Ancient ultra low noise narrow band preamplifier

[mikebg]

Llow noise narrow band preamplifier.
　
At sine induction, conventional smal sensing heads require BW (bandwidth) of preamp maximum 12Hz. At slow motion is sufficient even BW=4Hz. Attached circuit diagram represents an ancient transistor preamp stage with two tuned tanks. This stage was used during 50-years when Bulgarian metal detectors were transistorized. Coil L1 is wound on adjustable potcore. Gain Av and bandwidth BW depend on resistance of coil L1. The particular of the original design is too large collector current in operating point.

[mikebg]

The transistor Q1 was ancient germanium PNP type П1Д made in "Svetlana Leningrad" factory in St. Petersburg, Russia. Despite large dimensions shown below, maximal power is 50mW only. Transistor П1Д has about 18 dB noise, but because of the narrow band, the stage generates noises smaller than modern IC stages used in metal detectors. To experiment the stage with an low noise silicon transistor ZTX214C, I decided to calculate how much should be the collector current in operating point for minimum noise. I found a formula in the textbook "Applications of analog integrated circuits" by Sidney Soclof. According to formula, the low resistance of conventional RX coils, requires preamp transistor to work with collector current much stronger than used for operating point of input stages in IC opamps.

[mikebg]

My Components for test of ancient MT57 preamp:
Vee=5V, Vcc= - 5V. Total inductance of RX coils is calculated for resonance with total capacitance at frequency 6500Hz. Mutual inductace between RX coils is positive, but it is so smal that can be neglected by calculation.
Total resistance of coils RXa + RXb = 90 + 90 = 180 ohm. Measured stray capacitance 210pF.
L1 is made of adjustable potcore D=14mm, H=8,5mm, AL=400nH/turns^2. Resistance of L1<5 ohm.
C1 + C2 = 140pF, C1 is selected for resonance at middle position of C2.
C3 + C4 = 6800pF, C3 is selected for resonance at middle position of L1 ferrite core.
R1 = 5K11F, R2 = 4K02F
Q1 = ZTX214C, operating point : Ic=0,9mA, Ve = +0,6V, Vc = -1,6V.
IMPORTANT NOTE: This analog narrow band preamp is not suitable for conventional block diagram of metal detectors since they use TX coil voltage as reference signal for synchronous demodulation. The analog narrow band stage has unstable phase characteristic, which causes unstable discrimination. To use this circuit diagram in discriminating MD, the reference voltage for synchronous demodulation should be derived from preamp output (as the synchronization in oscilloscopes and televisions is derived from received signal).

[mikebg]

GIVEN: It is known that the use of sensors with small RX coil dimensions that are waving quickly RX must provide bandwidth BW = 30Hz. In the formulas, that we will use below for the design of the bandwidth, does not participate TX frequency. It is only necessary for the design capacitances of the two tanks, which we will not make here.

BW Design of RX tank
BW of preamp depends mainly on the Q factor of coil L1, because the Q factor of RX tank is less significant. There are two reasons for this:
1. RX coils can not generally have low resistance. They must have many turns to give a strong signal. For this purpose, capacitors C1 and C2 must have low capacity. However, resistance of RX coils must be less because it increases the thermal noise of preamp. This requirement increases the mass of coils. Trade-off is set for mass rating of RX coils and then is elected as large wire that to reach this mass.
2. Input resistance of the transistor amplifier is too low, leading to a damping of RX tank. Transistor Q1 should work with relatively strong current in the operating point for minimum noise. This leads to significant reduction of its input impedance h11. Although ZTX214C has too big Beta or h21, it is not Superbeta transistor. In this network can be used Superbeta transistor, because operating point is with very little collector voltage.
Therefore, the design of the BW NB preamp can ignore the influence of RX tank on bandwidth.

Design of tank in the emitter
To calculate coils timeconstant use the following formulas:
(1) BW = f / Q,
where
(2) Q = 2.pi.f.L / r.
From them it follows that we need potcore providing time constant of coil greater than
(3) L / r = 1 / (2.pi.BW) = 1 / (6,28.30) = 5,3 E-3s=5,3ms.
This time constant can provide potcore D14mm, H8, 5mm, having AL = 800nH/turn ^ 2.
In order not to increase the noise of the system, I choose a small value of coils resistance r = 4ohm. From formula (3) follows that
(4) L = r / (2.pi.BW) = 4 / (6,28.30) = 0,021 H.
The number of turns N is calculated by the formula
(5) N = sqrt (L / AL) = sqrt (0,021 / 800E-9) = 162 turns.
To have a minimum coil resistance, must choose a thicker wire so as to fill the whole window of reel. As is shown in the drawing below, the window has an area S = 8sq. mm. Hence we can calculate the maximum external diameter of the wire with insulation:
(6) d = sqrt (S / N) = sqrt (8 / 162) = 0,22 mm. From the table for standard wires should select less diameter than the calculated value. I chose wire with external diameter 0,17 mm, which has a nominal diameter of metal 0,14 mm.
Should check the resistance of the coil.
In the drawing is calculated the middle (average) diameter of a turn:
(7) Dmid = (Dmin + Dmax) / 2 = (7 +11) / 2 = 9mm.
Average length of a turn
( lmid = pi.D = 3,14.9 = 28,27 mm
The total length of wire is
(9) Lt = N.l = 162.28,27 E-3 = 4,6 m
The table of standard wires recognizes that the selected conductor with nominal diameter of 0.14mm has a linear resistance r '= 1,14 ohm / m. Hence we can calculate the resistance of the coil:
(10) r = Lt.r '= 4,6.1,14 = 5,24 ohm. Since the value is greater than necessary, should choose the next in table wire with larger nominal diameter than 0,14mm.

[mikebg]

For modernisation of ancient Low Noise Narrow Band Preamp MT57, added are:
1. Low ohm resistor R3 for inserting ABS (Automatic Balance Signal), according to patent US 3,500,175 of Vaino Ronka. Select resistance of
R3<r coil/10 to keep low noise level.
2. Voltage controlled resistor with Q2 for AGC and ABWC (Automatic Band Width Control).
R4 and R5 are about 300K.
Measured are: load resistance for RX tank (h11 at base of Q2):
More than 5Kohm at small or deep targets (saturated Q2).
About 1,8Mohm at strong signal (Q2 is off).
ADJUSTMENT:
Select C1 for resonance at middle position of C2.
Select C3 for resonance at middle position of L1 ferrite core.
Select R5 for minimal distortion at strong signal (with ABS off).

[mikebg]

RF amplifier with AGC

The AGC is applicable only for motion metal detectors.
The task of AGC is to maintain maximal not distorted signal at the output of U1B - pin 7. If the supply rail voltages are +5 V and-5V, an ideal rail-to-rail opamp would give output Vpp = 10V. However the not distorted output at TL072 is around Vpp = 5V only. When this is achieved, a DC voltmeter connected to test point tp1 shows DC voltage approximately minus 2V.
Simultaneously with such a large output, the RF amplifier should give as much as possible less gaint. Thus, the AGC of LN preamp (see above circuit) provides its high gain and the preamp operates in a narrow band with minimal thermal noise. For this purpose, reduce resistance of the resistor R3.
For convenience in the settings, potentiometer P1 is set at a minimum (maximal SPEED) to shorten the settling time of P-I controller. Selection of components starts at the very beginning without preamp. After connecting preamp with its AGC and both RX coils in TWIN LOOP configuration, select R3 until the voltage at test point tp2 be almost zero without TX signal. Palaver began when the TX is on :-)

[mikebg]

How RFA works

This thread contains the circuit diagram of most sensitive RFA (radio frequency amplifier) used in metal detector. The circuit diagram consists of two parts: preamplifier (posting #5) and end amplifier (posting #6). Similar RFA circuit diagram is used by (R)EMI group in most sensitive metal detector GLEANER.

Despite GLEANER differs from a conventional metal detector, it is not something new, but only a reinvention. It operates on a principle described by Vaino Ronka in his patents US 3,500,175 and US 3,614,600.
The principle differences between GLEANER and a conventional metal detector are:

1. Modulation index of target signal.
GLEANER uses slow acting ABC (Automatic Balance Control) to maintain maximal modulation index of target signal. In a conventional metal detector, each demodulator sees target signal as very small relative change (modulation) of a large carrier voltage (sum of AIR and GND signals). The ABC compensates (balances to zero) AIR and GND signals in input of preamp. Noise remains and RFA amplifyes it untill AGC senses that output reaches saturation threshold. Old metal detectors for demining and sensitive metal detectors have such compensation in front end, but it is manual; operator should balance input signal with two knobs MAGN and PHASE (or Re and Im). At ideal operating ABC, the gain of RFA is limited by input noise only. Below is calculated possible gain for this ideal case.

2. Amplification.
GLEANER uses slow acting AGC (Automatic Gain Control) to maintain amplification at maximal usable level, ie near to saturation thresholds in output. Conventional metal detectors use RFA with fixed gain at low levellimited by sum of AIR and GND signals in input of RFA. Below is calculated possible gain for the conventional case.
Since the ABC described in point 1, balances AIR and GND signals in the input to zero, the RFA amplifyes only noise and interference in input. The gain can be increased untill output level becomes near to saturation.
There is metal detectors where the gain is manual adjustable by operator at threshold of saturation with AIR and GND signal.
The attached figure illustrates 3 knobs used by operator in old mine detectors to tune front end to maximal sensitivity. In GLEANER this tuning is made by slow acting ABC and AGC.
(will be continued)

[mikebg]

3. Reference voltage for synchronous demodulation.
GLEANER uses a real recieived GND signal as reference voltage for synchronous demodulation. In conventional metal detectors is used voltage across TX coil to create with phase shifting an artificial reference voltage for GND elimination (and other reference voltages for DISC settings). However when search head moves, the phase of real GND signal changes because conductivity and permeability of soil are not equal in all places, but the phase of artificial reference voltages remains unchanged.

4. Operating at resonance of RX tank circuit
When the possible gain of RFA is limited by noise in input, to reduce noise we need a narrow band front end. The required bandwidth for metal detecting is very narrow, no more than 12Hz. However when we use RX tank circuit for this purpose, we operate in a region where the phase changes very quick with frequency: from 0 to 180 deg at parallel LC tank or from +90 to -90 deg at serial LC tank. Conventional metal detectors can not operate steadily in the resonance region because reference voltage is artificial. When reference voltage is received by RX tank, phase difference remains the same, ie GND eliminiation and DISC setting not need correction.

Let we calculate the gain of RFA limited by noise:
For example we have RFA delivering maximal output swing (just before threshold of saturation) 5,6Vp-p, as shown in the attached figures. When AIR and GND signals are set by ABC to zero, we can obtain maximal undistorted noise voltage in output Vn=1/8 of 5,6Vp-p or 0,7Vrms. If we have in input of RFA noise density 7nV/rtHz and bandwidth is 10Hz, then the input noise is Vn=7E-9*10=70nV rms. Then the possible operating gain (without saturation) is A=0,7V/70nV=1E7 or 10 000 000 times, ie 140 dB. This is not realisable in practice because ABC can not suppress the sum of AIR & GND signals to zero, but even 70dB gain is a good value.
Conventional metal detector often operates with AIR & GND signal in input of RFA more than 28mVp-p. That means the maximal possible gain of RFA is limited to A=5,6/28E-3=200 or 46 dB only.

The left oscillogram shows an output saturated by noise because RFA has very high gain without AGC. In the middle figure gain is reduced manual untill we can not see saturation. The noise seems as a band having width 1/6 from saturating swing Vp-p max. However there is invisible part of noise, which is discovered by amplitude detector when we operate with AGC. Automatic limited noise seems as 1/8 width of referenced limits.

[mikebg]

This quiz is: BAD DESIGNED CIRCUIT
Once upon a time, when there was an Eric Foster's PI Technology Forum at "www.insidetheweb.com", I made the following posting:

http://www.geotech1.com/forums/showt...eferrerid=2910

It relates to a bad designed circuit (see here below). The circuit was published by ZETEX, a company producer of transistors having excellent parameters. However if you are designer of excellent transistors, that not means you are able to design excellent applications with them.
Most interesting was, that when this bad designed circuit was published by the company in 90-ties, no need to use medium power transistor to obtain low noise figure because the company ZETEX produces an excellent low noise low power transistor ZTX214C. Despite now this model is obsolete, the circuit diagram shown above in posting #5 outperforms all low noise circuits with modern opamps.
The quiz is:
DO YOU SEE SOMETHING BAD DESIGNED IN THIS LOW NOISE AMPLIFIER?
NOTE: In the subsequent publications of this circuit diagram, ZETEX not shows values of resistors.

[mikebg]

The quiz is:
DO YOU SEE SOMETHING BAD DESIGNED IN THIS LOW NOISE AMPLIFIER?

NOTE: In the subsequent publications of this circuit diagram, ZETEX not shows values of resistors.

ANSWERS

1. NPN or PNP?
In the moment when ZETEX published the bad designed circuit (see posting #9), the company produces two excellent types transistors for low noise applications: ZTX348C and ZTX214C. They have equal parameters because are absolutely equal in design, but are made of different silicon material. Transistor 348 is NPN, and 214 is PNP. What follows from this difference? Transit frequencies are respectively 150kHz and 200kHz, noise figures are 4 dB and 2 dB and hFE are 250 and 350.
HINT: When you design an ultra low noise amplifier, you should use PNP transistors. However the long-tailed pair shown in posting #9 is made of NPN transistors.

2. "Short" or long-tailed pair?
(Term "Long-tailed pair" means difference amplifer. The term "short" is as opposite of it, ie amplifier with only one transistor:-)
The above circuit of differential amplifier contains two noise generating transistors instead one. That means it generates in principle 3 dB more noise than an amplifier with only one transistor.
HINT: When you design an ultra low noise amplifier, don't use long-tailed pair nor yet IC opamp with differential input. The best solution is only one transistor not only for low noise. The differential amplifier drains in operating point twice more current from battery. Even with one transistor, the optimal collector current for minimal noise is enough large.

3. Optimal collector current in operating point.
For minimum noise there is an optimal collector current in operating point Icopt. It is calculated by formula
Icopt=Ut*sqrt[ß/(rs^2+rbb'^2)]
where
Ut is thermal voltage kT/q = 25mV at room temperature,
ß is transistor parameter hFE,
rs is resistance of signal source, and
rbb' is secret transistor parameter "basis resistance". It is secret because producers avoid to publish it for SPICE simulation of noise.
The resistances of resistors in the circuit (posting #9) are so incompetent designed, that operating point of input transistor T1 is below 1 mA:
Ic=0.6V/R2=0.6mA.

4. Filtering in input.
Why input signal should be passed by HF filter C1-R1?
There is an answer: To increase noise generated in input!
Note that noise generating resistance of signal source is only 1 ... 2 ohm. The designer adds to it the noises from R1 and R2 and twice the "secret" transistor parameter rbb' of T1 and T2.
Note that the circuit in posting #1 has no additional noise generating resistors. Only coil resistances generate additional noise to rbb'. Using potcore for L1, its resistance can be made only 4 ohm as shown in posting #4.

[detectoman]

mikebg: i not understand large thing on your hig knolodges you always splain, but i appreciate very much your teaching, my congratulations for your sforces, and good intentions for we, on these fields, all looking your explanations thanks

[mikebg]

mikebg: i not understand large thing on your hig knolodges you always splain, but i appreciate very much your teaching, my congratulations for your sforces, and good intentions for we, on these fields, all looking your explanations thanks

Detectorman, please use machine translator to translate your above posting in your native language.
If you do not understand something, then ask the forum. There are participants in this forum who are not only experts in this field but also know how to explain things simple.
I will try in my next posting (in this thread) to explain visually how a home builder of metal detector project to see if the RF amplifier is designed properly for minimum noise. In this case a builder should have more knowledge than a designer.

[mikebg]

LOW NOISE PREAMP FOR METAL DETECTOR

In order a RX to be sensitive, the received signal from its antenna must be amplified before it to be demodulated. The amplifier that performs this task is marked as RFA (Radio Frequency Amplifier) because amplifies the radiated by TX frequency band.
The RX used in metal detector also needs RFA. The depth of detection depends largely on how much is allowed to gain RFA. However amplification of the AM signals is limited. The gain can be increased only until the signal at the output of the RFA begins to distort due to saturation. Some metal detectors have even an indication that saturation has occurred and operator knows how to adjust gain.
An RX used for metal detecting has in input besides its own thermal noise 3 more unnecessary signals:
-AIR due to the direct transfer of energy from TX to RX,
- GND due to energy reflected from the ground and
- EMI (interference) due to other sources such as power supply network.
These signals are relatively large compared to thermal noise so they limit possible gain of RFA.
The figure shows two cases for the development of RFA.
Above is the conventional block diagram of Front End. In addition to noise source, there are unnecessary sources of EMI, AIR and GND signls. It is shown slightly exceeded the permissible gain because the output begins to notice distortion.
As shown improvement in which manually or automatically compensate AIR and GND signals the entrance of the RX. Manual compensation is used in some mine detecting equipment. The operator has two additional buttons Re and Im (or Mag and Phase).
Since compensation in input allows to increase the allowable gain, it is added to one step LN (Low Noise) preamp as shown below. Displayed is output is situation when gain slightly exceeds the allowable value. It began to limit peaks in the noise signal. Visual impact in this case is that the noise seems like a band without a sharp borders, a width of about one sixth of the eligible scale in output.
CONCLUSIONS:
LN preamp is not needed for a conventional metal detector.
If an synchronous demodulators is properly designed, it has ability to clear the noise because it acts as a lockin amplifier or synchronous correlator.
It is useful an old metal detector to be modifyed with an indicator for overloading of RFA.
The LN preamp needed every metal detector which has the compensation of AIR and GND signal in input.
As compensation allows to increase the gain of the RFA, which increases the depth of detecting, every metal detector should have a manual or automatic compensation in input.
Since the AGC (Automatic Gain Control) allows to work allways with maximum gain, each metal detector should have a slow acting AGC.

[Tinkerer]

Hi Mikebg,

I agree with you that AGC can improve most detector's efficiency.

Detectors need a very wide dynamic range. The AGC can greatly help to achieve that.

Now the question is, how to implement it. There are many dedicated AGC chips available.

For example AD600.

There are also many AGC circuits available, but how do we make a choice between all these?

What are the parameters that matter most for metal detector use?

My guess is, and you mention low a slow acting AGC, would be good enough.

Could you make a specific suggestion? of a circuit? or of a method? or of a class of dedicated IC's?

AGC is probably the single most important improvement that can be made on any detector that does not already have it.

Tinkerer

[mikebg]

Could you make a specific suggestion? of a circuit? or of a method? or of a class of dedicated IC's?

AGC is probably the single most important improvement that can be made on any detector that does not already have it.

Tinkere, My suggestion was made above. See postings #5 and #6.

Follows page 2 of representation:

RESISTANCE OF RX COIL
Resistance of RX coil generates thermal noise. Each resistance connected in series to RX coil increases this noise. The circuit in Fig. 1 is an ideal configuration because there is no additional resistor connected in series to RX coil.
The equivalent circuit diagram of Fig. 1 is shown in Fig. 2. Resistance rbb’ is transistor parameter which can increase significant noise only if it is larger than coil resistance rs. However we can make rs too low despite this increases weight of RX coil.
An ancient method to reduce rbb’ is to connect in parallel two or more transistors as shown in Fig. 3. Now this method is not used because industry produces low noise transistors having rbb’<50 ohm. That means we can design RX coil for resistance 50 ohm as a good value for lightweight and low noise application.
However, if we have an RFA designed with IC NE5534, how to design optimal resistance rs of RX coil?

[mikebg]

Quiz:
Here are three circuit diagrams of RFA with operational amplifier.

Indicate which is the worst circuit configuration for metal detecting.

HINTS:
1. Bad circuit configuration is one that amplifies the EMI signal, ie the circuit amplifies the mains frequency and its harmonics.
2. Bad circuit configuration is one that amplifies unnecessary excessive bandwidth of frequencies. For metal detecting is sufficient bandwdth BW <16Hz.
3. Bad circuit configuration is one that generates a lot of additional thermal noise because contains resistors connected in series to RX coil.
4. Bad circuit configuration is that in which the operating point is less stabile because amplifies DC.
5. Bad circuit configuration is one that amplifies frequencies below 100Hz. In this region there are 1/f (flicker) noise.

[mikebg]

Preamp with only one transistor or with opamp IC?

The formula for spectral density of noise voltage generated by a resistance is u°=rt(4kTR) or expressed as square of voltage u°^2=4kTR.
For room temperature T=300K the formula becomes u°^2=1.6E-20*R.
Since resistance of RX coil generates noise, we can make a table like this:

Coil resistance - Spectral density of noise voltage
10 ohm - 0.405 nV/rtHz
20 ohm - 0.566 nV/rtHz
50 ohm - 0.905 nV/rtHz
100 ohm - 1.28 nV/rtHz
200 ohm - 1.79 nV/rtHz
500 ohm - 2.83 nV/rtHz
1000 ohm - 4.05 nV/rtHz
2000 ohm - 5.66 nV/rtHz

Note that when we increase coil resistance twice, the noise not increases twice, but only sqrt2=1.41 times. The notation u° means RMS value. When we calculate peak-to-peak density of noise voltage u p-p, we should multiply the RMS value by 6 to 8 as was shown visual in other postings.
If we compare with above table the noise density of NE5534 given in data sheet u°=4 nV/rtHz, we will see that this IC adds to input so much noise as a 1000 ohm coil or resistor. However our coil has much lower resistance, for example 50 ohm. Remember that a low noise transistor generates noise with its rbb'<50 ohm.
When we use IC operational amplifier, we should connect to its inverting input a noise generating resistor noted as R1 in the attached circuit diagram. How to calculate resistance of R1?
In ideal case, if the IC is noiseless, we simply can choose R1<rs and problem is solved easy. However with an IC which generates noise as a 1000 ohm resistor, we can use R1 almost 500 ohm and despite R1>rs, this will not increase significant input noise.
If we build differential amplifier with PNP low noise transistors, this will increase current drain and noise in compariso with only one transistor
CONCLUSION: If we can compensate AIR and GND signals in input of RX, it is preferable to add a low noise preamp builded with only one PNP transistor.
The next question is:
How to design the resistance of RX coil for a low noise front end?

[mikebg]

HOW TO DESIGN RESISTANCE OF RX COIL?

Increasing RX coil resistance we increase noise and bandwidth of its tuned LC circuit, but decrease the weight of RX winding and search head becomes more lightweight.
Let we analyse a situation when bandwidth BW depends only on Q factor of tuned RX coil. Above in posting #4 was given the formula for calculating the BW.
/1/ ________ BW=f/Q=r/(2пL)=0,16/T,
where
r and L are parameters of RX coil,
Q=2пfL/r is quality factor of RX coil, Q=2пf*T
and
/2/ ________ T=L/r
is timeconstant of RX coil.
Timeconstant of a coil is design parameter which determines its size, volume and weight (quantity of used metal for wire). From /1/ we obtain formula for calculating of design timeconstant:
/3/ ________T=0,16/BW.
For example, if we need the bandwidth of tuned RX coil to be BW=16Hz, we should design RX coil having design timeconstant T=0,16/BW=0,16/16=0,01s or 10ms. Such coil is not suitable for lightweight sensing head because need more metal, more volume and is too weighty.
From /2/ follows that resistance of RX coil is
/4/ _________ r=L/T.
The analysis shows that at given bandwidth BW, the design timeconstant T is given and we should use maximal possible inductance L to reduce weight of RX coil. An increased L is also good for sensitivity because more turns of RX winding means more voltage induced in RX coil and better S/N ratio. For maximal inductance we should use low equivalent capacitance of RX tank circuit. However in practice, the cable capacitance not allows to reduce equivalent tank capacitance below 900pF. Note that in WHITE'S RX tuned circuit, the capacitance is 15 times lesser than in TESORO RX tuned circuit. If the difference of capacitances was 16 times, we need 4 times more turns for RX coil, ie WHITE'S circuit receives near 4 times stronger RX signal or it has 12dB better S/N ratio if both circuits operate at equal TX frequency.
From calculation follows that we can not achieve necessery BW using only tuned RX coil. We need an additional LC tank having high Q as shown in above postings.
Let's design a RX coil for maximal S/N ratio in input. Because of minimal bandwidth of RX tank, it will generate minimal noise and receive minimum interferences.
DATA FOR CALCULATION
Given is TX frequency f=15kHz. To operate in wide temperatute range, we will increase bandwidth and calculate BW=16Hz for tuned RX coil, despite the target signal has maximum bandwidth about 4Hz. Let we use for calculation equivalent tank capacitance C=1nF as shown in WHITE'S patent for Coinmaster.
Calculation of coil inductance:
At given capacitance C of RX tank, for resonance we need inductance

/5/ ________ L=25,33E-3/(f^2.C)=25,33E-3/(15000^2.1E-9)=0,112H.

This is too large value despite the TX frequency is relative high.
Calculation of coil resistance:
From /1/ we obtain
/6/ _______ r=2п.BW.L=6,28.16.0,112= 11,2ohm.
CONCLUSIONS:
A) We calculated too small coil resistance for such large inductance 112mH, that means the RX coil will be too heavy. We should calculate the weight of metal and if coil is too weighty, we should increase bandwidth.
B) We calculated too small coil resistance relative to large noise resistance rbb'=50 ohm of used transistor.
C) From formula /3/ follows that design timeconstant of RX coil is independent on TX frequency and depends only on BW of tuned RX coil.
D) We need additional measures to realise lightweight RX coil for narrow band metal detector. One measure is to connect RX coil to Q-multiplier circuit, but this increases noise. Other measure is to connect in RFA an additional tuned LC circuit with high Q, for example using coil with potcore as shown in above postings.

At end of this analysis I should remember again that all circuit diagrams and calculations made in this thread are not so important nor suitable for a conventional metal detector for two resons:

1. The narrow bandwidth in input tank means steep phase characteristic. That causes not stabile discrimination and GND elimination of conventional metal detector, because reference voltage for synchronous demodulation is not extracted from RX signal, but from voltage across TX coil. For best results should be used the GND signal as reference.
2. The sensitivity of conventional metal detector is not limited by noise, but by large AIR and GND signals in input of RFA. To achieve maximal sensitivity, each metal detector should have two automatic controls: AGC and ABC, or 3 manual controls as shown in posting #7.
Two manual controls noted there as MAGNITUDE and PHASE, may be noted different as X and Y or as compensation of Real and Imaginery component.