Shockley Equation Verification

The 1N4148 Diode 

and the 

Shockley Equation.

My next project will be to design from scratch, a functioning and reliable high fidelity stereo headphone amplifier. Although based on a well established Class A-B model, in this project I will seek to derive all the related equations myself. But first there is a matter of investigating the 1N4148 diode (as one example), and its V-I characteristic.

There must be several such suitable diodes I could use to control the biasing of the eventual Class A-B push-pull headphone amplifier.

Below is a table showing forward bias I-V measurements I took, the input voltages varied from 0.575v to 0.835v.

I (mA)	0.393	0.644	1.060	1.747	2.680	4.160	6.340	9.520	13.900	19.540	26.485	30.614
V (volts)	0.575	0.600	0.625	0.650	0.675	0.700	0.725	0.750	0.775	0.800	0.825	0.835

Basic Diode Action

Under forward bias configuration (as shown) the
width of the contact potential is narrowed as electron current
is swept across the junction.

William Shockley Equation

According to the Shockley Equation, the relationship between forward bias diode voltage and diode current for a PN junction diode is ...

Where I_O = reverse saturation current,  Io = 25*10^-9 for the IN4148
V_T= KT/q ... thermal equivalant of voltage, 
K = boltzmann constant, T = absolute temperature, q = electron charge.  
And at 20C V_T ~ 0.025v

n= ideality factor, 
n=1 for a perfect diode, but often n = 1.. 2.5

From my empirical results, I wanted to establish n, and by ignoring the '-1' term, and rearranging I obtain ...

For the 1N4148 data above, I calculated the mean value of  n, that is ...

n = 2.351 

(to 4 significant figures)

With that I plotted the data vs the Shockley Equation model .... 

The correlation was remarkably good, better than I had anticipated! 

Matsushita DC Motor

AN6652 Matsushita 
DC Motor Circuit Investigation,
but No Modification.

I have another identical Matsushita motor where I decided to examine the motor controller circuit - this time based on the (later?) AN6652.

Before I took the back off, I had to de-solder a tag which connected the metal casing to the ground point. The casing is now electrically neutral to both ground (or -ve) and +Vcc. If in doubt, I would suggest we do this before attaching a similar motor to any cassette deck.

The circuit above is essentially the same as in the previous AN6651 configuration, except for: a signal diode, C=22uF, and here R1, R2, and R3 all effectively form part of Rs. 

Rs = R3+R1||R2 (R2 effective).

18/05/2020.

External Motor Controller Experiment

An External Motor Controller Experiment

Revised 15th May 2020.
(Apologies for the continual editing, the blogger has so many HTML editor bugs)

Traditionally, a brush-type DC 12v motor for a cassette deck has its controller tucked inside the housing of the motor assembly.

I hope to experiment again using the Panasonic or Matsushita AN6651 controller, and applying it to a stripped-out Matsushita DC motor, that is - remove the internal circuit and attempt to control speed externally.

All internal motor control components have been removed.

The motor in question here has the following measured internal parameters at approximately 2400rpm when under loaded conditions within the Sansui SC-1330 during Playback.

Vm=6.71v, Ra=18Ω, Ia=68.5mA

Vm = voltage across the motor brush terminals,
Ra = internal armature resistance, measured before loading.
Ia = armature current at approximately 2400rpm, under 'Play' loading conditons.

The widely accepted mathematical model is shown below.

Ea = generated 'back' electromotive force 'EMF' induced into the armature windings by the fixed internal field magnet. Note that Ea ∝ armature speed n.

From which Kirchoff's Voltage Law applied to the circuit, we have Vm=Ea+Ia*Ra, and from this I deduce that back-emf generated at approximately 2400rpm is Ea ~ 5.38v 

The AN6651 Block Diagram

Turning attention now to the AN6651 ...

Note: a series of 40 (in parallel?) output transistors, these will effectively restore the working or optimal Vm voltage across the brushes of the motor.

And in circuit ...

For the AN6651, the value of K is 40, this appears to be linked to the number of output transistors. Seemingly, every controller of this type has a value for K? The value of Vref for the AN6651, is nominally 1.0 volts - according to the datasheet.

Another similar motor controller is the LA5586, I examined its datasheet. As usual, the guides are very poorly written, but I believe I have managed to decipher how to calculate both Rt and Rs so as to get my Matsushita, or another similar DC motor working under the LA5586, or the AN6651. 

Note: The expression (Is+Ia)/K in the circuit diagram above is a guide and parameter that I have to use - it's highlighted in the LA5586 datasheet. I will assume that this can be applied to the AN6651.

The journey to find Rs begins with a voltage loop equation (from the datasheet) relating system parameters, Kirchoff's Voltage Law here states ...  

Eo + Ia*Ra = Rt(Is(1+K)+Ia)/K + Vref

(Vm = voltage divider/comparitor section)
 

where Is = current through Rs. 

Substituting Vm for Eo + Ia*Ra

and solving for Is gives ...

Is = (1/K+1)(K/Rt*(Vm-Vref) - Ia)

and since Rs = Vref/Is,  

Rs can be found.

The datasheeet also suggests that I must assign Rt < K*Ra otherwise the system will become unstable.

In this case: Rt < 40*18Ω or < 720Ω. Okay, so I'll go for Rt=680Ω or lower (I have plenty in stock!)

Calculating Rs 

According to the expression for Is, and setting Rt=680Ω, 

I obtain ..

Is = (1/41)(40/680*(6.71-1)-0.0685) = 0.00652 A.

Is ~ 6.50mA

As Rs = Vref/Is, and Vref=1v (for this controller), then

Rs ~150Ω

Okay, so I'll try Rs = 47Ω + 200Ω(variable)  or better try 100Ω + 100Ω(variable)?

I'll also set Vcc=12v, note that Vcc=Vm+V4.  

==========================================================

After putting this circuit into practice, I modified the ability to alter Rs by introducing a 'coarse' and fine' speed trimmer resistor.

AN6651 based circuit soldered on to 'stripboard'

Later the circuit was electrically isolated, moved away from the transformer and secured into place using plastic ties.

The good news is the motor controller works!, however - at this time other thoughts cross my mind...

Temperature Coefficient of Resistance?
There is another issue for me to factor in - an increase in motor resistance Ra with temperature as the motor begins to heat up! That is - what is the coefficient of temperature of Ra, and will this upset the running speed over a period of time?

Will I have to position Rs close to the motor so the ratio of Ra/Rs is maintained?, much like the wheatstone bridge principle. It doesn't appear so, but I'm not 100% certain.

Assuming Rs and the motor are not 'in close contact' - will I need to allow time for the motor to 'warm up' to calibrate its true stable speed? If so, then for how long?

Incidentally, I measured Ra ('cold'): 17.9Ω, and Ra (after 30 minutes of 100mA loading): 18.8Ω.

Findings, so far ...

Speed Calibration Using 3000Hz Test Tape

With a 3Khz test tape positioned at the middle of the tape,  I calibrated the tape speed.

At a 'cold' start up, I calibrated using my 'fine tune' trimmer for maximum stability, then noted the playback speed ... Playback frequency = 3000Hz ± 1, 2, 3Hz 

After 5, 10, and 15 minutes of running the motor in, so that Ra's temperature was effectively raised ... Playback frequency = 3000Hz ± 3,4,5,6 Hz. The median deviation in frequency was around 4Hz .. 5Hz. I didn't note if the motor was running fast or slow - just the steady difference.

Sometimes it would 'run' to about ± 8Hz/10Hz, but these were possibly down to wow and flutter wobbles, and that I was examining the deviations of speed for a longer period of time? The speed would soon run back to a stable difference of 3,4,5,6 Hz etc.

========================================

Later: I finally decided to have the motor running permantently, much like the Aiwa AD-F770. And after 20 minutes of warm up time, I aligned the speed of the motor to register 3000Hz ± 1,2,3 Hz etc. I am able to set the motor speed to playback exactly 3000Hz, but it is unrealistic to expect this to remain rigid throughout various tape positions. So far the Sansui SC-1330 and its 'new' motor/controller configuration is remaining within a ±0.5% tolerance of its motor speed target.   

Time will tell if having an 'external' ('offline') AN6651 motor controller works effectively?

Please note: Tape speed at the beginning, the middle, and at the end of a tape differ slightly. Calibrating at somewhere about the centre of the cassette tape is said to reduce overall errors. (I can't qualify the apparent speed variation from beginning to the end of a tape - it's based on general forum consensuses)

Power Consumption: The AN6651 is rated at 1.3 watts (1300mW) heatsinked on to conventional circuit board. At average use, the power consumed is largely through the output transistor stages which are employed to regulate Vm, and since Eo = Vm-Ia*Ra, then Eo and hence speed n is regulated. 

An estimation of the power dissipated by the AN6651 is approximately V4*Ia, or (12-6.71)*0.0685, that is 0.362w, and adding internal biasing into consideration, probably around 0.4W or 400mW.

Note: This article is finished, but subject to the correction of minor mistakes or amendments.
 

2/11/2020 - small addition.

***I tend to write in this www.blogger.com directly online, and not offline. The reason? - the blogger editor is full of software bugs, if I prepare and later paste a document into the blogger editor it is filled with formatting errors and colour quirks, hence it is easier to write this way.***
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