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: