Radiated Emission Analysis
Theory and Physics
Overview — What is Radiated Emission?
How do you simulate radiated emission? Do you mesh the entire enclosure?
Good question. To get straight to the point, meshing the entire enclosure all at once is not practical. PCB patterns are on the order of sub-millimeter detail, but the test distance for the 3-meter method is 3,000mm—the scales differ by more than three orders of magnitude. Trying to solve this with a single mesh would result in a huge problem with billions of cells.
Then how do you do it?
Leakage from enclosure slots and cable exit ports is dominant. So in practice, a multi-stage approach is used: first simulate the near field at the PCB level, then pass those results to an enclosure-level FDTD or MoM solver. The correlation accuracy with actual 3-meter method measurement results is ±6dB as the industry standard.
±6dB seems quite large, doesn't it? That's a 4x difference in power, right?
Sharp observation. In terms of voltage ratio, it's a factor of 2. In the EMC world, ±6dB is considered "practically reliable" accuracy. Variations in cable routing and connector contact resistance alone can cause fluctuations of 5–10dB even in mass-produced units. That's why in simulation, designing with a 6dB margin is the realistic solution.
Radiated Emission (RE) refers to electromagnetic waves unintentionally radiated from electronic equipment. Limits are defined by international standards such as CISPR 32 (for information technology equipment) and CISPR 25 (for automotive equipment), regulating the frequency band from 30 MHz to 6 GHz (varies by standard). Failure in EMC testing prevents market release, making simulation prediction during the design phase extremely important.
Radiation Mechanism — DM Radiation and CM Radiation
Where does the electromagnetic wave come from on the board? Is it from everywhere uniformly?
There are two main paths: Differential Mode (DM) radiation and Common Mode (CM) radiation.
- DM Radiation: Signal current and return current form a loop on the PCB, radiating as a tiny magnetic dipole. Radiation intensity is proportional to the fourth power of frequency, so high-frequency clock signals become more problematic.
- CM Radiation: The entire cable becomes charged in phase, acting as a monopole antenna. Even a few μA of common mode current can exceed CISPR limits; 80% of EMC test failures in the field are attributed to this cause based on practical experience.
What? A few μA of common mode current can cause failure? Such a tiny current?
Exactly. For example, a 5μA common mode current flowing through a 1m cable generates an electric field strength of about 40 dBμV/m at 100 MHz at a 3m distance. The CISPR 32 Class B limit is 30 dBμV/m, so it's easily exceeded. Therefore, accurately modeling the common mode current path is what determines simulation success or failure.
Governing Equations and Far-Field Transformation
The starting point for radiated emission analysis is Maxwell's equations.
For transformation from near field to far field, Far-Field Transformation using equivalent electromagnetic currents is used. From the electric field $\mathbf{E}_s$ and magnetic field $\mathbf{H}_s$ on a closed surface $S$, the far-field electric field $\mathbf{E}_{far}$ at distance $r$ is calculated by the following equation.
Here $k = 2\pi/\lambda$ is the wavenumber, $\eta_0 = 377\,\Omega$ is the free space impedance, and $\hat{n}$ is the surface normal vector. This transformation allows fast calculation of electric field strength at any measurement distance after solving within a small computational region near the enclosure.
Huygens Equivalent Principle (Huygens Equivalent Source)
Where is the Huygens equivalent principle used in the multi-stage analysis?
It's precisely the "adhesive that connects the stages." It's used to input the near field obtained from PCB-level analysis as the excitation source for enclosure-level analysis. Specifically, it converts the electric and magnetic fields on a closed surface surrounding the PCB into "equivalent current sources."
The equivalent current sources on the Huygens surface $S_H$ are defined as follows.
$\mathbf{J}_s$ and $\mathbf{M}_s$ are defined at every point on the surface, and these equivalent sources alone can completely reproduce the electromagnetic field outside the closed surface. This is the Equivalence Theorem (Love's Equivalence Theorem). In multi-stage analysis, the near-field data obtained from the inner analysis is converted into equivalent sources on the Huygens surface and passed to the outer analysis domain.
Cable Common Mode Radiation Model
Earlier you said "common mode current is the main cause of EMC test failure." What does the specific radiation model look like?
Clayton Paul's model is classic and practical. When a common mode current $I_{CM}$ flows through a cable of length $L$, the electric field strength at measurement distance $d$ can be approximated by the following equation.
Converting to dBμV/m,
Here $f$ is frequency [Hz], $L$ is cable length [m], $I_{CM}$ is common mode current [A], $c$ is the speed of light, and $d$ is measurement distance [m]. Resonance occurs when the cable length is an integer multiple of half-wavelength $L = n\lambda/2$, causing radiation to peak.
I see! Radiation increases or decreases based on the relationship between cable length and frequency. So shortening the cable is advantageous?
Exactly. Shortening the cable length is a fundamental EMC countermeasure. However, it only shifts the resonance frequency, so looking across a broad band, peaks may appear at other frequencies. The fundamental solution is suppressing the common mode current itself, for which inserting ferrite cores or shielding cable exit ports is effective.
Physical Meaning of Each Term in Far-Field Transformation
- $e^{-jkr}/r$ term: Represents spherical wave propagation. The electric field attenuates inversely proportional to distance $r$, and the phase rotates. The ~10 dB difference between the 3m and 10m methods is due to this term.
- $\hat{n} \times \mathbf{E}_s$ (Equivalent magnetic current): Tangential component of the electric field on the Huygens surface. Describes radiation from slots and apertures. This is why enclosure openings are the main leakage paths.
- $\hat{n} \times \mathbf{H}_s$ (Equivalent electric current): Tangential component of the magnetic field on the Huygens surface. Describes radiation from loop currents on the PCB or common mode currents on cables.
- $e^{jk\hat{r}\cdot\mathbf{r}'}$ (Phase term): Phase delay from each point on the surface to the observation point. Determines the far-field pattern (directivity). When slot dimensions become large relative to the wavelength, strong directivity appears.
Assumptions and Applicability Limits
- Far-field condition: Observation distance must satisfy $r > 2D^2/\lambda$ ($D$ is the maximum antenna dimension). For a 1m cable at 30 MHz, the strict far-field condition is over 20m, but the 3m method includes corrections for this.
- Linear medium assumption: Material permeability/permittivity does not depend on electric field strength. Nonlinear models are needed to handle saturation characteristics of ferrite cores.
- Perfect conductor approximation: Ignores losses due to finite conductivity of enclosures. For thin plates, consideration of transmitted waves is needed based on the relationship between skin depth and plate thickness.
- Thin-wire approximation for cables: Valid when cable cross-sectional dimensions are sufficiently small compared to wavelength. This approximation breaks down for thick cable bundles in the GHz band.
Dimensional Analysis and Unit Systems
| Physical Quantity | SI Unit | Notation in EMC Practice |
|---|---|---|
| Electric Field Strength $E$ | V/m | dBμV/m (1μV/m = 0 dBμV/m) |
| Magnetic Field Strength $H$ | A/m | dBμA/m |
| Common Mode Current $I_{CM}$ | A | dBμA (1μA = 0 dBμA) |
| Free Space Impedance $\eta_0$ | Ω | 377 Ω (= 120π Ω) |
| Wavenumber $k$ | rad/m | $k = 2\pi f/c$ |
| Skin Depth $\delta$ | m | $\delta = 1/\sqrt{\pi f \mu \sigma}$ |
Numerical Methods and Implementation
Multi-Stage Analysis Approach
How exactly do you divide the "stages" in the multi-stage approach?
Typically, it's three stages.
- Stage 1 — PCB Near-Field Analysis: Use circuit simulators (SPICE, etc.) or full-wave 2.5D analysis (Ansys SIwave, Cadence Sigrity, etc.) to obtain the PCB's current distribution and acquire the near field just above the board.
- Stage 2 — Enclosure-Level Analysis: Input the near field from Stage 1 as Huygens equivalent sources into the enclosure model. Calculate leakage from enclosure slots and cable exit ports using FDTD or MoM.
- Stage 3 — Far-Field Transformation: Perform far-field transformation from the Huygens surface data on the enclosure's outer surface to calculate electric field strength at 3m/10m distances. Compare with CISPR limits.
Doesn't accuracy degrade during data handover between stages?
Good point. If the sampling density on the Huygens surface is insufficient, high-frequency components are lost. As a guideline, sampling at intervals of λ/10 or less is needed to avoid accuracy degradation. At 1 GHz, that's 30mm spacing; at 3 GHz, 10mm spacing. CST Studio Suite has an "System Assembly and Modeling (SAM)" feature that automates handover between stages, widely used in practice.
FDTD Method (Finite-Difference Time-Domain)
The FDTD method calculates electric and magnetic fields staggered by half a step in time and space on Yee cells. Its greatest advantage in radiated emission analysis is that a single calculation can obtain a broad spectrum (30 MHz to several GHz).
Stability condition (CFL condition):
| Parameter | Recommended Value | Notes |
|---|---|---|
| Cell Size | ≤ λmin/20 | Based on wavelength of highest frequency |
| Absorbing Boundary Condition | CPML (8–12 layers) | More stable and broadband than PML |
| Excitation Source | Gaussian pulse / Modulated sine wave | Choose based on analysis band |
| Calculation Termination | Residual energy −30 dB | Insufficient termination causes error at low frequencies |
MoM (Method of Moments)
How is MoM different from FDTD? How do you decide which to use?
MoM discretizes only conductor surfaces, making it suitable for open-space problems like cables and enclosures. Since volume meshing is unnecessary, computational cost is significantly reduced for problems with large spaces. However, it's not good with non-homogeneous materials (like dielectric fillings). Altair Feko is a representative MoM solver, strong in predicting common mode radiation from cable harnesses.
Basic MoM formulation (EFIE: Electric Field Integral Equation):
Here $G(\mathbf{r},\mathbf{r}') = e^{-jk|\mathbf{r}-\mathbf{r}'|} / (4\pi|\mathbf{r}-\mathbf{r}'|)$ is the Green's function. MoM expands this integral equation using RWG (Rao-Wilton-Glisson) basis functions to obtain a dense matrix equation. For large-scale problems, the Multilevel Fast Multipole Method (MLFMM) reduces computational cost to $O(N \log N)$.
High-Frequency FEM
FEM is characterized by flexibility in handling complex shapes and non-homogeneous materials, with Ansys HFSS being a representative solver. It discretizes the vector Helmholtz equation using edge elements (Nedelec elements).
| Method | Strengths | Weaknesses | Representative Tool |
|---|---|---|---|
| FDTD | Broadband, transient analysis | Discretization of curved surfaces, resonant structures | CST Studio Suite |
| MoM | Cable radiation, open problems | Dielectrics, volume problems | Altair Feko |
| FEM | Complex shapes, multiphysics | Computational cost for broadband | Ansys HFSS |
Mesh Requirements and Computational Cost
The mesh for radiated emission analysis is different from structural analysis, right?
Completely different. In structural analysis, you refine areas of stress concentration, but in electromagnetic field analysis, the wavelength of the highest frequency is the benchmark. If analyzing up to 6 GHz, the wavelength is 50mm—so a cell size of 2.5mm or less (1/20) is required. For an enclosure of 300mm×200mm×100mm, that's 120×80×40 = about 400k cells. Adding the PML region brings it close to 700k cells.
700k cells just for that... Adding PCB patterns would be a nightmare.
That's exactly why the multi-stage approach is necessary. PCB patterns require sub-0.1mm detail, so including them would result in billions of cells, which is not practical. The standard practice is to calculate the PCB separately with a 2.5D solver and pass it as equivalent sources.
FDTD vs FEM Analogy
The FDTD method is like a "video camera taking continuous snapshots"—it calculates electromagnetic wave propagation sequentially over time, obtaining information for all frequencies in a single run. FEM, on the other hand, is like a "radio tuned to a specific frequency"—it finds a precise solution for one frequency. FDTD is advantageous for screening in broadband EMC testing (30MHz–6GHz), while FEM is better suited for detailed analysis at specific resonance frequencies.
Practical Guide
Analysis Flow
If you were to actually run a radiated emission simulation, what's the first step?
The procedure is as follows.
- Identify Noise Sources: List clock frequencies and their harmonics, switching power supply ripple frequencies. Cross-reference with CISPR standard limits to narrow down problematic frequency bands.
- Acquire PCB Near Field: Obtain PCB current distribution via circuit simulation or 2.5D electromagnetic field analysis. If near-field scanner measurement data is available, using it reduces model error.
- Enclosure
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