Crystal Radio Modeling
— David Wagner 2007/10/05 14:27
Formulæ
- ^{1)}
Component | Voltage Drop v(i) | dv/dt |
---|---|---|
Capacitor | ||
Resistor | ||
Inductor | ||
Diode | ^{2)} | |
Diode with Series Resistance |
“Some diodes, especially germanium and silicon junction diodes seem to have Is and n values which increase at very high currents (higher than those usually encountered in crystal radio set operation). In some of these diodes, the values of Is and n also increase at very low currents, harming weak signal reception. Is and n are usually constant in Silicon Schottky diodes, over the current range encountered in crystal radio set use.
- n = Ideality factor, sometimes called emission coefficient. This parameter is usually between 1.05 and 1.15 for silicon Schottky and germanium diodes commonly used in crystal radio sets.
- Vd = Diode voltage in Volts
- Id = Diode current in Amps
- Is = Diode Saturation current in Amps
- Rs = Diode parasitic series resistance in ohms (usually small enough to have no effect in Xtal sets)”
–A New Way to look at Crystal Radio Set Design.... by Ben H. Tongue
- V_{T}=0.026
- I_{s}: 45-1300 nA (consider 200 nA)
Frequencies of Interest
The frequency ranges of concern are widely separated.
DC | Direct Current | 0-100 Hz |
---|---|---|
VF | Voice Frequency Range, | 300 Hz -5 kHz^{3)} Hz (especially 512 to 2,048 Hz^{4)} or 300-3,400 Hz^{5)} |
BC | Broadcast Band | 500-1,700 kHz |
HF | High Frequency | >1,700 kHz (ignored) |
“1. AM stations mostly use amplitude compression which enhances the low-amplitude audio portions. According to station engineers a modulation factor of m = 0.65 is a good mean value for averaged music and speech contributions if amplitude compression and a peak modulation of m = 1 are employed.”–Diode AM Conversion Efficiency in Crystal Sets and Some Implicationsby Berthold Bosch, DK6YY
Hobbydyne
Redo the schematic shamelessly ripped from the author's site. Include the trimmer cap.
Note the four resonant circuit loops in the radio, including the Benny, but not counting the loop with the diode, the headphones, or the antenna tuner.
L1 L1 -i3-> +--UUUUU--+ +--UUUUU--+ | = = = | | -i1-> | E(~) | E(~) | | C1/ | | C <-i1-| C = C1 +(Ct^-1 +Cda^-1)^-1 +---||----+ +--||-----+ | / | - | / -i2-> | Cdb = Cd-Cda | / | / - | | / +---||---|||--+----> - | +--||--+----> Detector | /Ct /Cd | | /Cdb | | | | <-i2- | +------UUUUU--+ +------UUUUU-----+ | ===== | ===== <-i3- | Lrfc | Lrfc +------------------< +---------------------<
Note how the Hobbydyne is a capacitive transformer.^{6)}
+---------------->|--+ | | | +-UUUU-+ +-----VVV | | | | | +--||--+--||--+--||--+--||--+-----o | | | | | | +--UUUUUU-----+UUUUUUUUUUUUU+UU---o
Hobbydyne Eigenvalue Analysis
The Hobbydyne circuit has three current loops.
- (E)→(L_{1})→(C)
- (L_{rfc})→(C)→(C_{db})
- (E)→(L_{1})→(C_{db})→(L_{rfc})
These can be represented by three equations setting the sum of the voltage drops around each loop to zero:
- ,
- ,
- .
These equations can be differentiated and rearranged,…
- ,
- ,
- ,
…the currents assumed to be sine waves,…
- ,
- ,
- ,
…cancellations done and dE/dt set to 0 so ω represents the resonant frequencies,…
- ,
- ,
- ,
…and the result rearranged as a general eigenvalue problem of the form …
- ≡ ,
…or as a standard eigenvalue problem of the form …
- .
Note this is not the most efficient way to convert a general to a standard eigenvalue problem to solve it, but these are small matrices so performance is not an issue. However, there is an issue with these being singular matrices because they include the third loop, which is just the sum of loops one and two. There are ways to solve this, but conversion to a standard eigenvalue problem as shown above is not a valid method.
Besides using the above to work with resonant frequency response, by assuming
- E = -I cos(ωt + φ) so
- E' = dE/dt = I ω sin(ωt + φ),
the following can be used to characterize the response of the Hobbydyne circuit at any frequency:
- or
- .
In these equations, I_{2} + I_{3} represents the current through L_{rfc}. A sufficiently large impedance (Z_{det} » X_{rfc}) across L_{rfc} would load the circuit lightly enough to indicate the detector voltage potential without unduly effecting the tank and Hobbydyne response.
Capacitive Transformers
Rout +---------o--VV--+ | | o--+--||--+--||--+ | | C2 C1 | | | | | +----UUUUU----+ | | | 0----------------+--o------+ RT=40k L
For 40kΩ input impedance to R_{out}=16Ω output centered at 1500 Hz with a bandwidth of 1000 Hz.
Analysis and Applications of the Capacitive Transformer by Ramon Vargas Patron.
- The bandwidth is inversely proportional to the filter's Q: 1.5.
- 0.02.
- 0.2 μF.
- 10 μF.
- .
- Input Q:.
87 H ^{7)}
o-------------+ | +----||----+--||---+---o | C+(Ct&Cda) Cdb | | | +-------UUUUU------+ | L o--+----------------------o
Hobbydyne
Check out the Hobbydyne. It's also backwards. Take f0≈1 MHz, C1=C+(Ct&Cda)≈95 pF, C2=Cdb≈5pF, R≈40kΩ.
- 25.13
- 40 kHz
- 0.05
- 477
5 mH
References
Applied Numerical Methods for Engineers and Scientists, by Singiresu S. Rao, Prentice-Hall, Inc., 2002. Pages 276-277 have the circuit analysis used as a model for the Hobbydyne Eigenvalue Analysis.
Discussion