Evaluating the Shielding Effectiveness of Nonmagnetic Shields

No doubt the best materials for electromagnetic shields will have both high conductivity and high permeability, as does, for instance, Mumetal. Unfortunately, these types of materials are very expensive. Employment of nonmagnetic but cheaper materials to obtain the desired shielding effectiveness is generally felt to be acceptable in the light of this cost obstacle. This article develops that approach. It presents and discusses a circuit model along with experimental results.

Figure 1. Schematic setup of the physical shield model.

Figure 2. Physical-geometric model of the solenoids and the shield in the shield model of Figure 1.

The Shield Model

In the power frequency range (50 or 60 Hz and their lower harmonics), a well-known way of suitably modeling shielding problems involves a pair of coaxial solenoids with a shielding plate inserted between, as shown in Figure 1. In the transmitting solenoid T a sinusoidal current, I_G, of constant amplitude and frequency is impressed. The no-load induced voltage V_R is calculated or measured at the terminals of the receiving solenoid R. The ratio, or the related attenuation, of the induced voltage when the shield is on (V_R = V_sh) to the same voltage when the shield is off (V_R = V₀) can be correlated to the effectiveness of the shield.

A circuit model of the device is obtainable by decomposing both the solenoids and the shield into a set of circular coaxial parallel turns, mutually coupled, as schematized in Figure 2, where, for the sake of clarity, the transmitting solenoid T has only three turns, the receiving solenoid R has only four turns, and the shield S is decomposed into only five turns. In sinusoidal steady state, the phasors are as follows:

(The numerical values of the indexes refer to the deconstruction in Figure 2.)

The circuit parameters of the equivalent network (their calculus is developed and discussed below) are:

R_s = the resistances of the turns composing the shield.

L_s = the total (internal and external) self-inductance of the aforesaid turns.

m_s's" = the mutual inductances between the s'th and the s"th turns of the shield, where s' is not equal to s".

M_st = the mutual inductances between the sth turn of
the shield and the tth turn (t = 1–3) of the transmitting solenoid T.

m_rs = the mutual inductances between the rth turn of the solenoid R and the sth turn of the shield.

M_rt = the mutual inductances between the rth turn of the receiving solenoid R and the tth turn of the transmitting one.

The system of equations governing the unknown currents I_s in the shield is (with reference to Figure 2):

R₁I₁ + j( L₁I₁ + m₁₂I₂ + m₁₃I₃ + m₁₄I₄ + m₁₅I₅ ) + j( I_G ( M₁₁ + M₁₂ + M₁₃ ) = 0

R₅I₅ + j ( m₅₁I₁ + m₅₂I₂ + m₅₃I₃ + m₅₄I₄ + L₅I₅ ) + j( I_G ( M₅₁ + M₅₂ + M₅₃ ) = 0

Once the currents I_s are calculated, the induced voltages V_R in the four turns of the solenoid R are easily obtainable through a second system of equations.

V₁ = j [(m₁₁I₁ + m₁₂I₂ + m₁₃I₃ + m₁₄I₄ + m₁₅I₅ ) + j( I_G ( M₁₁ + M₁₂ + M₁₃ )]

V₄ = j( [(m₄₁I₁ + m₄₂I₂ + m₄₃I₃ + m₄₄I₄ + m₄₅I₅ ) + j( I_G ( M₄₁ + M₄₂ + M₄₃ )]

Also, the voltage available at the terminals of the solenoid R, when the shield is on, can be calculated as V_sh = V₁ + V₂ + V₃ + V₄. When the shield is off, System 2 is replaced by the equations

V'₁ = j( I_G ( M₁₁ + M₁₂ + M₁₃ )

and the corresponding available V₀ voltage is the sum of V'₁ + V'₂ + V'₃ + V'₄.

This procedure can be generalized by introducing the following matrices and vectors:

in which T is the number of turns of the solenoid T, R is the number of turns of the solenoid R, and S is the minimum number of turns of the shield S necessary to obtain a large enough disk, namely, one encompassing within it essentially all of the magnetic flux produced by the solenoid T.

Thus, if Z_S is posited as representing R_S + j(L_S, then the equation system 1 can be characterized as

and the currents induced in the shield are obtainable via the expression

Then systems 2 and 3 become

respectively, and the induced voltages at the terminals of the solenoid R have respectively the values

when the shield is on and

when the shield is off. The corresponding attenuation A is calculated as 20 log(V₀/V_sh).

This mathematical model accommodates a wide choice of geometrical and electrical parameters and can fit a large variety of physical models of the type depicted in Figure 1.

The Basic Model

The basic physical model used in the study reported here is depicted in Figure 3. There are two parallel coaxial turns of radius r₁ and r₂, whose conductors have circular sections of radius ₁ and ₂, respectively; their planes are a distance d from each other. The parameters that must be computed are the mutual- and self-induction coefficients.

With regard to the former, it is well known that, if the radii ₁ and ₂ are negligible with respect to the radii r₁ and r₂ and to the distance d, then the mutual-induction coefficient M of the two turns has the value

where E₁(k) and E₂(k) are respectively the first- and second-kind Legendre's elliptic integrals:

with

All the elements of the matrices M_ST, m_RS, and M_RT and all the nondiagonal elements of the matrix L_S were computed by means of Equation 4. These were also calculated when the turns were too close each other.

With regard to the total (internal and external) self-induction L of a single turn whose radius is r and whose conductor has an inner radius (with much less than r), a good approximation may be obtained through application of the formula

Again, all the diagonal elements of the matrix L_S were computed. When the condition ( << r was unfulfilled, as in the case of the more-internal rings of the shield (see Figure 2), it was also computed. Moreover, the square sections of the rings were considered as circular sections of equal area. In the proximity of its axis, the shield so decomposed does not comply with the geometric constraints imposed on various radii and distances for Equations 4 and 5 to be valid, but the current field density in this area is very weak, as the magnetic flux, here linked by the more-internal turns, is very small.

Finally, the turn resistances of the diagonal matrix R_S were evaluated by application of the formula for dc resistance.

Figure 3. Geometric references for a pair of parallel coaxial turns in the basic physical shield model.

Experimental Results

An experimental device was set up in order to check both the circuit model described above and the effectiveness of the shielding properties of various plates with different geometric and electric characteristics. This device was formed of:

• A transmitting solenoid T with a length of 78 mm (L_T) and radius of 42 mm (R_T), composed of 36 turns (T) whose conductors had a diameter
2₁ of 2 mm, and carried an impressed sinusoidal current, I_G, of 3 A (rms) of frequency f, chosen to be varied over the range 50–1000 Hz (a range large enough to take into account the usual distortions of industrial waveforms—so that the 20th harmonic could be reached—but narrow enough that a penetration depth great enough with respect to the shield thickness could be obtained).

•A receiving solenoid R of length L_R = L_T and radius R_R =R_T, with R = T turns whose conductors had a diameter 2₂ of 2 mm. (Its distance D from the solenoid T was 10 mm, and the voltages V_sh and V₀ were measurable at its open terminals.)

•A shielding plate of aluminum with a thickness s of 1.5 mm and resistivity of 30.0 nm, which could be inserted orthogonally in the space separating the solenoids, its dis-tances from the solenoids T and R (deliberately made the same) being respectively D_T and D_R, so that D = D_T + s + D_R. (It is important to note that, at a frequency of 1000 Hz, the aluminum skin depth is nearly twice the thickness of this shield.)

In order to carry out the calculations whose results are displayed in Table I, the shield was decomposed into S = 40 rings of square sections, whose sides equaled the shield thickness. So, 65 mm was chosen for the value of the external radius of the bigger ring and 15 mm for the value of the internal radius of the smaller one. The previously mentioned weakness of the fields in the central part of the shield justifies this choice, as the experimental data confirm.

Table I compares mathematical and experimental results. In the last column, the disagreement between the two approaches—due to inaccuracies of both the mathematical model and the measurements—is given as an algebraic difference of attenuations. Note that the calculated attenuation is always lower than the measured one, that is, that a design employing the proposed model suggests that actual shielding may be more effective than modeled values.

Conclusion

Taking into account all the noted inaccuracies of the model (the not-always-fulfilled geometric conditions in Equations 4 and 5 and the formal introduction into the model of a central hole not existing in the real shield), the results in the column of Table I seem acceptable. Therefore, this approach is basically correct.

Because the unavoidable inaccuracies chiefly relate to the decomposition of the shield, the model can be improved by decomposing shield thickness into two or more layers. This is only a matter of computing time and memory. Moreover, such further decomposition allows not only a better approximation of the actual case by employing Equations 4 and 5, but also the possibility of an upper limit for the excitation frequency, as the thickness of the layers remains negligible with respect to the skin depth.

Reference

Antonio Orlandi and Tommaso Scozzafava are on the faculty of the Department of Electrical Engineering of the University of L'Aquila in Poggio di Roio, Italy. They can be reached at orlandi@ing.univaq.it and tosco@ing.univaq.it, respectively.