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The Maths Behind

Transformer Design

Rule of Thumb: For pulse transformers use the minimum number of turns possible while avoiding saturation of the core.

Observation: For the average power levels associated with this project the transformer can be expected to be physically small.  For comparison think about the miniature audio matching transformers rated at about 0.5 W less than 1 cm cu, and switch mode power supplies with potted transformers with a total volume less than 2 cm cu rated at 12 W continuous with an operating frequency of around 50 kHz.

For pulse shape preservation we want high frequency performance (well above the 40 kHz resonant frequency for harmonics) therefore a ferrite core is desirable.

Toroids almost totally contain their magnetic fields, limiting coupling to other components. This is desirable.

As the coupling coefficient of the windings in a transformer approaches 1 the primary winding inductance approaches the magnetizing inductance, and the secondary inductance approaches the primary inductance multiplied by the square of the turns ratio. Therefore with tight coupling the inductance seen from the secondary will be close to the inductance of the secondary winding.

Too much inductance is preferable to too little. It is easier to add additional capacitance in parallel with the transducer than additional inductance to the transformer.

Limitation.

The most readily available ferrite toroidal core was an 18 mm diameter L8 core from Jaycar Electronics made by Electus Distribution.   The core has the following data sheet characteristics:

    Ur, Initial Permeability = 1500
    Tan d/u, Relative Loss Factor <2.8E-5 @ 0.3 MHz
    Bs, Saturation Flux Density = 2,550 Gauss
    Br, Residual Flux Density = 1,225 Gauss
    Hc, Coercive Force = 0.225 Oe
    Tc, Curie Temperature >120°C
    DF, Disaccommodation Factor = 11.8E-3
    d, Density > 5 g/cm cu
    p, Resistivity = 19 Meg ohm cm

The core dimensions were:

    Cross sectional area = 6.0 mm high x 4.0 mm wide = 24 sq mm
    Core inner diameter = 10 mm
    Core outer diameter = 18 mm

The sheet current inductance of a ferrite cored toroidal transformer is given by:

    L = 0.0046.u.N2.h.log(r2/r1) uH

where

    r1 = inner radius
    r2 = outer radius
    h = axial depth of winding in inches
    N = number of turns
    u = permeability of the core material

This simplifies to L = k N2.  By winding a few turns on the core and measuring the inductance k turns out to be 1.75 for the L8 core where L is in uH.

The transformer inductive reactance needs to equal the static capacitive reactance of the transducer at 40 kHz.  Cs = 1.8 nF.

So   L  = 1/((2*π*f)2.C)

       = 1/((2*π*40E3) 2*1.8E-9)

       = 8.8 mH

Using L = k N2, k = 1.75 we can determine the secondary number of turns.

       N  = (8.8E3/1.75)0.5

       = 71 turns

The desired drive voltage at the secondary is 140 V ppk. With a bifilar primary winding switched from a 6 V supply, and allowing 0.5 V drop for winding and MOSFET drain-source resistance in each of the bifilar primary’s, this gives an effective primary voltage of 11 V.

For a transformer the relationship between voltage and turns ratio is:

    Np/Ns = Vp/Vs

where Np and Ns are the number of primary and secondary turns, and Vp and Vs are the primary and secondary voltages respectively.

Solving for the primary turns:

    Np = Ns*Vp/Vs

           = 71*11/140

                 = 5.5

           ~ 5 (rounded down because by test circuit actually has a 5 V rail)

Now would the toroidal core saturate?  The fundamental transformer equation is:

    Bmax = Erms*1E8/(4.44*f*Np*Ae)

where Erms is the primary voltage, f is the frequency in cycles per second A is the core cross-sectional area in sq inches, and Bmax is the maximum flux density in Gauss.  (A latent memory from my school days suggests that the constant 4.44 should be 4 for a square wave drive but I can not substantiate this so I’ll run with 4.44.)

      = 5.5*1E8 / (4.44*40E3*10*(0.6*0.4))

      =  1,290 Gauss

The core should not saturate as the Bs is 2,550 Gauss and there should be no DC through the windings.

I have not considered the power rating of the core.  Both the average and peak power in this application are relatively low so that almost any core that I could physically wind would be expected to perform adequately. This was confirmed during testing. The core temperature did not rise perceivably above ambient temperature.

 

Transducer Range (in Air)

Transducer Specifications:

    Transmit SPL = 105 dB at 40 kHz referenced to 0dB = 0.0002 ubar at 30 cm.
    Receiver Sensitivity = -74 dB at 40 kHz referenced to 1 V/ubar.

SPL reduction at 1 m = 20 log (0.3/1.0)

           = -10.6 dB

Gain due to 70 V rms drive = 20 log ( 70 V rms / 10 V rms)

             = 16.9 dB

SPL at 1 m = 16.9 dB - 10.5 dB + 105 dB

                   = 111.4 dB

Equivalent pressure at 1 m = 10 111.4/20*0.0002

            = 74.3 ubar

Received voltage (assumed to be into open circuit because impedance levels were not specified for the transducer):

    Vr = 10-74 dB/20*1 V/ubar * 74.3 ubar

         = 14.9 mV

Measured received voltage ~ 17 mV at a distance of 1 m in air.

At a distance of 1.65 m the calculated receiver voltage was a meager 9 mV and the measured received voltage was fractionally over 10 mV. The signal level was not a problem but the broad band receiver noise (about 8 mV) potentially was.

Note that the transmission characteristics in water are expected to be significantly better than this because air is elastic and absorbs sound energy more readily than water.

 

Signal Analysis

The transmitted signal at the input to the transducer is an ultrasonic pulse with a rectangular envelope (see top waveform in the Figure below). The frequency spectrum envelope of this modulated pulse is a comb of delta pulses within a sinc function ( sin(x)/x ) envelope, centred about the carrier frequency (see centre illustration in the Figure below for the envelope).  The bandwidth of the transmit and receive transducers are relatively narrow about the carrier frequency, thus the higher and lower frequency components of the ultrasonic pulse are lost, giving rise to the signal shape measured here (see lower illustration in the Figure below).   If broadband noise and interference is introduced in the communications channel then some of this can be rejected by limiting the receiver amplifier bandwidth to only the frequencies that we expect to receive.  Therefore a bandpass filter response at the carrier frequency (40.3 kHz) with a pass band of approximately 1/(twice the pulse width) should reduce the reception of noise from brief high intensity sources, subject to the transient response of the filter.

 

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