Calculation of Electrical Parameters of Two-Wire

IEEE TRANSACTIONS ON ELECTROMAGNETIC COMPATIBILITY, VOL. 50, NO. 3, AUGUST 2008

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Calculation of Electrical Parameters of Two-Wire Lines in Multiconductor Cables Boris M. Levin

Abstract—A rigorous method for the calculations of the characteristics of two-wire lines (twisted pairs) located in a metal shield is considered. In this paper, it is first shown that the mutual coupling between lines in multiconductor cables results in the appearance of electromagnetic interference (crosstalk) in communication channels; second, the asymmetry of excitation and loads results in the appearance of common-mode currents in the cable. The voltage values (interference) in the loads placed at the beginning and the end of the adjacent line are determined at a given power in the main line. The effect of loads connected between wires and shield is examined. The proposed method allows generalization of the obtained results in the case of multiconductor cables with losses. Index Terms—Cables, communication channels, electromagnetic (EM) interference, lossy circuits, mutual coupling.

NOMENCLATURE a b c Cl Cn s0 d Dn s (Dn s )0 e, e1 Gn s0 in In k L L0 Mn s0 n, s N pn s R R0 Rn s0

Radius of the wire. Distance between wires of a two-wire line. Velocity of light. Linear (per-unit-length) capacitance of conductor. Mutual capacitance between wires n and s per unit of their length. Distance between axes of two twisted pairs. Distance between the wires n and s. Mean distance between the wires n and s. EMF of the generators. Leakage conductance between wires n and s per unit of their length. Current of the nth wire. Current at the beginning of the nth wire. Propagation constant of a wave in a medium. Length of a line wire. Wire inductance per unit length. Mutual inductance between wires n and s per unit of their length. Wire numbers. Number of parallel wires located inside a metal cylinder. Potential coefficient between wires n and s. Radius of a metal cylinder (a shield of a cable). Wire resistance per unit length. Loss resistance of wires n and s per unit of their length.

Manuscript received May 17, 2007; revised September 30, 2007 and December 31, 2007. The author is with the Holon Institute of Technology, Holon, Israel. Digital Object Identifier 10.1109/TEMC.2008.927924

un Un W Wn s z Z, Z1 , Z2 , Z3 α βn s ∆n s ∆, H ∆N = |pn s | ε γ ρn s

Potential of the nth wire. Potential at the beginning of the nth wire. Wave impedance of a line. Electrostatic wave impedance between wires n and s. Coordinate along a wire. Impedances of the line loads. Angular displacement of points along the section perimeter. Coefficient of an electrostatic induction between wires n and s. Cofactor of the determinant ∆N . Distances by which a wire is displaced. N × N Determinant. Permittivity of the medium inside the cable. Propagation constant of a wave along wires. Electrodynamic wave impedance between wires n and s. I. INTRODUCTION

HE DETERMINATION of the signal magnitude at the end of multiconductor cable located inside a metal shield required the calculation of the electrical characteristics of those lines. The mutual coupling between two-wire lines (twisted pairs) in a multiconductor cable results in the appearance of electromagnetic (EM) interference (crosstalk) in communication channels. The voltage values on loads placed at the ends of an adjacent line can be used as a measure of such distortions [1]; for example, [2] is devoted to the qualitative analysis of mutual coupling between lines. However, the two-wire line model considered in [2] is far from an actual twisted pair structure. The rigorous method of the calculation of the mutual coupling between lines enables to develop simple and effective methods of preventing interference. EM interference in communication channels (a cable unbalance) is caused by the cable asymmetry and also by excitation and load asymmetry, which provokes the appearance of common-mode currents in cables. The rigorous method of the calculation of the multiconductor cable electrical characteristics enables to determine the common-mode currents. Compensation of the common-mode currents permits to decrease the EM radiation and its susceptibility to external fields. In this paper, a rigorous method for the calculation is offered for the characteristics of a two-wire line located inside a metal shield and second for the mutual coupling between lines. The lines are considered as uniform ones. The EM waves are considered as transverse (TEM) waves, and the cable diameter is considered small in comparison to the wavelength.

T

0018-9375/$25.00 © 2008 IEEE

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general case, are determined by N Us 2Un sin kz − in = In cos kz + j Wn n s=1 Wn s un = Un cos kz + j

N

ρn s Is sin kz

(1)

s=1

Fig. 1.

Two wires inside cylinder.

Fig. 2.

Equivalent circuit of a single line inside a shield.

where In and Un are, respectively, the current and the potential at the beginning of the nth wire (at z = 0), k is the propagation constant of a wave in a medium, and Wn s and ρn s are the electrostatic and electrodynamic wave impedances between wires n and s. In this case 1/(cβn s ), n = s pn s Wn s = (2) ρn s = c −1/(cβn s ), n = s where pn s are the potential coefficients, βn s are the coefficients of an electrostatic induction, and c is the velocity of the light. The coefficients βn s and pn s are linked by the following relationship:

II. TWO WIRES INSIDE A SHIELD WITH A CIRCULAR SECTION A. Problem Statement A single pair of wires (twisted pair) inside the metal cylinder can be modeled as two wires of radius a, located at a distance b from each other inside a metal cylinder of radius R and length L (Fig. 1). In multiconductor cables, the wire radius a and the distance b are small in comparison with the cylinder radius R (a, b R), so the characteristic impedance of the line is constant along its length (when the axis lines of the twisted pair and the cylinder do not coincide, and the given inequation is not satisfied, the characteristic impedance varies along the line). We assume the wires to be straight and take into account twisting by increasing the length L of the equivalent line. Since the pitch of the helix followed by each wire is larger than helix diameter b, the inductance L0 per unit length varies slightly with the replacement of a spiral wire by a direct one. The wire capacitance per unit length also varies only slightly, i.e., the wire twisting does not change the characteristic impedances of a structure. The line asymmetry in a real cable can cause a change of the characteristic impedance and a change of the two-wire line input impedance. Another cause of the cable asymmetry is that each two-wire line is made in the form of a twisted pair (helix), a design that leads to a difference in the mean distances between different wires and to the mutual coupling (crosstalk) between two two-wire lines surrounded by a single shield, even if both the exciting electromotive force (EMF) of each line and the line load are symmetric. The input impedance of a two-wire line inside a metal shield is determined in the next section. B. Calculation of the Input Impedance of the Two-Wire Line The equivalent circuit of a single line inside the shield is shown in Fig. 2. The two-wire line is located above the ground (inside the metal cylinder). The theory of such lines has been worked out by Pistolkors [3]. The current and the potential of the nth wire of an asymmetrical line of N parallel wires located above the ground, in the

βn s =

∆n s ∆N

(3)

where ∆N = |pn s | is the N × N determinant and ∆n s is the cofactor of the determinant ∆N . For an asymmetrical line from two wires, we can write N =2

1 ρ22 = W11 ρ11 ρ22 − ρ212

1 ρ11 = W22 ρ11 ρ22 − ρ212

1 ρ12 = . W12 ρ11 ρ22 − ρ212

(4)

The boundary conditions for the currents and potentials in the circuit shown in Fig. 2 are i1 (0) + i2 (0) = 0 i1 (L) + i2 (L) = 0

u1 (0) = u2 (0) + i1 (0)Z u1 (L) = e + u2 (L).

(5)

Here, Z is the impedance of the line load (see Fig. 2). Substituting expressions (1) in the first and second equations of system (5), we find I2 = −I1

U2 = U1 − I1 Z.

From the third equation of system (5), taking into account expressions (4), we find that 1 1 W22 − W12 ρ11 − ρ12 = I1 Z U1 = I1 Z . 1 1 2 ρ11 + ρ22 − 2ρ12 W11 + W22 − W12 And from the fourth equation, we obtain e . I1 = [Z cos kL + j(ρ11 + ρ22 − 2ρ12 ) sin kL] The input impedance of a two-wire line inside a metal shield (the load impedance of generator e) is equal to Zl = e/i1 (L). Substituting the value of i1 (L) from expression (1) and using the relationships between e, I1 , I2 , U1 , and U2 , we find that Zl = W

Z + jW tgkL W + jZtgkL

where W = ρ11 + ρ22 − 2ρ12 .

(6)

LEVIN: CALCULATION OF ELECTRICAL PARAMETERS OF TWO-WIRE LINES IN MULTICONDUCTOR CABLES

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It is readily seen that the expression (6) coincides with the expression for the input impedance of a lossless two-wire-long line that is located in free space, characterized by wave impedance W , and loaded by impedance Z. The line asymmetry results in the difference of wires electrostatic (ρ11 = ρ22 ) and electrodynamic (W11 = W22 ) characteristic impedances. The calculation of currents i1 (z) and i2 (z) shows that the currents in a two-wire line are identical in value and opposite in sign jI1 Z sin kz. W In a wire pair, there are only differential-mode currents. A common-mode current in the wires is absent because the EMF and load impedance are placed between the line wires. The appearance of a common-mode current can be caused by the connection of an additional EMF or an additional load between one wire of a line and the shield. i1 (z) = −i2 (z) = I1 cos kz +

C. Calculation of the Characteristic Impedance For the determination of the potential coefficients pn s , it is appropriate to make use of [4]. It gives, in particular, formulas for the calculation of linear (per-unit-length) capacitance Cl of conductors in the form of an indefinitely long closed envelope of a circular section. In this case, the potential coefficients calculated with consideration for the mirror image in a perfectly conducting cylindrical surface equal 1 . (7) Cl To see this, if a system consists of two identical conductors (a wire and its image) and this structure is electrically neutral, the mutual partial capacitance coincides with the interconductor capacitance (see [4, expression (B-14)]). The mutual partial capacitance equals pn s =

C=

1 2 (p11 − p12 )

where p11 is the self-potential coefficient and p12 is the potential coefficient of the image. The conductor-to-ground capacitance is twice as much as the capacitance between two conductors Cl = 2C =

p11

1 1 = . − p12 p

For two wires of radius a, located at a distance b from each other, symmetrically located inside the metal cylinder of a radius R to the cylinder axes (see Fig. 1) in accord with (7) and [4, expression (4)–(20)], we can write R2 + a2 − b2 /4 1 ch−1 . 2πε 2Ra Here, ε is the permittivity of the medium inside the cable. If the wire radius a and the distance b are small in comparison with the cylinder radius R, then p11 = p22 =

p11 = p22 ≈

R 1 ln 2πε a

and in the air ρ11 = ρ22 ≈ 60 ln

R . a

(8)

Fig. 3.

Offset wires inside a cylinder.

Similarly, in accord with (7) and [4, expression (4)–(20)], we find R (9) ρ12 ≈ 60 ln √ ab i.e., the characteristic impedance b (10) a of a lossless two-wire line, symmetrically located inside the metal cylinder, is half of the characteristic impedance of the same line in free space. In a uniform line loaded by its characteristic impedance, the reflected wave is null, so the signal transmission in the absence of losses equals 1 and does not depend on frequency. In this regard, a two-wire line (twisting) inside the metal shield is different from nonuniform line considered in [5], the spiral two-wire line located along a metal plane. W0 = ρ11 + ρ22 − 2ρ12 ≈ 60 ln

D. Reasons of the Characteristic Impedance Change 1) Relative Displacement of the Wires: Let us consider possible reasons that a line’s characteristic impedance changes inside the shield. If wires inside the metal cylinder of a radius R are located asymmetrically, for example, they are displaced to the right by a distance ∆ (Fig. 3) p11 =

R2 + a2 − (b/2 − ∆)2 1 ch−1 2πε 2Ra

so at a, b R R 1 1 ∆(b − ∆) R ∆(b − ∆) p11 = ln + = 1+ ln 2πε a R2 2πε a R2 1 R ∆ (b + ∆) p22 = . ln − 2πε a R2 Then, the characteristic impedance of the line is 120∆2 . (11) R2 2) Increase of the Distance Between Wires: If the distance between wires is increased by value ∆, then if the distance ∆ is small in comparison with the distance between the wires (∆ b), then ∆ R R ρ12 ≈ 60 ln

≈ 60 ln √ − . ab 2b a (b + ∆) W = W0 −

4

Fig. 4.


Equivalent circuit of two coupled lines placed inside a shield.

Hence b+∆ ∆ = W0 + 60 . (12) a b As can be seen from (11) and (12), a change of distance between wires has more effect on the characteristic impedance of the line than the wire displacement relative to the cylinder axis. W = 60 ln

III. TWO WIRES PAIRS INSIDE A SHIELD WITH A CIRCULAR SECTION A. Problem Statement The equivalent circuit of two-coupled two-wire lines inside a shield is shown in Fig. 4. One of these lines is excited by the generator e and loaded by complex impedance Z1 , the loads Z2 and Z3 are connected to both ends of the other line. It is necessary to emphasize that such circuit has the most general character. If, for example, a generator e1 is connected at the end of the second line (in the point z = L), the currents and voltages created by the generator e can be determined by replacing the value Z3 by the input impedance of the generator e1 . Let us consider that a b d, R (here, d is a distance between axes of twisted pairs). In many cases, the diameter of a wire bunch is small in comparison with the diameter of the cable metal shield. When there are many wires in the bunch, its diameter is close to the shield diameter. However, it is necessary to take into account that the maximum mutual coupling exists between adjacent lines. Therefore, analyzing mutual coupling between them is possible by considering as a first approximation that d R. In the next section, it is shown that twisting is the reason of asymmetry.

Fig. 5. Distance between wires 1 and 4. (a) Usual winding of a wire 4. (b) Counter winding of a wire 4.

distance between wires 1 and 4 [see Fig. 5(a)] is equal to

b2 sin2 α D14 = (d + b cos α)2 + b2 sin2 α ≈ d + b cos α + 2d (here, α is the angular displacement of points 1 and 4 along the cross-section perimeter), i.e., the mean distance between these wires 1 π b2 (13) Ddα = d + (D14 )0 = π 0 4d differs from the value d. The potential coefficients and also the electrodynamic and electrostatic wave impedances vary accordingly. If, at the cable’s initial cross section, the leads of spirals 3 and 4 are shifted along the cross-section perimeter by π/2 and 3π/2 from the initial point accordingly, then the distance between wires 1 and 3 (or wire 4) equals b2 b (sin α ∓ cos α)2 . D13(4) ≈ d + (cos α ± sin α) + 2 8d Here, the top sign applies to wire 3, and the lower sign to wire 4. From this equation, the average distance between these wires is b2 b (14) D13(4) 0 ≈ d ± + π 8d i.e., the shift of the spiral leads of the cable by π/2 essentially changes the average distance between wires. Difference between (D13 )0 and (D14 )0 increases from value b2 /4d to 2b/π, at that b d. In order to make the average distance D0 between wires 1 and 4 undistinguishable from d, it is necessary to wind wire 4 counter to the other wires. In this case [see Fig. 5(b)] D = d + b cos α

D0 = d.

(15)

B. Twisting as the Reason of Asymmetry As was stated in Section I, the cable asymmetry results in mutual coupling (crosstalk) between two two-wire lines. The reason of such asymmetry is the fulfillment of each line as twisting (spiral). The placement of the lines conductors at the different variants of winding is shown in Fig. 5. If, at the cable’s initial cross section, the leads of spirals 1 and 3 are located in the same point of their section (we shall name it as initial one) and the leads of spirals 2 and 4 are shifted along the cross-section perimeter by π from this point, it means that the distance between wires 1 and 3 (and also between wires 2 and 4) along all their length equals D13 = D24 = d, whereas the distance between wires 1 and 4 (and also between wires 2 and 3) varies along wires from d + b to d − b. For example, the

C. Calculation of Electrical Parameters The electrodynamic characteristic impedances of this structure at usual winding become R ρ11 = ρ22 = ρ33 = ρ44 = ρ1 = 60 ln a R ρ12 = ρ34 = ρ2 = 60 ln √ ab R ρ13 = ρ24 = ρ3 = 60 ln √ ad R . (16) ρ14 = ρ23 = ρ4 = 60 ln

a(d + b2 π/4d)


In the case of lines arranged at a finite distance H from a cable axis in accord with formula (7) and [4, expression (4–20)], we find R 1 − H 2/R2 . ρ1 = 60 ln a The expressions for other magnitudes ρn remain valid ones. This means that the wave impedance of a lossless two-wire line located inside the metal cylinder at a distance H from its axis in accord with (10) is 2 b 1 − H 2/R2 W = 60 ln a i.e., as a result of a line displacement from a cable axis, its characteristic impedance decreases. When H is small and equal to ∆, we arrive at expression (11). According to (2) and (3), we find the electrostatic characteristic impedances  ∆N   ∆ , n=s ns (17) Wn s =    − ∆N , n = s ∆n s where ∆N = |ρn s | is the N × N determinant and ∆n s is the cofactor of the determinant ∆N . For a structure made up of four wires in accord with (16) and (17) W11 = W22 = W33 = W44 = W1 = ∆4 /∆11

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Knowing all parameters in expressions (1), it is possible to calculate the loading impedance of the generator e e Zl = i1 (L) =2

Z1 + 2j[ρ1 − ρ2 + A(ρ3 − ρ4 )]tgkL 2 + j[Z1 (1/W1 + 1/W2 ) − AZ2 (1/W3 − 1/W4 )]tgkL (21)

and the currents in the wires of the second (unexcited) line I1 [AZ2 (1/W1 + 1/W2 ) 2 − Z1 (1/W3 − 1/W4 )] sin kz

i3 (z) = I1 A cos kz + j

i4 (z) = −i3 (z).

(22)

The sum of the currents equal zero, i.e., as well as at placement of one line in the shield, the common-mode current is absent since the EMF and the loading impedances are connected only between wires of each line. The voltages across passive loads are equal to V1 = i1 (0)Z1 = I1 Z1

V2 = i3 (0)Z2 = I1 AZ2 1 V3 = i3 (L)Z3 = I1 Z3 A cos kL + j [AZ2 (1/W1 + 1/W2 ) 2 − Z1 (1/W3 − 1/W4 )] sin kL . (23)

W12 = W34 = W2 = −∆4 /∆12 W13 = W24 = W3 = −∆4 /∆13 W14 = W23 = W4 = −∆4 /∆14 .

D. Numerical Results (18)

The current and potential of the nth wire of an asymmetric line from N parallel wires located above the ground are determined from expression (1). The boundary conditions for the currents and voltages in the circuit shown in Fig. 4 are i1 (0) + i2 (0) = 0 i3 (0) + i4 (0) = 0 u1 (0) = u2 (0) + i1 (0)Z1 i1 (L) + i2 (L) = 0

u3 (0) = u4 (0) + i3 (0)Z2

i3 (L) + i4 (L) = 0

u1 (L) = e + u2 (L)

u3 (l) = u4 (L) + i3 (L)Z3 . (19)

Substituting expressions (1) in the equations of system (19), we find (analogously to Section II) e I1 = Z1 cos kL + 2j[ρ1 − ρ2 + (ρ3 − ρ4 )A] sin kL I3 = AI1

(20)

where A=

4(ρ3 − ρ4 ) + Z1 Z3 (1/W3 − 1/W4 ) . −4(ρ1 − ρ2 ) + Z2 Z3 (1/W1 + 1/W2 ) + j2(Z2 − Z3 )ctgkL

If ρ3 = ρ4 (and accordingly W3 = W4 ), then A = 0, the current at the beginning of the second line is null. In this case, the presence of the second two-wire line has no effect on the first line. This result obviously corroborates that the cable asymmetry results in a mutual coupling (crosstalk) between the two two-wire lines.

As an example, we consider the structure from two pairs of wires inside the shield with sizes (in millimeters): a = 0.2, b = 0.5, d = 2, R = 2. For the identical loads Z1 = Z2 = Z3 = 100 Ω, the ratio A of the currents at the beginning of the second (unexcited) and first line equals 0.13. If the values of the loads are equal to the characteristic impedance of the single twowire line inside the metal shield, i.e., in accordance with (10) Z1 = Z2 = Z3 = 55 Ω, the ratio of the currents is essentially increased (A = −0.76). The absolute values of the currents as functions of kz are plotted in Fig. 6. Here, k is the propagation constant of a wave in a medium, z is the coordinate along line (see Fig. 4). IV. LOADS BETWEEN THE WIRES AND THE SHIELD Let us consider the effect of the loads placed between the wires and the shield using a two-wire line as an example (Fig. 7). It differs from the circuit shown in Fig. 2 by connecting its wires at a line end (near the generator) with a shield via complex impedances Z1 and Z2 , whose values depend on the circuit of a line excitation. In a real circuit, the secondary winding of the transformer can act as the EMF e, exciting a two-wire line. In this case, parasitic capacities of this winding to ground (to cable shield) act as impedances Z1 and Z2 . The current and potential of the nth wire of an asymmetrical line of N parallel wires located above the ground are determined by expressions (1). The boundary conditions for the currents and

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page, and is (z) = j

eC sin kz . Z cos kL + j2 (ρ1 − ρ2 ) sin kL

C=

2(ρ1 − ρ2 ) − jZctgkL 2 − j(ρ1 + ρ2 ) ZZ 11 +Z −Z 2 ctgkL

(28)

where

Fig. 6.

Absolute values of the currents in the excited and unexcited wires.

Fig. 7. Equivalent circuit of the single line with the loads connected between the wires and the shield.

potentials in the circuit shown in Fig. 7 are i1 (0) + i2 (0) = 0 u1 (0) = u2 (0) + i1 (0)Z i1 (L) + i2 (L) +

u1 (L) u2 (L) + =0 Z1 Z2 u1 (L) = e + u2 (L).

(24)

Substituting expressions (1) in the equations of system (24), we find (analogously to Section II) the input impedance of the two-wire line, given in (25), shown at the bottom of the page, and the sum of the currents in the line wires as is (z) = i1 (z) + i2 (z) = jI1 [Z (1/W12 − 1/W22 ) + U1 /I1 (1/W11 + 1/W22 − 2/W12 )] sin kz

(26)

i.e., the loads connection results in the appearance of the common-mode current in the wires and the current along the inner surface of the cable shield, equal in value but opposite in direction. For two wires of the identical radius located symmetrically to the cylinder axes, given in (27), shown at the bottom of the

Zl =

Zl =

2Z 1 Z 2 Z 1 −Z 2

It is not difficult to make sure that at 1/Z1 = 1/Z2 = 0, the magnitude C is zero and the expressions for U1 and Zl coincide with the similar expressions for the circuit without loads between wires and shield. From the presented results, it is also easy to obtain expressions for the cases when there is only one from loads, for example 1/Z1 = 0. The analysis described before verifies that the reason of the appearance of the common-mode currents in a line wire is the asymmetry of its excitation caused by the connection of complex impedances, for example, parasitic capacitances of a secondary transformer winding to the ground (to a cable shield). A loads asymmetry at the line end distant from the generator (at z = 0) gives the similar results. The common-mode currents in the excited line induce the common-mode currents in the wires of the adjacent unexcited line, even if it is symmetric completely (about the ground and the excited line). At that, removal of the excitation and load asymmetry in the excited line results in the disappearance of the common-mode currents in wires of both excited and unexcited line. In order to decrease or eliminate the common-mode currents, it is necessary to annihilate this asymmetry, for example, to neutralize effect of parasitic capacitances to the ground (to the cable shield). To this goal in [6], it is offered to cancel the current through parasitic capacitance with the current equal in value and opposite in direction, which is created by the additional transformer winding. V. COUPLED LINES WITH LOSSES In the previous sections, for the calculations of the electrical characteristics of two-wire lines (twisted pairs) in multiconductor cables, the theory of asymmetrical electrically coupled lines developed by Pistolkors is used. This theory is based on the telegraph equations and on the relations between wires potential coefficients and coefficients of an electrostatic induction. For each wire of the structure, one can write two telegraph equations. One of them proceeds from the fact that the potential drop along the section dz of a given wire is a result of superposition of EMF induced by its own and other currents. Second equation is based on electrostatic expressions connecting charges to potentials, with consideration for the continuity equations. The

e Z cos kL + j (ρ11 + ρ22 − 2ρ12 ) = i1 (L) + u1 (L)/Z1 1 + U1 /I1 [1/Z1 + j (1/W11 − 1/W12 ) tgkL] + j [Z/W12 + (ρ11 − ρ12 )/Z1 ] tgkL

(25)

Z + 2j(ρ1 − ρ2 )tgkL 1 + j[ Zρ2 /(ρ21 − ρ22 ) ]tgkL + j/[ 2(ρ1 + ρ2 ) ][Z + (ρ1 + ρ2 )C]tgkL + (1/2Z1 )[Z + (ρ1 + ρ2 )C + 2j(ρ1 − ρ2 )tgkL] (27)


coordinate axis z is selected in parallel to wires (see, for example, Fig. 2), and the dependence of a current on coordinate z is accepted as exp (γz), where γ is the propagation constant of the wave along the wires. In the absence of losses in the wires and in the medium, in which they are placed, the electrostatic Wn s and electrodynamic ρn s characteristic impedances between wires n and s are real quantities determinable by equalities (2), and γ = jk is purely imaginary (k is the propagation constant of the wave in the medium). At that, the current and potential of the nth wire of an asymmetrical line of N parallel wires located above the ground are calculated from expressions (1). As follows from said, the electrodynamic wave impedances ρn s are proportional to the self and mutual inductances of wires sections, i.e., are proportional to reactances connected in series with wires circuits. The electrostatic wave impedances Wn s are proportional to the mutual capacitances between wires, i.e., to susceptances between them. Therefore, it is natural to connect in a circuit the resistance of losses in a wire (for example, the skin effect losses) in series with the inductance, and the leakage conductance—in parallel with the mutual capacitances. We shall take into account the losses in a medium and in wires considering that characteristic impedances Wn s and ρn s and the propagation constant k are complex values. If the inductance of the nth wire per unit of its length is L0 and its active resistance is R0 then its impedance per unit length is jρn n = jωL0 + R0 , i.e., the self-electrodynamic characteristic impedance of the wire with losses is equal to R0 ρn n = ρ0 1 − j (29) ρ0 where ρ0 = ωL0 is the electrodynamic characteristic impedance in the absence of losses and R0 is the total resistance of losses in the nth wire and in a metal shield per unit length. For the mutual electrodynamic characteristic impedance between wires n and s, we shall obtain Rn s0 ρn s = ρn s0 1 − j (30) ρn s0 where ρn s0 = ωMn s0 , Mn s0 is the mutual inductance between wires n and s per unit length, and Rn s0 is the loss resistance in both wires per unit length. Similarly, for the admittance between wires n and s per unit length, we find: jWn s = jωCn s0 + Gn s0 , i.e., the electrostatic characteristic impedance in a medium with losses equals Gn s0 (31) Wn s = Wn s0 1 − j Wn s0 where Wn s0 = ωCn s0 , Cn s0 is the mutual partial capacity between wires n and s per unit length, and Gn s0 is the leakage conductance per unit length. Thus, at the calculation of the electrical performances of the coupled lines with losses it is possible to use the results obtained for the lossless lines by substitution of the complex characteristic impedances into expressions obtained before in accord with (29)–(31). At that, both losses in wires and losses in an imperfectly conducting metallic tube (shield) are taken into account.

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VI. CONCLUSION A rigorous method for the calculation of the characteristics of two-wire lines inside the metal shield allows us in refining the mechanism of mutual coupling between lines in multiconductors cables. It permits to determine the voltage values (interferences) along impedances placed at the beginning and the end of the adjacent line at the given power on the main line. As it is shown in Section III, the reason of the crosstalk is the asymmetry of the wire arrangement (a difference in the average space between different wires), and accordingly, asymmetric characteristic impedances. The elimination of this asymmetry will reduce the crosstalk in multiconductor cables, i.e., will enable to increase the channel carrying capacity. That is also valid for multiconductor connectors. The reason of the appearance of common-mode currents in the lines of the multiconductor cable is the asymmetry of excitation and loads. Compensation of the common-mode currents offers to decrease the EM radiation and to reduce its susceptibility to external fields. The rigorous method for the calculation of the currents and voltages on the wires of a multiconductor cable enables to find methods to reduce the crosstalk and common-mode currents and to elaborate methods to compensate them. REFERENCES [1] C. Valenti, “NEXT and FEXT models for twisted-pair North American loop plant,” IEEE J. Sel. Areas Commun., vol. 20, no. 5, pp. 893–900, Jun. 2002. [2] X. Xu, S. Nitta, A. Mutoh, and S. Jayaram, “Study of electromagnetic interference of multiconductor twisted-pair wire circuit: The case of two-cored twisted-pair wires,” Electron. Commun. Jpn. 1, vol. 80, pp. 9– 16, Dec. 1997. [3] A. Pistolkors, Antennas. Moscow, Russia: Svyazizdat, 1947 (in Russian). [4] Y. Y. Iossel, E. S. Kotchannov, and M. G. Strunsky, Calculation of an Electrical Capacitance. Leningrad, Russia: Energoizdat, 1981 (in Russian). [5] J. A. B. Faria and M. V. G. Neves, “Analysis of the helical twisted-wire pair running above ground: Transfer function evaluation,” IEEE Trans. Electromagn. Compat., vol. 45, no. 2, pp. 449–453, May 2003. [6] D. Cochrane, “Passive cancellation of common-mode electromagnetic interference in switching power converters,” M.S. thesis, Virginia Polytechnic Inst. State Univ., Blacksburg, 2001.

Boris Levin was born in Saratov, Russia, in January 1937. He received the Graduate degree from Leningrad Polytechnic Institute, Saint Petersburg, Russia, in 1960, the Ph.D. degree in radio physics from the Central Research Institute of Automatic Devices, Leningrad, Russia, in 1969, and the Doctor of Sciences degree in physics and mathematics from Saint Petersburg Polytechnic University, Saint Petersburg, in 1993. From 1963 to 1998, he was with the Design Office “Svyazmorproyekt” of Russia Shipbuilding Department. From 2000 to 2002, he was with “MARS,” Holon, Israel. He has authored or coauthored three books, 76 original papers in technical journals, 38 papers in proceedings of international scientific conferences, and 37 abstracts of conference reports. He is the holder of 44 patents. His current research interests include electromagnetic theory, the theory of linear antennas and antenna optimization, and analysis, design, and developments of new antennas.