Time to market, tighter budgets and restrictive standards are all too common challenges facing today’s EMI/RFI engineers. These pressures are exerted not only at the system level and subsystem level but also at the circuit level. One such circuit, the noise suppression filter, is no exception.

EMI filters take up valuable space in today’s high-density electronic designs because the more stringent filtering requirements translate into increases in size and weight. With all these variables, it is important to determine filter performance very early in the design phase.

The best way to find out how a filter will perform is through simulation. A recently proposed technique demonstrates the response of an EMI filter using measured or simulated time-domain waveforms.

As electronic design progresses, different subsystems invariably come to fruition at different times in the development phase. As the subsystems become available, the measured response of the subsystems can be used to model and refine the EMI filter design.

Noise Characterization

The first step in designing any EMI filter is to characterize the noise emanating from the assembly. The spectral content of the noise sources must be determined to verify if the conducted emissions exceed a predefined level. The characterized conducted emissions dictate the required filter design. Once the filter is designed, its intended attenuation characteristic must be verified.

A method for characterizing the noise sources using time-domain data speeds up the process. The procedure imports the time-domain waveforms to PC-based software programs such as Mathcad or Pspice, which can perform an FFT on the data. The frequency-domain waveforms resulting from the FFT operations provide spectral content information about the data. The waveforms are also useful inputs to the filter models.

Various specifications dictate the allowable conducted emissions that a given equipment may induce into the power distribution system feeding it. A typical specification for the allowable conducted emissions on the input power lines of a switching power supply is shown in **Figure 1**.

The conducted emissions specification is given in the frequency domain. Consequently, the spectral content of the conducted emissions must be derived from the time-domain data to compare the measured data with the specification. This is done by performing an FFT on the time-domain data.

**Conducted Emissions Data Processing**

* **Data Acquisition***: **The time-domain measurement of the conducted emissions is performed with an oscilloscope. The oscilloscope must provide a method of permanently storing the waveforms so the modeled filter performance can be verified with actual test data.

Many digital oscilloscopes can store measured waveforms in ASCII code on a diskette. The stored data can be imported to PC-based software programs for editing, performing Fourier transform analysis and other necessary data processing. **Figure 2 **shows the time-domain current measurement on the input power lines of a switching power supply.

EMI Filter Response

To predict the performance of an EMI filter, it is necessary to determine how the filter affects the signals that pass through it. The portion of the frequency spectrum of a signal that exceeds the conducted emissions specification limit is the unwanted entity.

**Figure 3 **shows those portions of the frequency spectrum of a switching power supply that exceed the specification limit and the magnitude by which the conducted emissions exceed the specification. Only the portion of the switching power supply frequency spectrum that exceeds the specification requires attenuation by the EMI filter.

Filter Input Output Relationship

A filter that uses only passive components which do not vary with time can be considered a linear time invariant (LTI) system. For an LTI system with no initial stored energy, the output as a function of time is related to the input as a function of time by the convolution integral:

y (t) =where x(t) is the input to the filter, y(t) is the output and h(t) is the response of the system to the impulse function d(t) and referred to as the impulse response of the filter.

The Fourier transform of the filter impulse response, h(t), is referred to as the transfer function of the system and is written as H(f).

Convolution in the time domain is equivalent to multiplication in the frequency domain:

Y(f) = X(f)·H(f) (2)

where X(f) and Y(f) are the Fourier transforms of x(t) and y(t). The frequency domain functions X(f) and H(f) are, in general complex functions;

X(f) =The output Y(f) can, therefore be expressed as:

Y(f) =Taking the decibel of the magnitude of both sides, we get

½ Y½ dB = ½ X½ dB + ½ H½ dB (4)

where X(f) and Y(f) are the Fourier transform results of x(t) and y(t). The frequency domain functions X(f) and H(f) are complex functions:

This equation indicates that if we know the spectrum of the input signal, X(f), and the transfer function of the filter, H(f), then the magnitude of the output of the filter, Y(f), is found by adding the dB of the magnitude of the signal spectrum and transfer function. Equation 4 essentially transforms the complex convolution problem into a simple addition problem. This concept is the basis for this filter analysis.

Filter Transfer Function

In an LTI system, a sinusoidal input will product a sinusoidal output of the same frequency but generally with a change in amplitude and phase. Since all signals can be decomposed into a summation of sinusoids, the response of an LTI system to a signal represented as a summation of sinusoids is an output with sinusoids of the same frequencies as the original summation but with different magnitude and phase. This phenomenon provides a method for determining the transfer function of a modeled filter through simulation.

The Fourier transform of the impulse function has unity magnitude throughout all frequencies, meaning that the impulse function has a spectral density that contains every frequency with equal weight. This hints at a convenient method of finding the impulse response of an LTI system.

Since the delta function contains every frequency with equal weight, it is equivalent to a sinusoid of unity magnitude throughout all frequencies. Applying a unity magnitude sinusoidal signal throughout all frequencies is equivalent to inputting the delta function to the system.

The swept sinusoidal input is easily implemented in many circuit-analysis packages, such as Pspice or ECA, using the AC analysis feature. Because sinusoidal inputs to an LTI system result in sinusoidal outputs, the output due to the swept unity magnitude sinusoid results in the transfer function of the system, H(f), at the output.

The response of the EMI filter under investigation was determined by simulation using the ECA analog circuit simulator. The simulation results were stored in an ASCII file and read by Mathcad as an array. The array was then plotted in Mathcad as shown in **Figure 4**.

EMI Filter Output

Now that the spectral content of the conducted emission, X(f), and the filter transfer function, H(f), are known, the output of the EMI filter can be determined by equation 4. The results are shown in **Figure 5**.

Conclusion

The measurement and digital storage of time-domain waveforms, in combination with the FFT capabilities of many analytical software packages, present a very powerful tool for analysis. The results shown in Figure 5 demonstrate the benefits of this type of analytical approach.

The technique provides a simple method for determining both the frequency content of the conducted emissions and the response of a modeled EMI filter to the conducted emissions. The method differs from the more common method of conducted emissions measurements, which usually involve frequency-domain measurements using spectrum analyzers or EMI receivers.

This technique provides an alternative for analyzing conducted emissions with oscilloscopes because they are readily available to design engineers. Also, time- or frequency-domain waveforms can be manipulated with various software packages. This type of measurement has many advantages over spectrum analyzer measurements which do not posses the signal-phase information.

Time-domain measurements result in real-time signals that posses all the phase information as time shifts. As a result, the FFT of the time-domain signals result in complex functions containing the phase information necessary for performing many complex operations.

**References**

1. Fluke, J.C., *Controlling Conducted Emissions By Design,* Van Nostrand Reinhold, 1991.

2. Munoz, R.G., “Electromagnetic Interference Filter Design for the Control of Conducted Emissions,” MSEE Thesis, Loyola Marymount University, December, 1994.

3. Poularikas, A.D., and Seely, S.,

Signals and Systems,PWS-Kent Publishing Co., 1991.4. Ramirez, R.W.,

The FFT, Fundamentals and Concepts, Prentice Hall, 1985.

Acknowledgment

This article is based on “Simulation of Response of Modeled EMI Filters to Measured or Simulated Time Domain Waveforms,” by Raul G. Munoz, Allied Signal Aerospace Equipment Systems, and Tai-Wu Kao, Loyola Marymount University, which appeared in 1995 IEEE International Symposium on Electromagnetic Compatibility, Atlanta, GA, Aug. 14-18, 1995, pp. 272-275. Ó 1995 IEEE.

Figure 1: Specification Limit

Figure 2: Switching Power Supply Time Domain Current Waveform

y (t) =Figure 3: Switching Power Supply Conducted Emissions

where x(t) is the input to the filter, y(t) is . . .

Convolution in the time domain is equivalent to multiplication in the frequency domain, therefore,

Y(f) = X(f)·H(f) (2)

where X(f) and Y(f) are the Fourier transforms of x(t) and y(t). The frequency domain functions X(f) and H(f) are, in general complex functions;

X(f) =The output Y(f) can, therefore be expressed as:

Y(f) =Taking the decibel of the magnitude of both sides, we get

½ Y½ dB = ½ X½ dB + ½ H½ dB (4)

This last equation indicates that if we know . . .

Figure 4: EMI Filter Differential Mode Transfer Function

Figure 5: Switching Power Supply, Filtered Conducted Emissions

Copyright 1995 Nelson Publishing Inc.

December 1995