Parent Category: 2016 HFE

*By Bill Garner, Steve Rosasco, and Larry Burgess *

RF circuits with operating frequencies from MHz to GHz constructed using lumped or distributed technologies almost always require an enclosure for mechanical and electrical integrity. The foremost electrical criteria are grounding and shielding – shielding to contain radiation from the circuit or to prevent external signals from interfering with the circuit.

Another important consideration, however, is the effect the enclosure has on RF circuit performance. An improperly dimensioned enclosure may permit the propagation of RF energy coupled into the enclosure from the circuit which can cause oscillation or amplitude or phase response vs. frequency perturbations. In the following, waveguide theory is applied to select the enclosure dimensions to avoid this problem and to predict the attenuation that can be achieved for coupled fields. This approach may avoid the need for the inclusion of RF absorber material to prevent propagation of energy at the circuit operating frequency and its low order harmonics. The design method is demonstrated with a practical RF amplifier example.

**Waveguide Theory**

Considering the interior of the enclosure as a waveguide allows the selection of enclosure cross-guide dimensions to prevent energy propagation along the length of the enclosure at frequencies below a selectable cutoff frequency. By selecting the cutoff frequency to be above the maximum circuit operating frequency, the enclosure will not propagate energy within the circuit operating frequency band. The enclosure acts as a high pass filter for fields coupled into the enclosure by attenuating fields at frequencies below the enclosure high pass band edge as defined by the cutoff frequency. The enclosure is modeled as a rectangular waveguide with the larger cross-guide dimension (broad wall) denoted by a, and the shorter dimension by b. The RF circuit is typically mounted on the broad-wall surface.

The waveguide propagation mode with the lowest cutoff frequency (longest cutoff wavelength) is the first-order TE_{10} (transverse electric) mode. The cutoff wavelength λ_{1} for a general TE_{mn} mode is given by [1]

λ_{1} = 2a/[m^{2} + (an/b)^{2}]^{1/2}

where m and n are the first and second subscripts describing the propagation mode using the TE_{mn} notation and a and b are the longer and shorter cross-guide dimensions respectively. Using the TE_{10} m and n subscripts in the above equation yields λ1 = 2a which is independent of dimension b because n = 0. Since λ1 = c/f_{1}

f_{1} = c/2a(1)

where c = speed of light in air (for a in inches use c = 1.1811x10^{10} in/sec) and cutoff wavelength and frequency are denoted by λ_{1} and f_{1} respectively. Equation (1) allows the cutoff frequency f_{1} to be set by selection of a.

If energy from the RF circuit or its input or output couple into the enclosure volume and propagate along the length of the enclosure, perturbations of the circuit amplitude or phase vs frequency response or circuit oscillation may occur. To prevent this the enclosure should be designed so the enclosure cutoff frequency f_{1} is above the circuit maximum operating frequency thereby preventing propagation at frequencies in the circuit operating band. Under this condition E and H fields set up within the enclosure do not propagate as a traveling wave, do not transfer power along the length of the enclosure and are attenuated exponentially with distance along the length of the enclosure from the point where they are coupled into the enclosure in accordance with the relation [2]

α(dB/unit length) = (54.6/λ_{1})[1 –( λ_{1}/λ)^{2}]^{1/2}(2)

where λ is the operating wavelength with f = c/λ the corresponding operating frequency.

The length units are the same as the units of dimension a. Since λ1 = 2a for the TE_{10} mode, equation (2) becomes

α(dB/unit length) = (27.3/a)[1 – (2a/λ)^{2}]^{1/2}(3)

Expressing this in terms of frequency results in

α(dB/unit length) = (27.3/a)[1 – ( f/f_{1})^{2}]^{1/2}(4)

When f = f_{1}, α = 0 dB/unit length and for f << f_{1} and a = 1 inch, α= 27.3 dB/in.

The equations above allow the calculation of attenuation once the dimension a and the cutoff wavelength or frequency is known; for enclosure design it is convenient to have an expression that allows direct computation of a for the desired attenuation α(dB/in). Such a formula can be found from equation (4) by substituting f_{1} = c/2a and manipulating to isolate a. This yields

a(in) = 1/[(2f/c)^{2} + (α/27.3)^{2}]^{1/2}(5)

where the units of a are inches if the units of α are dB/in and c are in/sec.

**Practical Application**

The signal attenuation of equations (2), (3) or (4) is experienced by E and H fields coupled into the enclosure at circuit frequencies below f_{1}. Because the frequency is below f_{1}, no traveling wave in either direction along the length of the enclosure will be set up from the point where the fields are coupled into the enclosure. Most RF circuits mounted in an enclosure obtain their external signals from coaxial connectors mounted in an end wall. The center pin of the connector is either hard-wired to a lumped circuit element or mates to a micro-strip line (printed line on a PC board) and if these connections are short with good RF return to the shield of the connector, the RF energy coupled into the enclosure volume is significantly attenuated. RF lumped elements such as air inductors, however, can inject or intercept fields in the enclosure volume, therefore for design margin and simplicity it is advisable to assume 100% coupling. The following example illustrates how equations (4) and (5) are used in the design of the enclosure.

A 500 MHz, four stage, 60 dB gain amplifier is to be housed in an enclosure 6 inches long. Designing for 90 dB attenuation over the 6 inch length of the circuit provides sufficient reduction of high level amplifier output signals from coupling to the circuit input at a level that could cause circuit oscillation or frequency response perturbation. (Additional attenuation is provided by the expected less than 100% efficient coupling between the circuit and the enclosure.) Over the 6 inch length of the enclosure this requires attenuation of 15 dB/in. Using equation (5) with attenuation, α = 15 (dB/in), maximum circuit operating frequency f = 500 MHz, and c = 1.1811x10^{10} in/sec yields the cross-guide width a = 1.799 in. The attenuation can be checked by using equation (4) and inserting the cutoff frequency f_{1} = c/2a = 3.283 GHz, f = 500 MHz and a = 1.799 in which verifies the desired attenuation.

The attenuation of E and H fields coupled into the enclosure from RF circuit harmonics below cutoff for this example can be calculated by use of (4) and are shown in the table below. It is assumed that the fundamental frequency is 0.5 GHz. Note that equation (4) can only be used for frequencies below the enclosure cutoff frequency.

Harmonic no. |
Harmonic |
Attenuation |

2 |
1.0 |
14.5 |

3 |
1.5 |
13.5 |

4 |
2.0 |
12 |

5 |
2.5 |
9.8 |

Figure 1 • Table of Harmonic Attenuation for Amplifier Example.

The table below contains data for enclosure dimension a and cutoff frequency f_{1} for a set of 3 desired attenuations at four operating frequencies. Note that as dimension a is reduced for a selected operating frequency the attenuation increases. Conversely dimension a must be reduced in order to obtain the same attenuation as the operating frequency is increased.

Figure 2. Table of Enclosure Dimensions a and Cutoff Frequencies f_{1} for Different Operating Frequencies f and Attenuations α(dB/in).

The reduction in cross-guide dimension a for increasing attenuation or frequency may be a problem since it will limit the area available to mount the circuit. If a minimum total attenuation over the circuit length is required as in the example above, some increase in a may be achieved by increasing the length of the enclosure to reduce the required attenuation per unit length which will provide some increase in enclosure width a. Evaluation of equation (5) can readily be automated in a spread sheet to compute the increase of a.

If it is not possible to increase the enclosure length to increase cross-guide width as described above, the required α(dB/in) provided by the waveguide cutoff effect can be reduced by adding RF absorber within the enclosure. Absorber materials are available in sheets of various thicknesses with adhesive backing that permit permanent attachment to enclosure interior walls or the lid. Absorbers with a wide range of attenuation characteristics are available. Generally the attenuation of these materials increases with frequency and thickness. Alternate approaches include the use of partitions or shields mounted on the circuit board where good grounding can be arranged. This approach may be appropriate for components or circuits that efficiently couple energy into the enclosure or are highly susceptible to fields in the enclosure.

**References**

[1] Orfanidis, Sophocles J. *Electromagnetic Waves and Antennas,* Chapter 9, equation (9.5.8). (Available at http://www.ece.rutgers.edu/~orfanidi/ewa/orfanidis-ewa-book.pdf)

[2] Terman, F. E. *Electronic and Radio Engineering, *McGraw-Hill, New York, 1955, Sec. 5.8 is a reference for waveguide attenuation below the cutoff frequency.

**About the Authors**

**Bill Garner** received a BSEE from Valparaiso University and did Graduate Studies in Mathematics at Penn State University.

*Publications:*

1. Designing Antennas for Cellular Phones, Microwaves & RF, May 1999

2. Bit Error Probabilities Relate to Data-Link S/N, Microwaves & RF Nov. 1987

3. Quick Design of Symmetrical ‘T’ and ‘H’ Pads, EDN, July 1968

4. Nomograph: Designs Resistive Pi Networks, EDN, Nov. 1967

*Patents***:**

1. Mobile telephony standards converter—US06181951

2. Low voltage variable gain amplifier with feedback—US06052030

3. Dual band cellular/PCS antenna—US05974302

4. Fixed dual frequency band antenna—US05963170

5. Channel usage monitoring arrangement for base station—US05890056

6. Dual frequency band antenna system—US05717409

7. Band-pass filter and support structure therefor—US04940956

**Steve Rosasco** received a BSEE from Villanova University, an MSEE from New York University and a Ph.D. in Electrical Engineering from the University of Pennsylvania. He has been involved in the design, development and test of EW, communication, adaptive array, interference cancelation and radar systems covering frequencies from DC to Ku band.

**Larry Burgess** received a BS and MS from MIT and a Ph.D. from University of Pennsylvania, all in Electrical Engineering. He has designed and evaluated antennas, transmitters, and receivers at frequencies from 2 MHz to 6 GHz for both military and commercial applications. After working at large companies and startups for several decades, he resides in Silicon Valley and is the Principal at Wireless Consulting Services.

### March 2020

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