**Abstract **

There are two commonly used methods for approximating the attenuation of shielding barriers. This approximation is defined as shielding effectiveness (SE) for shielding materials used in the design of shielded enclosures. Both methods use wave theory and quasi-stationary assumptions. One of the methods uses Maxwell’s equations to estimate the shielding, and the other uses the correlation between transmission lines and radiated waves. This article proposes a third method based on circuit theory (Kirchhoff’s Law) as an applicable method of approximation worthy of consideration.

**Introduction **

The two common methods of estimating the shielding effectiveness of material used in the design of shielded enclosures require the understanding and use of wave theory and Maxwell’s equations. Very few working engineers understand, and therefore properly use wave theory and Maxwell’s equations. Therefore they find it difficult to evaluate the materials used in the shielding of electromagnetic waves for compliance to the various commercial and DoD EMC requirements.

A method of estimating the shielding quality (SQ) of materials used in the design of shielded enclosures using circuit theory (Kirchhoff’s Law) is included in this article. The advantages of using a circuit theory analogy are: (1) the ease by which the average design engineer can understand the variables and application of the theory; (2) these advantages will greatly assist the design engineer in selecting the proper material for meeting specific shielding requirements; and (3) the approximate magnitude of both the E and H fields emanating from a shielding barrier material can be easily obtained.

The paragraphs that follow will describe:

**1. **The generation and propagation of an electromagnetic wave.

**2. **The development of the attenuation factors associated with specific shielding materials.

**3. **Development of equations for estimating the shielding quality of specific barrier materials for both the E and H fields of an electromagnetic wave.

**4. **Boundary conditions and constraints associated with the theory.

**5. **Comparative analysis of shielding materials using wave theory and the circuit theory contained herein.

**Generation and Propagation of EM Fields **

The undergraduate courses on electromagnetic theory introduce the concept of an electromagnetic (EM) field by driving a pair of parallel plates with an AC voltage source as illustrated in *Figure 1*. The current that flows through the wire comes from the top plate and is stored in the bottom plate. The over presence of the electrons on the bottom plate is illustrated by plus symbols (+) and the absence of electrons on the top plate is illustrated by minus symbols (-). This creates an electromagnetic field which is illustrated in *Figure 2*. As is illustrated, a field exists between the plates. The magnitude of the E field is equal to the voltage differential between the plates divided by the distance between the plates in meters. The resultant E field is in volts/meter (e.g., we use a set of parallel plates for performing E field susceptibility testing to MIL-STD-461/462).

As is illustrated in *Figure 2*, the lines of flux in the center of the plates are straight and flow from the bottom to the top plate. At the edges they bow out, where the fields or lines of flux repel each other, forcing the bowing. The field that bows out is an EM field where the E vector quantity is equal to the voltage divided by the length of the force line in meters (i.e., if the point of concern is one meter from the set of plates, the E field would be the voltage across the set of plates divided by the circumference of the circle or approximately E/3.1). The magnetic or H field is approximated by the following equation:

**Suppression (Shielding) of EM Waves **

When we place a shielding barrier in the path of the EM field, the force of the field causes current to flow in the barrier. As is illustrated in *Figure 3, *the excess electrons in the bottom plate create a force on the electrons in the barrier. This force causes the electrons to flow away from the point of contact. In a similar manner, the lack of electrons on the upper plate will create an excess of electrons on the barrier at the upper point of contact. This current flow in the barrier is called the “surface current density” (Js) in amperes/meter, and is equal to the H field incident on the barrier when the field is perpendicular to the barrier. The current flowing in the barrier is attenuated by the skin effect.

The current on the transmitted side is equal to JSI e-d/δ (i.e., the current on the incident side attenuated by skin effect). The impedance of the field emanating from the barrier is equal to the impedance of the barrier. The values of ET and HT are as illustrated in *Figure 3 *and are as follows:

**Shielding Quality of Shielding Materials **

The definition of shielding quality as used herein is the difference in dB between the E field and H field of the wave incident on the barrier and the wave emanating from the barrier on the opposite side, i.e.,

From *Figure 3 *we know that the E field in the barrier on the incident side is equal to the H field (i.e., Js) times the impedance of the barrier. Therefore, we can conclude that the ratio of the E field of the incident wave to the E field in the barrier on the incident side is:

We also know that the impedance of the incident wave is equal to:

Substituting EI/ZW for HI in *Equation 6 *we can conclude that the ratio of the E fields in the incident wave and the E field in the barrier on the incident side equals:

From *Figure 3 *we also note that the E field in the barrier is attenuated by the skin effect, i.e.,

From *Equations 4, 8 *and *9 *we can conclude that the shielding quality of material used in a shielding barrier for the E field is:

Using the same logic and the information of *Figure 3 *we can conclude that the shielding quality of a material for the H field is:

**Comparative Analysis **

A comparative analysis at 1 MHz has been performed comparing the results of the shielding effectiveness of an aluminum shield using the accepted SE = R + A + B formula derived from wave theory and the shielding quality equations derived from circuit theory (see *Appendix A *for analysis).

The conditions used for the comparative analysis are consistent with the test conditions of an earlier paper entitled “Shielding Effectiveness Test Results of Aluminized Mylar.” These conditions are as follows:

**1. **The aluminum shield is aluminized Mylar having a dc resistance of 1.4 ohms/square. The thickness of the aluminum (based on the resistance) is 2 x 10-8 meters and has a theoretical impedance of 2.0 ohms.

**2. **The impedance of the wave at the shield radiating from the loop antenna is 4.0 ohms.

**3. **The impedance of the wave at the shield radiating from the electric dipole antenna is 3500 ohms.

The results of this analysis along with the results of the test contained in the earlier paper are illustrated in *Table 1. *These results are as follows:

**1. **Attenuation to E field. The analysis using equations derived from wave theory and circuit theory yielded a close approximation to the E field from both the electric and magnetic dipole antennas.

**2. **Attenuation to H field. The analysis using the equations derived from circuit theory gave a very close approximation. However, the analysis using the equations derived from wave theory resulted in an error of more than three orders of magnitude using the electric dipole antenna as the radiating source.

We can conclude from the results of the “SE=R+A+B” equations derived from wave theory that the equations were intended to predict the attenuation of only the E field through a shielding barrier.

The comparative analysis contained in the appendix contains a significant amount of information. Of particular concern are the results of the analysis contained in *Table A-l *(of the Appendix) using the wave impedance consistent with the magnetic dipole (loop) radiation source (4 ohms) and the thickness of the shield of 2 x 10-8 meters. From the explanation contained in the books and papers on shielding theory using R + A + B we learn that the reflection coefficient “R” represents a ratio of power reflected from the shield material to that which penetrates into the shield material. The 66.5 dB means that if 1 watt of power is incident on the shield, 2113 units are reflected for each unit that penetrates into the barrier (99.95% is reflected and .05% or .5 milliwatts penetrate the barrier). The shielding effectiveness level of 3.1 dB implies that 20% of the 1 watt (or 700 milliwatts) is observed on the secondary side of the shield material. This means that the re-reflection coefficient amplifies the energy which penetrates the shield by 140,000%. This amplification is obviously not possible and means that the explanation is faulty. It can also be noted using the equations of SE = R + A + B derived from wave theory that the impedance of the barrier ZB is calculated to be 4 orders of magnitude less than the actual impedance using a resistance bridge when the barrier was 2 x 10-8 meters thick (i.e., the impedance of the barrier is the same regardless of the thickness of the barrier).

**Conclusion **

The shielding quality equations which have been derived from circuit theory provide a close approximation of the attenuation of a wave through a barrier and are far easier to understand by the average design engineer than the presently used shielding effectiveness equations. The equations also provide information that is more appropriate to the design engineering community, i.e.,

The voltage induced into a circuit is a function of the wave impedance and the impedance of the circuit. If a design engineer uses 377 ohms instead of the 2 ohms emanating from the aluminized Mylar shield in performing a susceptibility analysis of a piece of electronic equipment, the calculated induced voltage can be off by more than two orders of magnitude.

Shielding quality as a measure of the attenuation characteristics of a shield is considered a more appropriate term. Shielding effectiveness is a well-defined term and possesses a specific connotation within the engineering community. However, the definition is not well understood. For example, suppose an engineer performs a susceptibility test on equipment circuits and discovers that he needs 40 dB of shielding to comply with his requirements. He selects a shield that renders 60 dB of shielding effectiveness using the shielding effectiveness equations. Upon retest after manufacturing his shield, he finds he still need 20 dB of shielding.

The term shielding effectiveness implies a level of shielding the engineer is going to obtain. In the above case the results are a level 40 dB less than is expected. There is nothing associated with the equations that can explain the results to him where the problem could easily be the distance from the shield material to the circuits being affected by the radiated field. The term of shielding quality defines the attenuation of a field by the shield material, and that definite information with regard to the field of the incident wave as well as information associated with the susceptibility of the circuits is required. Once the required information is available, a ready solution can be obtained.

The use of the shielding quality equations derived from circuit theory are more consistent with the principles associated with the engineering discipline than are the shielding effectiveness equations, especially when the shielding barrier is in close proximity to the EM source (near field).

**Selected Bibliography **

**1. **Hallen, Erik, “Electromagnetic Theory,” John Wiley and Sons Inc., New York, 1962.

**2. **Halme, Lauri and Amnanpalo, Jaakku, “Screening Thoery of Metalic Enclosures,” 1992 IEEE International Symposium on EMC, Anaheim 1992.

**3. **Hershberger, “Introduction to Electromagnetic Theory,” Class Notes, ENGR 117 A/B, UCLA, 1961.

**4. **Hyat, William H., Jr., “Engineering Electromagnetics,” McGraw-Hill Book Company, New York, 1958.

**5. **Johnson, Walter C, “Transmission Lines and Networks,” McGraw-Hill Book Company, New York, 1950.

**6. **Kunkel, George M., “Shielding Theory (A Critical Look),” IEEE, International Symposium on EMC, Cherry Hill NJ, 1991.

**7. **Ott, Henry W., “Noise Reduction Techniques in Electronic Systems,” John Wiley and Sons, 1976.

**8. **Rashid, Abul, “Introduction to Shielding – Boundary Conditions and Anomalies,” 1992 IEEE International Symposium on EMC, Anaheim 1992.

**9. **Schulz, R.B., “Shielding Theory and Practice,” Proceedings of the Ninth Tri-Service Conference on EMC, 1963.

**10. **White, Donald R.J., “Electromagnetic Shielding Materials and Performance,” Don White Consultants, Inc., 1975.

**Appendix A **

**Shielding Effectiveness Versus ****Shielding Quality Analysis **

Included is an analysis for estimating the shielding effectiveness of aluminum shielded barriers using the equations consistent with wave theory and R + A + B technology and estimating the shielding quality of the same barriers under the same conditions using the equations contained in the body of this article.

The shielding effectiveness equations of concern associated with wave theory are:

The results of the analysis are shown in *Table A-l*.

The equations used for calculating the shielding quality of the shielding material using circuit theory and contained in the body of this article are:

The results of the analysis are shown in *Table A-2*.

* This shielding effectiveness estimate is for both E and H fields (i.e., the shielding effectiveness for both fields is stated to be the same).

** The ZB equation and value in the shielding effectiveness equations assumes the barrier is infinitely thick.

where δ thickness in meters is applicable for an infinitely thick barrier.

The equation for a barrier of any thickness is:

**Appendix B **

**Shielding Effectiveness Approach ****to Shielding Theory **

The use of Maxwell’s equation to obtain the Shielding Effectiveness (SE) of a shielding material requires compliance to “Stokes Function” (the sum total of all power entering or leaving a given area must equal zero unless there is a sink or source of power). This method if properly applied will provide the engineering community with values of “SE” and the attenuation for the E and H field that are useful to the design engineer.

The wave theory approach (as presently interpreted) does not meet the requirements of “Stokes Function”. The present interpretation stipulates that the power loss to an H field inside the barrier is equal to the power loss associated with an E field being reflected at the incident side of the barrier. This does not occur for the following reasons:

**1. **Broaddus and Kunkel did not detect an H field loss.

**2. **When the barrier is thick, skin effect prevents the EM wave from reaching the back side of the barrier. This fact eliminates the possibility of an H field reflection.