3.4 Radiation from Step Function Waves
In logic structure, the waves are step functions and not sinusoids. A spectrum of frequencies with different amplitudes makes up this wave, and the radiation at each frequency is different. The higher frequency components are more efficient radiators. It is difficult to relate this type of signal to wave impedance. We can assume that the fields that are generated by loops will have a high H field content. It is correct to say that loops generate fields that have a low wave impedance character.
The ratio of E/H has units of volts per ampere or simply ohms. There is no available way to measure wave impedance directly.
It is interesting to note that characteristic impedance and wave impedance are both measured as the ratio of field intensities. In transmission lines, characteristic impedances are usually below 300 ohm. Near dipoles, where the H field is very small, the wave impedance can be well over 5000 ohm.
At a distance from a sinusoidal radiator, the ratio of E field to H field intensity in space is 377 ohm. These waves are referred to as plane waves. The distance from a sinusoidal radiator where the wave is considered to be a plane wave is λ/2π, where λ is the wavelength of the sine wave signal. This length is called the near-field/far-field interface distance. Beyond this distance, both fields attenuate proportional to the distance, and the ratio of field intensities is constant at 377 ohm. Near loops, the wave impedance can be a few ohms. This type of field is called an induction field. The fields near a radiating dipole are of high impedance.
The impedance of a field says a lot about how it can be shielded. In general, high impedance fields can be shielded by a sheet of conducting material. Utility power generated fields are usually of low impedance, and they are hard to shield. On a circuit board with logic traces where the radiation comes from loops, the wave impedances are considered to be low.
For nonsinusoidal fields, it is useful to assign a single frequency for an analysis. This frequency is equal to 1/πτr, where τr is the rise time of the waveform.2 This frequency can be called the rise-time frequency. This frequency is sometimes called the maximum pulse frequency. To obtain a measure of how a circuit might respond to a pulse or step function, this single sinusoid can be used as an input. The peak value of the sinusoid selected should be equal to the peak value of the step function or pulse. The response of the circuit or system to this sine wave signal can be an indicator of what to be expected when an actual transient takes place. For transmission lines, the wave impedance near the line can be considered the characteristic impedance of that line. Typically, this impedance is about 50 ohms.3 For logic with a rise time of 100 ps, the rise-time frequency is about 3 GHz. The near-field/far-field interface distance using this frequency is about 2 cm. Beyond this distance, the wave impedance for a wave is 377 ohm. This means that it is relatively easy to shield fast logic if it is needed.
The radiation from a circuit board comes from identifiable loop areas. The loops include the spacing between the traces and their signal returns, connections to decoupling capacitors, the connections to components, and the loop areas associated with vias. The individual loop areas are small, but their numbers can be large. It could be argued that the orientation of these loops is random, and this limits the total radiation. In any worst case analysis, it is a good idea to consider all radiation as additive.
An approximate equation for calculating radiation from a single loop is
where E is in decibel-microvolts per meter, I is in milliamperes, A is the loop area in squared centimeters, r is the distance to the radiator in meters, and f is the rise-time frequency in megahertz. This equation provides an rms measure of field strength without regard to waveform. For an entire board the total value for E is the sum of the radiation from each loop. Different logic families will require different rise-time frequencies. In this situation, the total radiation is the rms sum for each frequency group. Note that the E field falls off proportional to distance not as the square. The energy density falls off as the square of distance.
The loop areas formed by traces on outer layers must be considered first. The traces between conducting planes do not radiate directly into space. If inner traces do couple to energy, they can radiate this energy when they cross to an outer layer (see the discussion in Section 3.11).
Interference fields that are created by other circuits or external hardware can couple to entering cables. This interference can be carried to all parts of the board, including the space between conducting planes. Because transmission lines are two-way streets, connecting cables can also carry interference to the outside world.
Coupling processes occur in areas where waves are in transition. Even though there is energy in motion behind a wave front, there is no coupling where the wave is not changing. If the length of a transmission line is equal to the distance, a wave travels during its rise or fall time, then all points on the trace will radiate at the same time. Note that in Equation 3.1, the loop area is limited to the area where the wave is in transition. For a 100-ps rise time, the loop length is only 3.5 cm.