# Common Mode Analysis of Ethernet Transformers

In this letter, a distributed model for Ethernet transformers subject to common mode signals is developed. Explicit formulas for transfer functions are derived and illustrated by numerical examples.

In this letter, a distributed model for Ethernet transformers subject to common mode signals is developed. Explicit formulas for transfer functions are derived and illustrated by numerical examples.

SECTION I

Modern computer Ethernet networks universally use wideband ferrite core Ethernet transformers as interfacing elements. These Ethernet transformers are designed to filter out common mode (noise signals) and transmit with minimal distortions differential mode (information carrier) signals in the frequency range from 0.1 MHz to 1 GHz [Pulse 1999]. To provide this distortion-free transmission, the performance of Ethernet transformers must mimic the performance of an ideal transformer, which has flat transfer characteristics for differential mode signals. This requires the minimization of leakage inductances of primary and secondary windings, which can be achieved when these windings are wound together (in bifilar manner) around ferrite cores (see Fig. 1). This close proximity of the primary and secondary windings results in appreciable cross-winding capacitance that serves as a channel for common mode (noise) signals. To suppress this channel, the midpoints of the primary and secondary windings are grounded (see Fig. 2). This results in only low-inductive impedances of the primary and secondary windings to common mode signals. It is well known that the differential mode has odd symmetry with respect to the permutation of connecting wires, while the common mode has even symmetry. The different mode symmetries require different treatment. The differential mode analysis and testing of Ethernet transformers was previously presented in Bowen et al. [2009a, 2009b, 2010] (see also Ribbenfjärd [2007]). The purpose of this letter is to develop the common mode analysis of Ethernet transformers. Since the primary and secondary windings of Ethernet transformers are closely intertwined, the cross-winding and intra-winding capacitances are distributed in nature and will require distributed analysis. In the letter, a novel distributed model of Ethernet transformers for common mode signals is developed and explicit formulas for transfer functions are derived and numerically illustrated.

SECTION II

We shall first derive the differential equations for the distributed model and then we shall formulate the boundary conditions. To this end, consider an infinitesimally small element of coupled primary and secondary windings shown in Fig. 3. It is apparent that magnetic coupling between these windings due to their ferrite core is very weak and practically nonexistent. This is because (see Fig. 2) the common mode current in each winding flows in opposite directions toward tapped and grounded middle points and create magnetic fields, which cancel out one another. For this reason, these windings are mostly coupled through distributed cross-winding capacitances. By taking this fact into account and by applying Kirchhoff's voltage law and Kirchhoff's current law to the infinitesimal element shown in Fig. 3, we derive TeX Source $$\eqalignno{{{d\hat v_1 (y)}\over {dy}} &= Z\hat i_1 (y)&{\hbox{(1)}}\cr{{d\hat v_2 (y)}\over {dy}} &= Z\hat i_2 (y)&{\hbox{(2)}}\cr{{d\hat i_1 (y)}\over {dy}} &= - {{d\hat i_2 (y)}\over {dy}} = Y\left[{\hat v_1 (y) - \hat v_2 (y)} \right]&{\hbox{(3)}}}$$where TeX Source $$ Y = j\omega \tilde C. \eqno{\hbox{(4)}} $$

Here, the adopted notations have the following meaning: and are voltage phasors along the primary and secondary windings, respectively, and are current phasors along the same windings, *Z* is a per unit length impedance of the primary (and secondary) winding consisting of per unit length resistance in series with leakage inductance which are in parallel with per unit length intra-winding capacitance, is the per unit length cross-winding capacitance.

Next, we shall formulate the boundary conditions for and . To this end, we shall consider the “upper” half of the winding
TeX Source
$$0 \le y \le {l\over 2}\eqno{\hbox{(5)}}$$where *l* is the winding length. Voltages and currents can be extended to the “lower” half of the winding (− *l*/2) ≤ *y* ≤ 0 as even and odd functions, respectively. It is apparent from Fig. 2 and the aforementioned symmetry that the currents through the grounding impedances are twice of and , respectively. For this reason
TeX Source
$$\eqalignno{\hat v_1 (0) &= 2Z_g \hat i_1 (0)&{\hbox{(6)}}\cr\hat v_2 (0) &= 2Z_g \hat i_2 (0)&{\hbox{(7)}}}$$where *Z*_{g} is the impedance between the ground and midpoint of primary (or secondary) windings. By using differential equations (1) and (2), we can formulate the boundary conditions (6) and (7) only in terms of voltages
TeX Source
$$\eqalignno{\hat v_1 (0) &= {{2Z_g }\over Z}{{d\hat v_1 }\over {dy}}(0)&{\hbox{(8)}}\cr\hat v_2 (0) &= {{2Z_g }\over Z}{{d\hat v_2 }\over {dy}}(0).&{\hbox{(9)}}\cr}$$

Finally, at the terminals of the primary and secondary windings, we shall use the following boundary conditions, respectively, TeX Source $$\eqalignno{\hat v_1 \left({{l\over 2}} \right) &= \hat V_1&{\hbox{(10)}}\cr{{d\hat v_2 }\over {dy}}\left({{l\over 2}} \right) &= 0&{\hbox{(11)}}}$$where is the common mode terminal voltage, while the boundary condition (11) follows from the fact that the terminals of the secondary windings are assumed to be open.

Our goal is to derive an explicit analytical expression for the transfer relation , where stands for the terminal voltage of the secondary winding. To start the derivation, we shall first observe that according to (3), we have TeX Source $$\hat i_1 (y) + \hat i_2 (y) = \hat I = {\rm const}{\rm .}\eqno{\hbox{(12)}}$$

Next, by adding (1) and (2) and then integrating, according to (12), we find TeX Source $$\hat v_1 (y) + \hat v_2 (y) = Z\hat Iy + {\rm const}{\rm .}\eqno{\hbox{(13)}}$$

This implies that TeX Source $$\hat v_1 (0) + \hat v_2 (0) = {\rm const}.\eqno{\hbox{(14)}}$$

Now, by using boundary conditions (6) and (7) as well as (12), we find TeX Source $${\rm const} = 2Z_g \hat I.\eqno{\hbox{(15)}}$$

By substituting (15) into (13), we end up with TeX Source $$\hat v_1 (y) + \hat v_2 (y) = \left({Zy + 2Z_g } \right)\hat I.\eqno{\hbox{(16)}}$$

Next, from (1) and (2), we obtain TeX Source $${{d\left[{\hat v_1 (y) - \hat v_2 (y)} \right]}\over {dy}} = Z\left[{\hat i_1 (y) - \hat i_2 (y)} \right].\eqno{\hbox{(17)}}$$

By differentiating (17) and taking into account (3), we arrive at TeX Source $${{d^2 \left[{\hat v_1 (y) - \hat v_2 (y)} \right]}\over {dy^2 }} = 2YZ\left[{\hat v_1 (y) - \hat v_2 (y)} \right].\eqno{\hbox{(18)}}$$

A general solution of (18) can be written in the form TeX Source $$\hat v_1 (y) - \hat v_2 (y) = Ae^{\gamma y} + Be^{ - \gamma y}\eqno{\hbox{(19)}}$$where TeX Source $$\gamma ^2 = 2YZ.\eqno{\hbox{(20)}}$$

Now, by using the boundary conditions (8) and (9), we find TeX Source $$A + B = {{2Z_g \gamma }\over Z}(A - B)\eqno{\hbox{(21)}}$$which leads to TeX Source $$B = \left({{{2Z_g \gamma - Z}\over {2Z_g \gamma + Z}}} \right)A.\eqno{\hbox{(22)}}$$

By substituting (22) into (19), and taking into account that TeX Source $$e^{ \pm \gamma y} = \cosh (\gamma y) \pm \sinh (\gamma y)\eqno{\hbox{(23)}}$$we derive TeX Source $$ \hat v_1 (y) - \hat v_2 (y) = A{{4Z_g \gamma \cosh (\gamma y) + 2Z\sinh (\gamma y)}\over {2Z_g \gamma + Z}}.\eqno{\hbox{(24)}}$$

By adding and subtracting (16) and (24), we find TeX Source $$\eqalignno{ \hat v_1 (y) &= \left({{{Zy + 2Z_g }\over 2}} \right)\hat I + A{{2Z_g \gamma \cosh (\gamma y) + Z\sinh (\gamma y)}\over {2Z_g \gamma + Z}} \qquad&{\hbox{(25)}}\cr \hat v_2 (y) &= \left({{{Zy + 2Z_g }\over 2}} \right)\hat I - A{{2Z_g \gamma \cosh (\gamma y) + Z\sinh (\gamma y)}\over {2Z_g \gamma + Z}}.&{\hbox{(26)}}}$$

Now, we can express *A* in terms of . To this end, we substitute (26) into the boundary condition (11) and obtain
TeX Source
$$\left({{Z\over 2}} \right)\hat I - A{{2\gamma ^2 Z_g \sinh \big({{\gamma l}\over 2}\big) + Z\gamma \cosh \big({{\gamma l}\over 2}\big)}\over {2Z_g \gamma + Z}} = 0\eqno{\hbox{(27)}}$$which leads to
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$$A = {{Z\left({2Z_g \gamma + Z} \right)\hat I}\over {4\gamma ^2 Z_g \sinh \big({{\gamma l}\over 2}\big) + 2Z\gamma \cosh ({{\gamma l}\over 2})}}.\eqno{\hbox{(28)}}$$

By inserting (28) into (25) and (26), we end up with expressions for and with appearing as a common factor. Then, by using the boundary condition (10), the fact that
TeX Source
$$\hat V_2 = \hat v_2 \left({{l\over 2}} \right)\eqno{\hbox{(29)}}$$and somewhat lengthy but simple algebraic transformation, we derive the following final expression for the transfer relation:
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$${{\hat V_2 }\over {\hat V_1 }} = {{a\tanh \big({{\gamma l}\over 2}\big) + b}\over {c\tanh \big({{\gamma l}\over 2}\big) + d}}\eqno{\hbox{(30)}}$$where coefficients *a*, *b*, *c*, and *d* are specified by
TeX Source
$$\eqalignno{a &= - 2Z^2 + 2Z_g \gamma ^2 (Zl + 4Z_g)&{\hbox{(31)}}\cr b &= Z^2 \gamma l&{\hbox{(32)}}\cr c &= 2Z^2 + 2Z_g \gamma ^2 (Zl + 4Z_g)&{\hbox{(33)}}\cr d &= 8ZZ_g \gamma + Z^2 \gamma l.&{\hbox{(34)}}}$$

The previous derivation can be extended to the case when the secondary winding is connected to a balanced-load impedance *Z*_{L} (equal impedance to ground). The transfer relation in this case has the same form as in (30), however, the expression for *c* and *d* coefficients are modified as follows:
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$$\eqalignno{c &= 2Z^2 + 2Z_g \gamma ^2 (Zl + 4Z_g) + {{2Z_s^3 l + 8Z_s^2 Z_g }\over {Z_L }}&{\hbox{(35)}}\cr d &= 8ZZ_g \gamma + Z^2 \gamma l + {{4Z_g Z_S^2 l\gamma + 16Z_g^2 Z_S \gamma }\over {Z_L }}.&{\hbox{(36)}}}$$

In practice, the per unit length series impedance (*Z*) is formed by the series connection of per unit length leakage inductance () and per unit length line resistance () in parallel with per unit length intra-winding capacitance (). The cross-winding admittance (*Y*) is just due to the per unit length cross-winding capacitance (). To satisfy low frequency isolation requirements, the ground impedance is simply a large grounding capacitor (*C*_{g}) (see Fig. 2). The extraction techniques for these parameters were discussed in Bowen et al. [2009a, 2009b, 2010], and result in total values of the parameters. The values used in our calculations were approximately the values extracted from existing Ethernet transformers divided by unit length (*l*). The values quoted in the figures are total values (*L*_{l}, *R*, *C*_{i}, and *C*, respectively).

By using (30–34), we computed the transfer relation as a function of frequency for various values of parameters of the primary and secondary windings, respectively. Sample computational results are shown in Figs. 4–6. These figures reveal that the common mode signals can be suppressed in the frequency range of up to 100 MHz. Above 100 MHz, strong resonances may appear which are followed by a constant (but somewhat appreciable) tail. The resonances are the result of the series impedance that consists of leakage inductance and line resistance in parallel with intra-winding capacitance. Increasing the line resistance reduces the *Q* of the resonance peaks, making them broader with smaller amplitude as can be seen in Fig. 4.

Fig. 5 shows how increasing leakage inductance lowers the frequency of the resonances as well as increases the height. As the inductive reactance becomes larger in Fig. 5, the line resistance becomes smaller in comparison, effectively increasing the resonance *Q* and making narrower, taller peaks. Also, in Fig. 5, the 10 and 100 nH curves show the beginning of a bifurcated resonance. It has been verified that increasing leakage inductance further causes a decaying train of resonances to appear; a phenomena that comes from the distributed nature of the model.

The origin of the high-frequency flat coupling is due to the fact that for very high frequencies the Ethernet transformer operates as a voltage divider formed by effective cross-winding and intra-winding capacitances connected in series. The resonances and the high frequency constant tail can be suppressed (as evident from Figs. 4 and 6) by increasing winding resistance and intra-winding capacitance, respectively.

The use of ferrite cores for Ethernet transformers is problematic because of degradation of their magnetic permeability with increase of frequency beyond 10 MHz where the transformers essentially become air-core devices. It is worthwhile to note that the suppression of resonances by increasing winding resistance can be beneficial for highly resistive air-core (MEMs) designs of Ethernet transformers. Such air-core designs of Ethernet transformers are attractive for high bit rate communication networks where differential mode transfer characteristics must be flat in the frequency range to 1 GHz and above [Hamidy 2007].

Corresponding author: D. Bowen (dbowen1@umd.edu)

**1**. On design of air-core Ethernet transformers

D Bowen, I D Mayergoyz, C Krafft, D Kroop, M Beyaz

*J. Appl. Phys., Vol. 105, 07A307, 2009a, 10.1063/1.3063071*

**2**. Modeling and testing of Ethernet transformers

D Bowen, I D Mayergoyz, Z Zhang, P McAvoy, C Krafft, D Kroop

*IEEE Trans. Magn., Vol. 45, issue (10), pp. 4793–4796, 2009b, 10.1109/TMAG.2009.2023918*

**3**. Electromagnetic modeling of Ethernet transformers

D Bowen, I D Mayergoyz, C Krafft

*IEEE Trans. Magn., Vol. 46, issue (2), pp. 563–569, 2010, 10.1109/TMAG.2009.2033203*

**4**. Designing magnetics for 10-Gbit Ethernet

F Hamidy

EE Times, United Business Media, London: U.K., 2007 www.eetimes.com/showArticle.jhtml;?articleID=199703895

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Pulse, White Paper G019, Pulse Engineering, San Diego: www.pulseeng.com/file.php?3100, 1999

**6**. A lumped element transformer model including core losses and winding impedances

D. Ribbenfjärd

A lumped element transformer model including core losses and winding impedances, Licentiate thesis, Electromagn. Eng., Royal Institute of Technology, Stockholm, Sweden, 2007

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- This paper appears in:
- IEEE Magnetics Letters
- Issue Date:
- DECEMBER 2010
- On page(s):
- 0500204 - 0500204
- ISBN:
- 1949-307X
- Print ISBN:
- N/A
- INSPEC Accession Number:
- 11228829
- Digital Object Identifier:
- 10.1109/LMAG.2010.2045882
- Date of Original Publication:
- 14 Apr, 2010

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