# Wideband transformers III (practical applications)

As a last article about designing transmission-line transformers I present a practical example aiming at adapting a whip antenna to a 50ohm/100W system using a Ruthroff 1:9 transformer and a common-mode RF choke. I have chosen the well known Amidon FT140-43 core  for transformer and FT240-43 for RF choke.

Ruthroff 1:9

1. Core parameters taken from vendor’s catalog for FT140-43

 $l_e(cm)$ $A_e(cm^2)$ $V_e(cm^3)$ 9.02 0.807 7.280

Now we could calculate form factor F:

$F=\mu_0 \frac{A_e}{l_e}=1.12*10^{-9}$

2. Material parameters:

 $\mu_i$ $A_L(mH/1000turns)$ $B_{sat} (Gauss)$ 850 2750 952

Complex permeability is given by a plot of $\mu'$ and $\mu''$ versus frequency. Then we calculate $\mu_c$ with formula $\mu_c=\sqrt{\mu'^2 + \mu''^2}$ and $Q=\frac{\mu'}{\mu''}$

 $f(mhz)$ 1.5 4 7 10 15 20 30 40 50 $\mu'$ 600 400 310 270 200 140 95 65 48 $\mu''$ 170 280 270 250 210 200 170 140 120 $\mu_c$ 624 488 411 368 290 244 195 154 129 $Q$ 3.53 1.43 1.15 1.08 0.95 0.7 0.56 0.46 0.4

Amidon offers a table with maximum inductance (flux density) versus frequency:

 $f(mhz)$ 0.1 1 7 14 21 28 $B_{max} (Gauss)$ 500 150 57 42 36 30

The equation for the curve is:

$logB_{max}=-0.48299logf+2.17609$ which allows us to calculate the induction at any missing frequency in the table:

 $f(mhz)$ 0.1 1 1.8 3.5 7 14 21 28 $B_{max} (Gauss)$ 500 150 113 82 57 42 36 30

3. Calculating impedance

Using the 4x rule we could calculate the impedance for the low impedance winding:

Z=4*50ohm =200ohm

4. Calculating inductance at lowest frequency of operation (1.8mhz)

$L=\frac{Z}{2\pi f}=17.7\mu H$ the the number of turns $n=1000*\sqrt{\frac{L}{A_L}}=5$

5. Calculating maximum inductance (flux density) at lowest frequency

$E_{RMS}=\sqrt{PZin^2}=71V$ then $B_{max}=\frac{E_{RMS}*10^8}{4.44 n A_e f}=219Gauss$

We see that this value is above the maximum value recomended by the vendor and therefore we need to increase the number of turns. Therefore we will use 7turns.

With this we could now calculate inductance, reactance, loss and modulus of impedance:

$L=n^2 \mu' F$ $X_L=\omega n^2 \mu' F$ $r_f = \omega n^2 \mu'' F$ $Z_c = \omega n^2 \mu_c F$

6. Calculating maximum voltage due to dissipation

We will allow maximum raise in temperature of $\Delta T=30$ Celsius, thus we calculate:

$P_{max}=\Delta T*a*\sqrt{V_e}= 3.65W$ and the maximum voltage allowed for maximum dissipation:

$U_{dissipation}=\sqrt {K*P_{max}(Q/6+1/Q)X_L}$

The above formula is for continue power and could be improved in the case of manual keying and depending on the modulation type. For SSB the multiplying factor is K=3.2.

7. Calculating maximum voltage due to induction

With the new chosen number of turns (7 turns) we could now calculate the maximum allowed voltage due to induction.

$U_{induction}=4.44B_{max} n A_e f$

All calculations are presented in the table below.

 f(mhz) n L(uH) XL(ohm) rf(ohm) Zc(ohm) ULdiss(volt) ULind(volt) 1 7 41.16 258.62 22.41 259.59 80.32 37.58 2 7 32.93 413.79 117.24 430.07 66.91 53.77 4 7 24.70 620.68 282.75 682.05 79.55 76.94 7 7 17.01 748.26 555.16 931.72 94.75 102.76 10 7 13.72 862.05 724.12 1125.83 105.41 123.57 15 7 10.98 1034.46 1034.46 1462.95 122.40 152.39 20 7 8.23 1034.46 1241.36 1615.88 131.12 176.83 30 7 4.94 931.02 1551.70 1809.57 142.89 218.07 50 7 2.47 775.85 1810.31 1969.56 152.18 283.98

It is easily seen that lines corresponding to first two frequencies in the table gives maximum voltage due to induction less than the voltage for maximum power (71V) and therefore we should increase the number of turns further or use a bigger core.

8. Calculating characteristic impedance

Remember that for Ruthroff 1:9 we calculated optimum impedance of the upper transmission-line as being 150ohm and for lower transmission-line 75ohm. Let’s see how close we are if we use closely coil three wires on the core.

$Z_{characteristic}=\frac {276}{\sqrt{\epsilon}} lg \frac{2D}{d}$

The calculation with various isolation gives the following result:

 PVC Teflon(PTFE) Polyethilene Enamel epsilon 3.20 2.10 2.20 5.10 D 0.90 0.90 0.90 1.67 mm d 0.50 0.50 0.50 1.64 mm Zch 85.93 106.07 103.63 37.88 ohm

We could see that there is a mismatch between the optimum characteristic impedance and the impedance of the cables.

An Excel file with all calculations in this article could be found in wideband_transformers.xls.

This transformer has been realized in practice and used extensively on the field with a whip antenna.