# 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.

# Wideband transformers II (types)

In the previous post I derived formulas that could be used to calculate inductors on ferrite and iron powder cores. In this chapter I present various types of transmission-line transformers. The goal in designing a good transmission-line transformer is to make it usable in matching generator with the load over a broad range of frequencies, therefore making it a wideband transformer.

From the functioning principle point of view there are two main categories of transformers:

– transformers based on induction in the core.

– transformers based on transmission line techniques that use the core only to isolate the output from input.

Induction transformer

One could employ a regular transformer design with primary winding and secondary winding on a ferrite core so that the impedance is matched between the generator and the load but it turns out that the more the number of turns is, the more parasitic capacities are and therefore the cutoff frequency for the transformer is lowered. A core with high permeability could be used to reduce the number of turns and therefore to reduce the parasitic capacities.

Since this type of transformer is based on the magnetic flux inside the core, any leakage in the flux will be seen as a loss. Coupling between primary winding and secondary winding is much more important and one could use different winding techniques in order to make the coupling better. For example primary and secondary winding could be put together when rolled on the core.

Magnetic Longwire Balun

This is a transformer made by RF-systems in 1988 and registered to that name (MLB). They claim to use a special ferrite to obtain a broad frequency range (0.1 to 40Mhz) in matching a longwire antenna. It is actually a 1:3 voltage UnUn.

Current induction transformer

This type of induction transformer is based on principle of two currents producing same value of induction in the core but with oposite directions so that inductions will cancel each other. This is exactly what happens inside a transmission line and therefore it is also known as a transmission-line transformer.

Transformers based on transmission lines

Model of a tranmission line

Inside a transmission line the electromagnetic fields of the two conductors cancels each other so that there is no radiating outside. Therefore it is not important for the differential electromagnetic field between the two conductors what is the form of the transmission line. However if we coil up the transmission line on a core we will make a much bigger impedance for the common-mode currents, allowing us to isolate the input from the output, thus allowing us to connect input to output in various ways. This method is used in designing transmission-line transformers and a model of transmission-line in which common-mode impedance $Z_{sleeve}$ is seen is presented below:

Therefore if we have a decoupling between input and output then we have both conductors of the transmission line floating at the end,  we could feed a symetrical load like the following, making a BalUn:

This design was called by Guanella (see below) Basic building block.

The 4x rule

We will see that for some designs the equivalent $Z_{sleeve}$ impedance will be connected in parallel with the generator. In order to minimize its influence on the generator impedance, we will use the rule saying that $Z_{sleeve-equivalent}$ will be four times the $Z_{in}$ impedance:

$Z_{sleeve-equivalent}=4*Z_{in}$

Characteristic impedance

If we have a transmission line with $Z_{in}$ – generator imput and $Z_L$ impedance load, we could calculate a maximum power transfer between the generator and the load when characteristic impedance is:

$Z_c=\sqrt{Z_{in} Z_L}$

Guanella and Ruthroff designs

These designs uses the following rules:

–  Currents in the coupled conductors of a transmission-line are equal in magnitude but opposite in phase (total coupling rule).

–  For every transmission-line the output voltage will be identical (magnitude and phase) to the input voltage (no loss rule).

– There is no other connection between the transformer input and output other than through the transmission-line (total decoupling rule).

In 1944 Guanella extended the previous design in an article called ‘Novel Matching Systems for High Frequencies‘ by adding two transmission-lines fed by the generator whose output are connected in series
to double the voltage(1:2 voltage, 1:4 impedance):

In 1959 Ruthroff analysed the design of Guanella and came up with an article called ‘Some broadband transformers’ in which the output of the transmission line is added to the input to produce a double voltage (1:2 voltage, 1:4 impedance)

Since the output is partially added to the input ( Bootstrap effect), this kind of design is more prone to errors when there is a unbalance in the load.

The various designs based on Guanella and Ruthroff will be discussed further on in this chapter.

Guanella 1:4 impedance

The following pictures presents the Guanella 1:4 UnUn and BalUn respectively together with relevant currents and voltages:

For both Guanella UnUn and Balun we could calculate:

$Z_{in}=\frac{u/2i}{2u/i}*Z_L=\frac{Z_L}{4}$ $Z_c=\frac{u/i}{2u/i}*Z_L=\frac{Z_L}{2}$

It can be seen that $Z_{sleeve}$ is at potential so that it become in parallel with the generator.

$Z_{sleeve}=4*Z_{in}=Z_L$ – using the rule of 4x

It is interesting that the $Z_{sleeve}$ is given by the upper transmission-line since the lower transmission-line will have no differential voltage between its ends. The two windings could be done also on the same core.

Example:

$Z_{in}=50 ohm$ and $Z_L=200 ohm$

$Z_c=100 ohm$ $Z_{sleeve} = 200 ohm$

Ruthroff 1:4 impedance

The following pictures presents the Ruthroff 1:4 UnUn and BalUn respectively together with the relevant currents and voltages:

For both Ruthroff UnUn and BalUn we could calculate:

$Z_{in}=\frac{u/2i}{2u/i}*Z_L=\frac{Z_L}{4}$ $Z_c=\frac{u/i}{2u/i}*Z_L=\frac{Z_L}{2}$

It can be seen that $Z_{sleeve}$ is at potential so that it become in parallel with the generator.

$Z_{sleeve}=4*Z_{in}=Z_L$ – using the rule of 4x

Example:

$Z_{in}=50 ohm$ and $Z_L=200 ohm$

$Z_c=100 ohm$ $Z_{sleeve}=200 ohm$

Guanella 1:9 impedance

The following pictures presents the Guanella 1:9 UnUn together with relevant currents and voltages:

$Z_{in}=\frac{u/2i}{3u/i}*Z_L=\frac{Z_L}{9}$

$Z_c=\frac{u/i}{3u/i}*Z_L=\frac{Z_L}{3}$

It can be seen that Z across input is $2*Z_{sleeve}$ in parallel with $Z_{sleeve}$ giving $\frac{2Z_{sleeve}}{3}$ thus

$\frac{2*Z_{sleeve}}{3}=4*Z_{in}=\frac{4}{9}Z_L$ – using the rule of 4x, thus:

$Z_{sleeve}=6*Z_{in}=\frac{2}{3} Z_L$

Example:

$Z_{in}=50 ohm$ and $Z_L=450 ohm$

$Z_c=150 ohm$ $Z_{sleeve}=300 ohm$

Ruthroff 1:9 impedance

The following pictures presents the Ruthroff 1:9 UnUn together with relevant currents and voltages:

$Z_{in}=\frac{u/3i}{3u/i}=\frac{Z_L}{9}$

For the $Z_c$ we observe that the two lines have different characteristic impedance:

$Z_{c-upper}=\frac{u/i}{3u/i}*Z_L=\frac{Z_L}{3}$ $Z_{c-lower}=\frac{u/2i}{3u/i}*Z_L=\frac{Z_L}{6}$

For optimum transfer we need transmission lines with different characteristic impedance.

$Z_{sleeve}=4*Z_{in}$

What we could see from the picture is that the outer conductor of the two transmission-lines are connected together, therefore they could be put on the same core.

Example:

$Z_{in}=50 ohm$ and $Z_L=450 ohm$

$Z_{c-upper}=150 ohm$ $Z_{c-lower}=75 ohm$ $Z_{sleeve}=200 ohm$

Models using winding impedance:

When constructing Guanella transmission-line transformers using coils, the following is the equivalence with the transmission-line version (only BalUn is presented):

When Ruthroff transmission-line transformers are built using coils, the following is the equivalence with the transmission-line version (only BalUn is prezented):

Modeling characteristic impedance:

When making transmission-line transformers and not using transmission lines but cables either twisted or not, it is hard to make characteristic impedance resulted from the above formulas. One way to know the characteristic impedance is to calculate it with the following formula which gives the impedance of a bifilar cable:

$Z_c=\frac{276}{\sqrt{\epsilon}}lg\frac{2D}{d}=\frac{120}{\sqrt{\epsilon}}ln\frac{2D}{d}$ where

D – distance between center of the cables

d – diameter of the cable

$\epsilon$ – permeability of the cable dielectric (we’ll discuss it in the next chapter).

If D become larger then the characteristic impedance will raise. Therefore one can improve impedance matching by using isolator tubes around winding cables.

# Wideband transformers I (magnetic core theory)

I write a series of articles about transmission-line transformers with the intention to eventually build a UnUn and current choke for a whip antenna.

The first article will present the theory of magnetic core used in transmission-line transformers, the second article will present transmission-line transformer types together with their characteristics while the third article will present practical calculation of the UnUn and current choke using easy to find materials.

Transmission-line transformers theory:

A transmission-line transformer aims at adapting from a transmission line with one impedance to a transmission line with another impedance.

Generally there are two types of materials that are largely used for constructing transmission-line transformers:

-ferrite (MnZn composition, NiZn composition)

– iron-powder (electrolytic powder-iron, carbonyl powder-iron)

Each of the materials presented above have different proprieties that will recommend it for a specific type of application. I will present the differences when this is necessary. Magnetic materials have couple of parameters among which some of interest are:

$\mu$ – permitivity

$A_l$ – inductance factor

$f_r$ – resonance frequency

$T_{co}$ – temperature coeficient

$T_{max}$ – maximum temperature

$B_{sat}$ – saturation induction (flux density)

$A_e$ – effective area

Permeability, magnetic field, magnetic induction:

Magnetic field

The ampere law says:

$H=\frac{ni}{l}$ where

n – number of turns

i – current [A]

l – current path[m]

Magnetic induction (flux density)

$B=\mu H$ where:

H – magnetic field [A/m]

$\mu$ – permeability [Henry/m]

B – induction (flux density) [Tesla, Gauss]

The dependency B=B(H) is usually plotted like in the following image:

$B_r$ – remanent induction (induction in the material at zero magnetic field).

$H_c$ – coercitive magnetic field (magnetic field at zero induction.

$B_s$ saturation induction (maximum induction in the material).

Inductance

$L=\frac{\mu n^2 A_e}{l}$ where

$A_e$ – area of a turn

Ferrite producers offer the inductance coefficient $A_L$ which could be used to calculate inductance L:

$L=n^2 A_L$ where $A_l$ is in $H/sp^2$

From this formula one could calculate the number of turns given the inductance L and $A_L$:

$n=\sqrt\frac{L}{A_L}$

One important note regarding the $A_L$ is that it is normally calculated at low frequency but since it is dependent of permeability $\mu$ we should not expect it to be constant at all frequencies. I will discuss dependency of $\mu$ versus frequency in more details later in this post.

Calculation of maximum inductance in the core

$e=n\frac{d\phi}{dt}=nA_e\frac{dB}{dt}$ therefore integrating on both sides for half period:

$B_{max}=\frac{E_{med}}{4nA_ef }$

We see that this depends on the frequency, number of turns, effective area of core.

We can derive the formula for couple of signal types:

Sinusoidal signal:

$B_{max}=\frac{E_{RMS}10^8}{2\pi\sqrt{2}nA_ef}=\frac{E_{RMS}10^8}{4.44nA_ef}$

Square signal with D duty cycle:

$B_{max}=\frac{DE_{pk}10^8}{nA_ef}$ therefore for 50% duty cycle:

$B_{max}=\frac{E_{pk}10^8}{nA_ef}$

Pulse signal with $T_{on}$ and frequency f:

$B_{max}=B_r+\frac{T_{on}E_{pk}10^8}{nA_ef}$ where

$B_r$ – remanent induction in Gauss

$A_e$ – effective area in $cm^2$

$f$ – frequency in Hz

$B_{max}$ – induction in Gauss

If the signal has a DC component then another component is added to $B_{max}$:

$B_{max}=B_{AC}+B_{DC}$ with $B_{AC}$ from the above formula and

$B_{DC}=\frac{Li}{nA_e}$

Some ferrite vendors offer a plot or a table of maximum inductance versus frequency. In order to verify that core is suitable for the application, one should calculate $B_{max}$ at minimum frequency and compare with the value in the table offered by the vendor.  If $B_{max}$ is larger than the value in the table then a core with a larger $A_e$ should be chosen or the number of turns should be increased.

Permeability at higher frequencies

$A_l$ is offered normally at lower frequency and is supposed to not be dependent of the frequency. However, at higher frequencies the $\mu$ is frequency dependent and has a complex form like this:

$\mu=\mu'-j\mu''$

therefore we  could calculate the impedance:

$Z_L=j\omega L=\frac{\omega n^2A_e\mu''\mu_0}{l}+j\frac{\omega n^2\mu'\mu_0A_e}{l}=r_f+jX_L$ where

$r_f=\frac{\omega n^2A_e\mu''\mu_0}{l}$ is called inductor loss

$X_L=\frac{\omega n^2\mu'\mu_0A_e}{l}$ is the core reactance

Here we will define:

$tg\delta=\frac{\mu''}{\mu'}$ – tangent of loss angle

$Q=\frac{X_L}{r_f}=\frac{\mu'}{\mu''}=\frac{1}{tg\delta}$ – quality factor of the core

We could now define the impedance taking in consideration the complex permeability:

$Z_c=\sqrt{r_f^2+X_L^2}=\omega n^2 \mu_0\frac{A_e}{l}\sqrt{\mu'^2+\mu''^2}=\omega n^2 \mu_0\frac{A_e}{l}\mu_c$

where $\mu_c=\sqrt{\mu'^2+\mu''^2}$            (1)

Some ferrite vendors offer graphics for $\mu'$ and $\mu''$ versus frequency. From these tables one could calculate $\mu_c$ and $Z_c$.

Inductance at higher frequencies

We could now derive the inductance and impedance of a core using the complex permeability and see how this could be calculated using the information that is usually provided by the core vendors. We will try to obtain another factor that is not dependent of frequency.

$L=\frac{\mu n^2 A_e}{l}=\mu_0\mu_c \frac{n^2 A_e}{l}=\mu_c n^2 F$ where $F=\mu_0\frac{A_e}{l}$ is a form factor not frequency dependent.

$F=\frac{A_L}{\mu_i}$ – F could be calculated from $A_L$ and initial permeability $\mu_i$ (usually offered by yhe vendor) or

$F=\mu_0\frac{A_e}{l}$ – directly from the dimensions of the core.

Now we could calculate magnitude of complex impedance of the winding:

$X_L=\omega n^2 \mu' F$ $r_f=\omega n^2 \mu'' F$ $Z_c=|r_f+jX_L|=\omega n^2 \mu_c F$

Power loss in ferrite

Thermal resistance:

Increase in temperature can be calculated as follows $\Delta T=P*R_{th}$ where $R_{th}$ is thermal resistance of the core.

$R_{th}=\frac{1}{a\sqrt{volume}}$ with an scaling factor. Therefore we could calculate maximum power dissipation if we know scalling factor ‘a':

$P_{max}=\Delta T*a*\sqrt{V}$. In this formula V is the volume of the core.

Some ferrite and iron powder vendors offer a formula to calculate temperature raise from max power dissipation and input power . We’ll use that in the last article when practically calculating a UnUn transformer.

Power dissipation:

We determined maximum dissipated power for a specific raise in temperature. Now we can calculate the maximum allowed voltage to produce the maximum power dissipation.

$P_{max}=\frac{U_L^2}{Z_c^2} r_f=\frac{U_L^2 r_f}{r_f^2+X_L^2}=\frac{U_L^2}{(Q+1/Q)X_L}$ therefore:

$U_{Ldisippation}=\sqrt{P_{max}(Q+1/Q)X_L}$

We have now two constrains for the maximum allowed voltage, one from maximum dissipated power and the other from maximum allowed induction:

$U_{Linduction}=4.44B_{max}nA_ef$

Which one to choose between $U_{Ldissipation}$ and $U_{Linduction}$? The smallest one in order to be on the safe side.