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

From Faraday law of EMF:

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.

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