different diodes, that tends to experimentally back up the validity of equation #5 and those following it.

Assumptions used in the following discussion:

The Q and L/C ratio of tuned circuit T are assumed to be high and low enough, respectively, so that the 'stored energy effect' of T prevents any appreciable clipping of the positive voltage wave form peak by diode D1.
The value of C2 is assumed to be high enough so that a negligible amount of RF voltage appears across it.
The diode parameters Is and n are known from measurement or a Data Sheet. A simplified method of estimating Is is given in Section 2, Article #4, but the parameter n has to be estimated. A method for measuring both Is and n is given in Article #16. The effect of the series parasitic resistance of the diode is assumed to be negligible - as it is at low signal levels for most all detector diodes. Diode back leakage current from either 'parasitic leakage' or operation with voltage swings reaching into the 'reverse breakdown current' region is assumed to be negligible. The diode temperature will be assumed to be 25 degrees C.

Approach: The RF signal input power range is divided into two regions and one point; impedance and power relationships are determined. Refer to Figs. 2 and 3. Two Cases will be considered. In Case A, R1=R2=Rx=0.0256789*n/Is. In Case B, R1=Rx/sqrt2 and R2=Rx*sqrt2.

The low power region: Here, the relation between output power and input power approaches 'square-law'. That is, for every one dB change in input power there is about a two dB change in output power. The detector input and output resistances approximate Rx.
The high power region: Here, the relation between output power and input power approaches 'linear'. That is, for every one dB change in input power there is about a one dB change in output power. The detector input and output resistances are no longer equal. The detector input resistance is equal to about half of R2. The detector output resistance is about twice R1.
The point where the two areas overlap equally: This is the 'linear-to-square-law crossover point' (LSLCP). At this point there is a 10*log(sqrt2) dB change in output power for every 1.0 dB change in input power (slope of about 1.5). If R1 and R2 are both equal to Rx, in Case A the detector input resistance is about 12% less than Rx and its output resistance is about 12 % greater than Rx.

Transition from the linear to the square law region: All good diode detectors, at high input power levels, if well impedance matched at input and output, have a low insertion power loss (a fraction of a dB). If the input power is reduced, at first the output will drop approximately dB for dB in step with the input. If the input is further reduced, the output will start to drop faster (in dB). This can be thought of as the onset of noticeable 'detector insertion power loss'. The insertion power loss at the LSLCP, in two SPICE simulations (see Cases A and B below), is about 5 dB. Put another way, at the LSLCP the output power is about 0.3 times the available input power. This power loss figure changes by less than 0.1 dB between Cases A and B.

Most crystal radio sets can deliver a readable signal at an input of Plsc(i) Watts. It would obviously be desirable to lower the input power at which the LSLCP occurs so that more of the weak signals would be closer to the linear mode of operation, experience less insertion power loss and therefore be louder.

Example SPICE simulation of a diode detector at 25° C.: Figs. 2 and 3 show power relations at various power levels. The LSLCP is shown by a red arrow. In the SPICE simulations for these graphs, the source and load resistances, R1 and R2, are equal to Rx (Case A). The DIPL values for Case B (Fig.3) are within 0.4 dB of those of Case A. The main difference is the lower DIPL values in Case B at high input powers. For instance, the insertion power loss at an input of -48.912 dBW is 0.76 dB for Case A and 0.30 for Case B. The loss figures from the equations that follow are quite close to those that occur in a SPICE simulation of both Case A and Case B.

At input power levels several times or more below the LSLCP, the impedances of the input and output ports of the detector both approach Rx for both Cases, A and B. At input power levels several times or more above the LSLCP, the detector approaches operation as a peak detector having a low insertion power loss. In this condition the input RF resistance of the detector approaches half the output load resistance and the output resistance of the detector approaches twice the RF source resistance. Summary: In Case A, the detector input and output ports both approach an impedance matched condition when the signal power is several times lower than that at the LSLCP. At signal power inputs several times greater than that at the LSLCP, a moderate impedance mismatch exists at both the input and output ports. In Case B, conversely, the detector input and output ports are both are moderately impedance mismatched when the signal power is several times lower than that at the LSLCP. At signal power inputs several times greater than that at the LSLCP, both input and output ports approach an impedance matched condition.

Simulated diode detector output power and insertion loss vs input power, Case A.
This diode has a low saturation current, compared to that used in the average
crystal set, Is=38 nA, n=1.03 The LSLCP is shown by the red arrow.

Unpublished information from Xavier Le Polozec indicates that the optimum practical compromise values for R1 and R2, for input powers well below the LSLCP to well above it is: R1=Rx and R2=Rx*sqrt2.

The following equations are developed for the Class A termination condition of R1=R2=Rx. They give insertion power loss values to within a fraction of a dB of those provided by SPICE simulation for both Cases A and B. The output current, I2, is different for Class A and B.

Observation of a curve of output vs. input power (in dB), from SPICE simulation of a Class A terminated detector reveals a slope of 1.5 at an input power value of -78.91 dBW, for the particular diode used. This is the LSLCP for input power. Another observation is that the rectified diode current I2 at the LSLCP point appears to be very closely two times the Is of the diode. This two times figure appears to be apply to all diodes. Note: The Plsc(i) of -78.91 dBW occurs when the detector parameters are R1=R2=Rx and the diode parameters are: Is=38 nA, n=1.03 and temperature is 25° C. In general, the 'two times' figure does not hold for termination conditions other than R1=R2=Rx.

Calculation of diode detector output power when it is operating at its LSLCP:

Differentiating the Shockley diode equation with respect to the diode junction voltage yields:

Some obvious relations: Output power=Po=(I2^2)*R2 Watts. The output load R2 has been specified as equal to Rx, Rx is defined in equation (0). I2 is 2 times Is at the LSLCP (See equation 1). Substituting into the equation for Output power Po, we get, for the output power at the LSLCP:

Plsc(o)=0.102716*Is*n. (2)

Calculation of the diode detector DC output voltage [V2lsc(o)] when it is operating at its LSLCP:

Some simple manipulation of equations (0) and (2) results in the relation: V2lsc(o)=0.051*n Volts (2a)

Calculation of the ratio of output power to input power (DIPLR) at any power level:

A proper relation between Po, Pi and I2 obviously requires that I2 approach zero as Po/Pi approaches zero, that Po/Pi approach proportionality to I2 as I2 becomes low (the square law relation) and that Po approach Pi as I2 becomes very high. Also, at an output power of Plsc(o), I2 must equal 2*Is, as stated earlier. Curve fitting suggests this relationship: Po/Pi=(I2/(I2+4*Is)=DIPLR (3)

Interesting note: Since, at the LSLCP, I2=2*Is (eq. 1), Plsc(i)=Plsc(o)*3. (4)

Calculation of the diode detector input power when it is operating at its LSLCP:

Substituting the value of Plsc(o) from equation 2 into equation 4 results in: Plsc(i)=0.308148*Is*n for the input power at the LSLCP (4a)

Various relationships between input power, output power and diode parameters:

A lot of mathematical manipulation of the relations given above results an equation that fits the simulation data quite well over the whole range of the graph in Fig. 2.

Pi=Po+2*sqrt(Plsc(o)*Po) Watts. Equation (5r) normalized to Plsc(o), by using equation (2). (5rn)

From equation (5r), at low output power power levels, the input power required to produce a given output approaches:

An equation that fits the detector Insertion Power Loss Ratio (DIPLR) is obtained by dividing equation (5) by Pi:

DIPLR={sqrt[1+Plsc(o)/Pi]-sqrt[Plsc(o)/Pi]}^2 Normalized to Plsc(o) by dividing right side of eq. (5n) by Pi (6n)

The Insertion power loss (DIPLR) at which a diode detector is operating can be determined by noting the DC rectified output voltage:

Adjust the DC component of the diode load to equal its axis-crossing resistance (0.0256789*n/Is ohms).** Measure the DC voltage V2 developed across the DC load. Some simple manipulation of equation (3) above results in equation (7).

** See articles #16 and #27 for info on determining the Is and n of diodes as well as measurements on some diodes.

Equation (5r) seems important. It shows that the input power required for a specific output power is reduced if n and/or Is is reduced. At low input powers, the required input power for a specific output power approaches direct proportionality to the square root of n and/or Is, as shown in equation (5Li). The product n*Is can be considered to be a 'figure of merit' for diodes as weak signal detectors, provided input and output impedance matching exist and losses from passive components remain unchanged. The ideality factor (n) and saturation current (Is) of the diode are important parameters in determining ultimate very weak signal sensitivity. If all other diode parameters are kept the same, the weak signal input and output resistances of a diode detector are directly proportional to n and inversely proportional to Is. Assume a diode with a value of n equal to oldn is replaced with an identical diode, except that it has an n of newn, and the input and output impedances are re-matched (the new impedances are doubled). The result will be a detector insertion power loss change of: 10*log(oldn/newn) dB. That is, a doubling of n will result in a 3 dB increase in insertion power loss, assuming the input power is kept the same and input and output impedances are re-matched. The result is a 3 dB reduction of output power (volume). A similar effect occurs if Is of the diode is increased except that this change reduces the impedances that must be re-matched instead of increasing them.

Interesting note: A simple manipulation of Equations #0, 2 and 4 shows that, at the LSLCP, the RMS value of the AC signal at the input to the diode is: (0.08895*n) Volts, and it is independent of Is.

One final equation: One can combine two equations: Rx=0.0256789*n/Is, as given in the "Definition of Terms" at the beginning of this Article and equation (5Lo) to come out with a very useful result relative to actual crystal set design. For a given input power, equation (8) gives the output power as a function of the diode axis-crossing resistance and ideality factor (for weak signals and matched conditions).

Let us review the conditions that apply to equation (8) before discussing it. 1) The signal power to the detector diode is well below the LSLCP (weak-signal reception). 2) Detector source and load impedances are well matched to Rx, the axis-crossing resistance of the diode (see "Definition of Terms"). Equation (8) indicates that the maximization of output power, when receiving weak signals, requires using a diode of low n because output power is shown to be inversely proportional to the square of n. Also, with other factors being fixed, higher source and load impedances increase output power in direct proportion to their increase in value (remember they are both kept equal to Rx). A real-world problem occurs when one manipulates the RF input and audio output impedance transformations when attempting to optimize them to get greater weak-signal volume. Their internal power losses can change and this is not accounted for in equation (8). The RF input impedance transformation loss probably won't change a lot if the ratio of unloaded to loaded Q of the tank doesn't change, and will not be treated here. Audio transformer loss is worth investigating. An audio transformer usually has a Manufacturer-stated loss that applies when it is used between the Manufacturer-stated impedances. When used between other impedances (higher, in the usual case with crystal sets), transformer loss will most always be greater. Equation 8 shows that the transformer winding connected to the audio output side of the diode detector should present the highest impedance possible but, then that is done, transformer power loss usually increases. A compromise is needed.

Warning: Don't use two diodes in series if you want the best weak signal sensitivity. The result of using two identical diodes in series is the emulation of an equivalent single diode having the same Is but an n of twice that of one original diode.

Experimental measurements on eleven different diodes used as detectors is shown in Article #27. Close correlation between these equations and actual measurements is demonstrated.

Quantitative insights into Diode Detector Operation derived from Simulation in SPICE, and some Interesting new Equations relating diode parameters to weak signal sensitivity