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Click on the following bookmarks for further details of a Periodic Wave Tool for emc analysis:

Fourier Analysis of Trapezoid Waves
Fourier Analysis of Sinusoid Waves
Fourier Analysis of Pulse Sequences
Discrete Fourier Transforms
EMI Suppression Filters
The Common Mode Choke
Decibel Conversions


Fourier Analysis of Trapezoid Waves

    Digital waveforms with fast rise and fall times, which can be as little as a few ns, can be significant sources of radiated and conducted interference. An initial step in identifying the source of particular peaks in emissions for example is to consider the harmonic frequencies produced by suspect waveforms. Any time periodic waveform can be expressed as a series of discrete sinusoidal frequencies, occurring at multiples, or harmonics, of the original digital waveform. The Fourier analysis tool automates this analysis.

    The Fourier analysis data entry form is shown below. A spot analysis is available, in addition to a plot facility. Note that the number of harmonics to be plotted can be set; plotting time can be lengthy on 486 based computers if numbers much in excess of 100 are specified.


FourierT_form.gif (7372 bytes)


    The spot analysis can be made on any harmonic, and the harmonic magnitude is expressed both as an actual value, and as a dB value referred to the peak value.

    Note that the period, risetime and fall times can be set independantly. An entry made in the period box automatically results in the equivalent frequency appearing in the frequency box. Similarly, a frequency entry automatically produces a period entry. If the sum of the rise and fall times exceed half the period, a warning message results.

    Manipulation of the rise and fall times can allow the analysis of square waves, trapezoidal waves, triangular waves and sawtooth waves.

    An option also exists to apply a suppression filter characteristic to the series of harmonics, by selecting the Apply Filter Curve option. This applies the last plotted Model One suppression filter curve, reducing the harmonics by the appropriate dB insertion loss figure of the filter at each particular harmonic frequency. The plotted graph then shows the original harmonic magnitude in one colour, and the reduced (filtered) value in aother colour.

    The original harmonic is plotted using Model 5 colour, and the reduced harmonic using Model 2 colour. These colours can be set using the Options...Graph Colour form. The filter insertion loss curve is also drawn on the graph area.

FourierT_graph.gif (8829 bytes)


    An example plot of a 2MHz trapezoidal waveform, with 20ns rise and fall times. Significant signal levels are present up to at least 100MHz.



Fourier Analysis of Sinusoid Waves

The data form for the Fourier analysis of a chopped sinusoidal waveform is shown below, and this is similar to the trapezoidal waveform analysis. Note that the number of harmonics to be plotted can be set; plotting time can be lengthy if numbers much in excess of 100 are specified.

Fours_foprm.gif (7526 bytes)

   
    The spot analysis can be made on any harmonic, and the harmonic magnitude is expressed both as an actual value, and as a dB value referred to the peak value. A filter curve can also be applied to the harmonic series.

    Note that the period, delay time and transition times can be set independantly. An entry made in the period box automatically results in the equivalent frequency appearing in the frequency box. Similarly, a frequency entry automatically produces a period entry. If the sum of the delay and transition times exceed half the period, a warning message results.

Fours_graph.gif (8117 bytes)


    An example plot of a 1kHz waveform, switched on .2ms from the beginning of the cycle, with a transition time of 2 microseconds.


Fourier Analysis of Pulse Sequences

 

    Digital waveforms are found in an infinite variety of combinations of pulse patterns, perhaps comprising bursts of short pulses, or sequences of variable width pulses. The harmonic content of such pulse trains is more complex than that of a constant width rectangular waveform, and the Fourier Analysis: Pulse Sequence Tool allows analysis of the Fourier harmonics of a repeating sequence of up to 50 variable width pulses.

The Pulse Sequence Tool is shown below:

390_1.bmp (483686 bytes)

 

Successive pulses are defined in terms of high, or on, time, and low, or off, time. The period and frequency of the entire sequence is calculated automatically from the sum of the individual pulse durations. Defined pulses can be deleted, and new pulses can be inserted into the sequence. The entire sequence can be deleted using the Clear All button.

The Pulse Voltage defines the on voltage for all pulses, and the Rise Time and Fall Time defines those values for all pulses.

Individual harmonics can be calculated using the Analyse button, whereby the actual harmonic value and its value relative to the Pulse Voltage is displayed.

The Fourier spectrum can be displayed using the plot button; an example of a series of 5 pulses of 5 microsecond duration, with a sequence frequency of 1.835kHz is shown below:

390_2.bmp (404478 bytes)


Discrete Fourier Transforms

    The Discrete Fourier Transform Tool can be used to obtain an approximation to the Fourier transform of a User Defined Waveform. One cycle of the waveform is defined either by pointing and clicking on the graph area, or by entering the data points to a Table. The Table is accessed by the Table button.

    The Mirror button can be used to produce an inverse of the first half of the waveform in the second half. The Finish button will connect the last point to be defined with the waveform end, with a straight line. The Undo button simply removes the previous point; Undo can step back to the beginning of the graph. Note that the printed output includes a sketch of the defined waveform. Values for the peak amplitude and the frequency or period of the waveform must be entered before a waveform can be drawn. The New button erases the current waveform.

    The calculation performed when the Analyse button, or the Plot button, is pressed, firstly takes a number of samples of the amplitude of the waveform. Samples are taken at fixed intervals, equal to the Period divided by the number of samples.

    The number of samples will affect the accuracy of the transform, but a large number of samples will increase the calculation time. The maximum analysable frequency is displayed, calculated as the fundamental frequency times half the number of samples. Above this frequency the spectrum is folded, and consists of an image of the spectrum below the maximum frequency.

    The Fourier coefficients A(m) and B(m) are calculated as follows:


        A(m) = 1 / N * [SUM n = to N-1 [ x(n) * cos ( 2 * Pi * m * ( n / N )) ]]



        B(m) = 1 / N * [SUM n = to N-1 [ x(n) * sin ( 2 * Pi * m * ( n / N )) ]]

   
    where N is the number of samples, n is the nth sample, and m is the mth harmonic.

    The Discrete Fourier Transform Tool form is shown below:

Discretef_form.gif (10610 bytes)


    The DFT Tool input form. The magnitude of individual harmonics can be calculated, and a plot of the series of harmonics can be made. The Offset button toggles the zero amplitude line from the centre of the graph to the base. When the zero line is in the centre, the Mirror button is enabled.

    The Tabular input form is shown below. This allows point by point input of time and amplitude, or editing of existing graph points.

Discrete Fourier Table.gif (6199 bytes)


EMI Suppression Filters

    The L-C filter shown below is used to illustrate the analysis principle used in the emi suppression filter tool. Inductors are represented as a series inductor and series resistor, shunted by a capacitor Cs to represent self-capacitance.

L_c.gif (2380 bytes)

    Capacitors are represented as an ideal capacitance in series with an inductance and resistance. For feedthrough filters, Le can be taken as being practically zero, whilst the residual resistance Rc is typically 5 mohms in value. The inductance Le dominates the rf performance of leaded capacitors, and is due to construction and lead inductance. A typical value of 5nH produces a significant resonance in the insertion loss characteristic.

Fil_form.gif (7983 bytes)

    The EMI Suppression Filter Data entry form.

    Insertion loss, the basic assessment of emi suppression filter performance is defined as the ratio of the voltage across the load Rl before addition of the filter to the voltage across Rl after the addition of the filter. (Expressed in dB).

    In addition to the L-C Filter shown above, circuit models are also provided for a Capacitor, an Inductor, a C-L Filter, a Pi Filter (two shunt capacitors separated by a series inductor), and a T Filter (two series inductors separated by a shunt capacitor). Circuit diagrams are viewable on screen; although capacitors are depicted as feedthrough types, examination of the models will enable most filter types to be modelled.

    Mains filters must be modelled as two separate filters, with different performance for symmetric and asymmetric interference. Thus the asymmetric or common mode interference (i.e. line to ground) is normally suppressed with a series inductor and capacitor to ground, usually called the 'Cy' capacitor. Symmetric or differential mode interference (i.e. line to line) is often suppressed with a single capacitor between live and neutral, usually called the 'Cx' capacitor. The series inductor is normally wound so that fluxes due to the line current are self-cancelling, and thus it has little effect on differential mode interference.

    Up to five sets of data can be defined and plotted, and the tool can be used in a 'what-if' mode. It is particularly useful for assessment of the effects of load resistance on insertion loss performance, as shown in the example plot below.

Fil_graph1.gif (7592 bytes)

    Insertion loss plot of a 10nF capacitor, in various impedance systems. Curve 2 shows the conventional 50 ohm source and load. Curve 3 shows how the capacitor performs between a 5 ohm source and load, where insertion loss at any particular frequency is much reduced. Curve 1 shows the increased loss when the system source and load impedance is increased to 500 ohms.

    A series inductor would show the inverse performance, with insertion loss falling as system impedance increases. Filters consisting of series and shunt elements show more resilience against system impedance changes.

    The curves shown below illustrate the effect of series inductance in the capacitor ground circuit. Above the resonant frequency, the insertion loss falls rapidly.

Fil_graph2.gif (7438 bytes)

    Curve 1 depicts a 100nF capacitor with no ground inductance; this plot would be typical of a feedthrough capacitor. Curve 2 shows a 100nF capacitor with 5nH of ground inductance, such as may be obtained with a leaded capacitor.



The Common Mode Choke

    The Common Mode Choke consists of two inductors wound on a common core (often toroidal), and is used, as the name suggests, to suppress common mode noise. The windings are arranged so that flux due to the differential mode signal currents cancel out, and a low insertion loss is offered to normal signal currents.

    Common mode currents however flow in the same direction through the choke, and do not cause flux cancellation. Insertion loss for common mode currents is therfore much higher.
    There are several aspects of this apparently simple arrangement which should be understood if correct application is to be made.

LOW FREQUENCY CUT-OFF

    The circuit diagram on the form shown below indicates a return ground path between the two circuits. It is undesirable for signal currents to flow via the ground path, and all signal current will return via the choke if the frequency is greater than 31.4 x RSeries / L. This means that the series resistance due to the choke winding and cable resistances for a given choke inductance should be minimised.

    The signal insertion loss will also be increased below the cut-off frequency. The CM Choke tool also calculates and plots the ratio of the noise voltage at the load to the common mode noise voltage in the ground loop. The cut-off frequency again determines whether or not any common mode noise reduction occurs. Below the cut-off, no attenuation occurs.

Choke_form.gif (13722 bytes)

    The Common Mode Choke data form. Three different types of plot are provided.
    The signal insertion loss is shown in the graph below.

Sigloss.gif (6964 bytes)

    Low frequency signal insertion loss for a 1mH common mode choke. The insertion loss is constant above a cut-off frequency, determined by the series resistance and the choke inductance. Curves 1 to 5 show the effect of increasing the resistance from 1 ohm through to 15 ohms, each increase in resistance raising the minimum frequency where the choke is effective. The insertion loss itself is entirely due to the series resistance, not the inductance.

    The circuit analysed for the signal insertion loss is shown below.

Cm_circ1.bmp (35398 bytes)

    The ratio of the noise voltage at the load to the noise voltage in the ground loop is plotted below, for the same parameters as used in the graph above. The circuit for the noise voltage analysis is also shown below.

Cm_circ2.bmp (42886 bytes)

Noise.gif (8311 bytes)

    This plot shows that the there is little or no attenuation of common mode noise below the cut-off frequency.

HIGH FREQUENCY BEHAVIOUR

    The signal insertion loss analysis, and noise voltage ratio analysis both assume a 'perfect' choke, with no interwinding capacitance. As typical winding capacitance is of the order of a few pF, this assumption is good at frequencies up to a few hundred kHz for typical inductances of fractions of a mH. At higher frequencies however, the choke inductance and self-capacitance resonate, and the attenuation provided to common mode currents above resonance becomes dependant only on the capacitance. The inductance has almost no effect on insertion loss above resonance.

    The tool plots the ratio of common mode current without the choke to the cm current with the choke in place. The load impedance in this case should be the common mode load impedance Zcm, rather than the differential mode impedance Rl. At higher frequencies the cabling is in antenna mode, and Zcm is typically in the range 40 to 400 ohms.

    The circuit for high frequency analysis is shown below.

Cm_circ3.bmp (51742 bytes)

Hfnoise Plot.gif (8674 bytes)

    A collection of common mode current insertion loss plots for a common mode choke with varying self-capacitances. These range from 1pF in Curve 1 to 15pF in curve 5, all for a constant choke inductance of 1mH.

    The self-resonant frequency falls as the inter-winding capacitance increases. Below resonance, the insertion loss is determined by the inductance level. Above self-resonance, the insertion loss is almost entirely determined by the self-capacitance.


Decibel Conversions

This tool offers a range of conversions around the decibel ratio, using the form shown below:

Decibel Form.gif (7818 bytes)

    A decibel (dB) is the unitless logarithmic ratio of two quantities, and may be expressed as:

            dB = 10 * Log (P1 / P2 )

    where P1 and P2 are two powers. Where the powers are dissipated in a constant impedance, and as power is proportional to voltage squared, the voltage ratios can be expressed in dB's as:

            dB = 20 * Log (V1 / V2 )

    Current ratios in a constant impedance can also be expressed in the same way. The power or voltage can also be referenced to a given level, for example 1mW. In this case the units are dBm, representing the signal ratio with respect to 1mW. Common referred units include dBuV (dBmicrovolts), dBuA (dBmicroamps), dBuV/m (dBmicrovolts per metre).

    The Decibel Conversions Tool allows the following conversions:

a) Between dB's and actual power and voltage ratios. Values can be entered as dB's, Power or Voltage ratios, and the other two figures are calculated.

b) Conversion of Referred Values to actual values. For example, dBm values can be converted to milliwatt values, and vice versa.

c) A conversion from a dBuV figure to a dBm figure for a given impedance is provided, and vice versa. This calcuation manipulates the follwoing equation:

        Power (dBm) = V (dBuV) - 90 - (10 * Log 10 ( R ) )

    where R is the load resistance.

d) Summation of referred power values, adding for example 5dBm to 25dBm. This uses the following equation:

        Total Power (dBm) = Larger Power + 10 * Log 10 ( 1 + 10 - D / 10 )

    where D is the absolute value of the difference between the two powers. The maximum value of the increase is 3dB, when both powers are equal in value.

e) Summation of referred voltage or current values, adding for example 0dBuA to -5dBuA. This uses the following equation:

        Total Current (dBuA) = Larger Current + 20 * Log 10 ( 1 + 10 - D / 20 )

    where D is the absolute value of the difference between the two currents. The maximum value of the increase is 6dB, when both currents are equal in value.

 

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