2-4. Harmonics in digital signal
As described in Section 2-3, one type of the noise sources generated by digital circuit is harmonics. If you can control harmonics well, you can efficiently implement noise suppression for digital circuits. This section will describe the basic nature of the harmonics contained in digital signals.
2-4-1. Nature of harmonics in terms of noise
(1) Digital signal is made out of harmonics
Generally, all waves with a constant cycle period can be broken down into a fundamental wave with the cyclic frequency and the harmonics that have frequencies of integral multiples of the cyclic frequency. [Reference 2]
The multiple for the fundamental wave is called harmonic order.
In the case of accurately repeating waves, there is no other frequency component apart from these. Many of digital signals have a repeating waveform. Therefore, when the frequency distribution (called “spectrum”) is measured, it is precisely broken down to harmonics, showing a discretely distributed spectrum.
(2) Measuring the harmonics of a clock signal
Fig. 2-4-1 shows an example of the harmonics of a 33MHz clock signal measured by a spectrum analyzer. The sections that are projecting upwards like a needle are the harmonics and are observed accurately at the interval of 33MHz. You can see that the odd harmonics and even harmonics have a different trend. The bottom part around 40dB or lower indicates the background noise of the spectrum analyzer.
(3) How to find noise source from noise frequency
The above mentioned nature of harmonics comes in useful when looking for a noise source based on the noise frequency. Measuring the noise spectrum interval allows us to analogically think the cyclic frequency of the signal that is causing noise. For example, let's say we have observed noise such as shown in Fig. 2-4-2 in an electronic device. The interval of the frequencies with strong noise seems to be at 33MHz. Therefore, this noise is considered to be caused by a circuit that operates in synchronization with the 33MHz clock.
Even if this electronic device concurrently uses circuits with a near-by cyclic frequency such as 33.3MHz or 34MHz, such a frequency can be separated if you can accurately measure the noise frequency and interval. For example, if noise exists at 330MHz in Fig. 2-4-2, we can suspect that it is caused by the circuit with 33.0MHz not 33.3MHz. This is because neither signal of 33.3MHz nor 34MHz contains the harmonic of 330MHz.
(4) Contains no frequencies other than integral multiples
In addition, a cyclic waveform does not have any frequency component that is lower than the fundamental frequency. For example, 100MHz signal will never generate noise of 20MHz, 50MHz or 90MHz. If you find noise in any of these frequencies, it is considered to be caused by a frequency-divided signal not the original signal.
Digital circuits often operate in synchronization with the clock signal, and many of them operate at a frequency of 1/N of the clock signal (called “frequency division”). In this case, the harmonics are the integral multiples of frequency-divided signal frequency. However, if two or more circuits are operating at the same clock signal that has been frequency-divided, the harmonics of the clock signal and the harmonics of the frequency-divided signal overlap with each other, making it difficult to tell them apart.
2-4-2. Composite waveform of harmonics
(1) Adding up sine waves makes it closer to digital waveform
How the digital signal waveform and the included harmonics are related to each other? Fig. 2-4-3 shows the changes in waveform when adding small order harmonics to the fundamental wave. You can see that the sine waveform of the original fundamental wave is getting closer to a rectangular waveform as each harmonic is added to it.
(2) Higher-order harmonics have less influence on waveform
In contrast, when subtracting the higher-order harmonics from an ideal rectangular waveform, it is getting closer to sine wave. However the change is gentle. As an example, Fig. 2-4-4 shows waveforms, wherein the highest harmonic is taken away one by one from the waveform that has been added up to the 17th order harmonic.
(3) Waveform of duty 50% has strong odd harmonics
When making a waveform of duty ratio 50%, only the odd harmonics are added up. If you are making a waveform that does not have duty ratio 50%, you also need the even harmonics as described in Section 2-4-5. Here, duty ratio means a ratio of the signal level “High” in one cycle.
In the real-world waveforms, the duty ratio will never be just 50%. So the even harmonics are also included as shown in Fig. 2-4-1.
(4) Reduce noise by cutting off higher-order harmonics
As above, the relatively lower frequency (lower-order) components among the harmonics of digital signal are important to maintain the signal waveform, while the higher frequency (higher-order) components can be considered as less important.
However, as described in Section 2-3-6 Harmonics in signal, the higher-order harmonics have higher frequencies and thus have a nature of being easily emitted and causing noise. Therefore, noise suppression is implemented by eliminating the higher-order harmonics to the extent of not causing any problems to the signal waveform. Usually up to the 3rd to 7th harmonics are maintained and any higher-order harmonics are eliminated. Fig. 2-4-5 shows measurement results of waveforms and noise when harmonics are eliminated by a low-pass filter. The digital signal without harmonics has a waveform with rounded corners like this instead of having decent square corners.
(5) Eliminate harmonics by EMI suppression filters for signals
EMI suppression filters for signals are the filters that are used for this purpose. In Fig. 2-4-5, an EMI suppression filter with a cut-off frequency of 150MHz has been used for the signal of 20MHz. Therefore, the waveform in the figure (b) contains up to the 7th harmonics (140MHz). EMI suppression filters will be further described in later sections.
2-4-3. Trend of harmonic frequencies
(1) Nature of harmonics of trapezoidal wave
Let's look at the trend of the levels of harmonics included in digital signal. If the voltage waveform of digital signal has a perfect trapezoidal wave as shown in Fig. 2-4-6, you can find several trends.
Figure 2-4-6(b) shows an envelope of the harmonics included in the trapezoidal wave. As shown in the figure, if the frequency is plotted in a logarithmic axis, the envelope of the harmonics forms a simple polygonal line with two inflection points (A, B). [Reference 2]
A is a frequency point determined by the signal pulse width t p
. The narrower the pulse width becomes, the more A shifts towards the higher frequency side.
B is a frequency point determined by the signal rise (fall) time t r
. The shorter this period becomes, the more B shifts towards the higher frequency side. (Provided that the rise time and fall time are the same for the sake of simplifying the trend)
(2) To control the levels of harmonics
The envelope of harmonics tends to have a constant level from DC to Point A (Region a), while decreasing in accordance with frequency at a rate of 20dB/dec. (20dB for every decade of frequency) from Point A to Point B (Region b), and then steeply decreasing at 40dB/dec. from Point B towards the higher frequency side. (Region c). Therefore, it is desired to shift Points A and B to the lower frequency side from the viewpoint of noise suppression.
Please refer to the reference
, which shows the theoretical formula that expresses this trend.
(3) Compare the theoretical curve with the actual measurement
The above frequency characteristics merely indicate the general trend. The individual harmonic levels can be affected by duty cycle etc. and may be slightly smaller than this envelope (a particular harmonic can be very small).
Fig. 2-4-7 shows an example of comparing the trend of Fig. 2-4-6 with the actual measurement. Fig. 2-4-7 (a) shows a case of duty ratio 50% while (b) shows a case of duty ratio 20%.
The voltage waveforms measured by an oscilloscope are shown on the left side of the figure, while the spectrums measured by a spectrum analyzer are shown in the middle. The same harmonics as those indicated in Fig. 2-4-1 have been observed. In the case of duty ratio 20% in Fig. 2-4-7(b), you can see that the levels of the even harmonics are almost as large as those of the odd harmonics.
On the right side of the figure, the frequency axis of the spectrums in the middle is converted into logarithmic axis so that those can be compared with the envelope of Fig. 2-4-6. For your reference, red lines indicate the theoretical envelopes. You may say that the envelope of Fig. 2-4-6 corresponds well to the actual measurements in the frequency range below 100MHz. In the higher frequency range above 200MHz, the actual measurements are smaller than the theoretical values. This is considered to be because the signal generator used for the experiment could not output accurate trapezoidal waves due to its upper limit in the frequency generation.
(4) To design electronic devices with less noise
The following trends are derived from the envelope shape shown in Fig. 2-4-6(b).
The higher the cyclic frequency of the signal becomes, the narrower the pulse width becomes. Therefore, Point A moves towards the higher frequency side, resulting in more noise
As the rise time becomes shorter (the signal speed becomes faster), Point B moves towards the higher frequency side, resulting in more noise
In order to design a circuit causing less nose, it is advantageous to avoid these trends and shift Points A and B towards the low frequency side. If you cannot avoid the above trends in your design, it will be easier to perform noise suppression if the signal line is provided with pads to attach EMI suppression filters.
When observing harmonics of an actual digital signal, it is hard to observe Region a. This is because many of digital signals have a duty ratio of close to 50% and thus Point A comes to the lower frequency side of the fundamental frequency.
2-4-4. Influence of signal rise time
(1) Change the rise time of a 10MHz clock signal
Fig. 2-4-6 showed that slowing down the rise speed of waveform moves Point B towards the lower frequency side and thus suppresses the harmonic levels. Fig. 2-4-8 shows an example that confirms this trend by calculation.
Here, the harmonics are calculated based on cyclic frequency of 10MHz, duty ratio of 50%, and voltage magnitude of 1V. The left side of the figure shows the assumed signal waveform, and the middle shows the calculation results of the harmonic spectrum. Just like Fig. 2-4-7, the right graphs show the results with the frequency axis converted into logarithmic axis. The right graphs show each spectrum with dots and superimpose the envelope shown in Fig. 2-4-7. The level of spectrum has been calculated in root-mean-square value on the assumption of using a spectrum analyzer for measurement. The same is applied to all of the following data.
(2) Point B appears at 30MHz by rise time of 10ns
Fig. 2-4-8(a) shows the case of fast rise (
=0.1ns), while (b) shows the case of slow rise (
=10ns). Point B of the envelope calculated from the formula in Fig.2-4-6 is approx. 3GHz under the condition (a), which significantly deviates from the display range (up to 1GHz) of the graph. The one under the condition (b) is approx. 30MHz.
The calculation results of Fig. 2-4-8(a) shows that the harmonic spectrum tends to simply decrease at the rate of 20dB/dec. In addition, it was confirmed that Point B is not visible within the display range of the graph (up to 1GHz).
In contrast, the calculation results of Fig. 2-4-6(b) shows that the harmonics decrease steeply at the rate of 40dB/dec. above the frequency range over 30MHz. An inflection point, which is considered as Point B possibly exists around here.
(3) Decreases by 20dB or more at 500MHz
Comparing the spectrums in the middle with each other, (b) with a slow signal rise has smaller harmonic levels than the others in the entire frequency range except for the very small range in the lower frequency side. The difference reaches as large as over 20dB at 500MHz.
From the above calculation results, it is clear that slowing down the signal rise speed is effective for suppressing the harmonics. In order to create a circuit that causes less noise, it is effective to choose an IC that is as slow as possible to the extent of not causing any problems to the circuit operation. It is also effective to equip EMI suppression filters for the signal.
2-4-5. Influence of waveform duty ratio on harmonics
(1) Changes the duty ratio of a 10MHz clock signal
One type of typical digital signals that easily cause noise is clock signal. Clock signals usually have a waveform with a duty ratio of around 50%. As mentioned before, if the duty ratio is close to 50%, the signal comprises strong odd harmonics while the even harmonics tend to be weak. The levels of the even harmonics are disposed to change significantly in accordance with duty ratio. (The changes in the odd harmonics are also large in the high frequency range, where the orders of harmonics are high). Fig. 2-4-9 shows an example that confirms this trend by calculation.
(2) Harmonics are grouped into the odd group and even group
The figure compares between harmonics wherein the duty ratio has been gradually changed from 50% (a) to 49.9% (b) and then to 49% (c) based on the ideal fast rise digital signal shown in Fig. 2-4-8(a). These calculation results show that the even harmonics and odd harmonics line up along the green line and yellow line respectively, indicating a different trend between the even orders and odd orders.
In Fig. 2-4-9(a), which has a duty ratio of 50%, the odd harmonics line up along the envelope shown in Fig. 2-4-6, while no even harmonics are observed.
(3) Change in duty by 1% can cause difference of 10dB
In contrast, Fig. 2-4-9(b), which has a duty ratio of 49.9% shows the even harmonics, even though the levels are still low. Fig. 2-4-9(c), which has changed the duty ratio to 49% shows increased levels of the even harmonics, which is even higher than the odd harmonics in a certain frequency range. When observing the higher frequency range over 1GHz, or calculating a case wherein the duty ratio significantly deviates from 50%, you can observe a trend that the relative relationship (higher/lower) of the levels of the even and odd harmonics switches over in a cycle. Please use MEFSS to confirm this trend.
As above, even 1% change in the duty ratio, which is hard to tell by an oscilloscope can cause a change of several tens of dB in the levels of the even harmonics and higher-order harmonics. There is no large change in the general shape of the spectrum, which is still in line with the envelope shown in Fig. 2-4-5. However, the influences seem significant when looking at each spectrum separately. You need to note this disposition since it can cause a serious influence on the reproducibility of noise measurement.
Regarding the decision of pass or fail for noise regulation, it is considered to be fail even if only one part of the spectrum exceeds. Careful measurements are required if such a component of significant variation is close to the limit.
2-4-6. Voltage harmonics and current harmonics
(1) Compare the voltage waveform with the current waveform
The above mentioned disposition of harmonics is based on the premise that the voltage waveform has a rectangular waveform. You need to note that even though an actual circuit has a rectangular waveform for the voltage waveform, the current waveform may be different. That means the noise emission can show a different trend depending on whether it is mainly from the voltage or the current.
Fig. 2-4-10 shows the results of the waveforms and spectrums calculated by MEFSS for the case of setting 5pF capacitor as the load assuming a C-MOS digital circuit. The voltage waveform is close to an ideal digital pulse, and the values of the harmonic spectrum are close to those of the envelope shown in Fig. 2-4-6 (the shape is slightly difference due to the capacitance load, indicating the minimum point at around 500MHz).
(2) The current contains more harmonics
In contrast, the current only flows at the rising and falling moments as shown in the figure. The spectrum of such waveform has a constant level up to a high frequency of several 100MHz (depending on the rise time) as shown in the figure. Therefore, if there is a noise emission due to the current, the noise is likely to be caused by high frequencies. In this way, MEFSS is also capable of calculating the spectrum of current waveform.
In the results of noise measurements shown in Fig. 2-3-14, barely any voltage spectrum is seen above 500MHz in (b), while the emission noise spectrum in (c) shows a strong emission. Therefore it can be considered that the cause of noise emission in this experiment has been the electric current as one of the causes of indicating differences in the frequency distributions between the noise source and the emitted noise. (As opposed to this experiment, there are cases that voltage causes a noise emission)
(3) Current has a needle-like pointy waveform
In Fig. 2-4-10, the reason for the current harmonics not being attenuated up to a high frequency is understandable, if you think that it is because the current waveform is in a fine spike shape. Considering the trapezoidal wave model in Fig. 2-4-6, the spike shape waveform just like the current waveform can be viewed as a trapezoidal wave with a very small duty ratio. For the envelope of a trapezoidal wave with a small duty ratio, Point A shifts to the high frequency side, keeping a constant level up to a high frequency. Therefore, the harmonics of the current waveform can be observed up to a high frequency without being attenuated.
Please note that the trapezoidal wave model in Fig. 2-4-6 is different from the current waveform since the spikes of the current waveform are pointing up and down. Therefore, a trapezoidal wave model with a small duty ratio has stronger even harmonics as it shifts Point A. However, this trend is weaker in the current waveform.
2-4-7. Influence of pulse waveform change by resonance
(1) Resonance causes distortion in pulse waveform
Since the above explanation premises that the digital signal pulse waveform is an ideal rectangular waveform, a correction is needed if the waveform deviates from a rectangular waveform due to the circuit conditions. One of the reasons for the pulse waveform being distorted is resonance of the driver IC, receiver IC and wiring. This section describes examples of changes in spectrum when the waveform is distorted by resonance.
If the influence of wiring is disregarded, C-MOS digital circuit can be considered as a very simple circuit as shown in the model diagram of Fig. 2-4-10, allowing us to obtain an almost ideal pulse waveform in simulation.
(2) Example of increasing noise caused by ringing due to long wiring
How does the waveform look like if the influence from wiring is added to this circuit? The calculation results are shown in Fig. 2-4-10. Fig. 2-4-11 compares the waveforms between a circuit with wire and one without wire by assuming a long wire (20 cm) so that the influence becomes obvious. When there is a wire, a large ringing occurs to the signal waveform. Accordingly you can see a trend of significantly increasing the harmonics around 150MHz. (In order to observe the ringing, the voltage has been measured in a range wider than Fig. 2-4-10)
(3) Confirm ringing by experiment
Such ringing is often seen in actual digital circuits. Fig. 2-4-12 shows an example of measurement result wherein a wire of 20 cm is connected. Although it is not as strong as the one from the simulation in Fig. 2-4-11, ringing occurs at a similar cycle showing the trend of significantly increasing the harmonics around 150MHz. Therefore, if a digital circuit connects a longer signal line, the signal waveform is more likely to suffer from ringing. In this case, the ringing frequency can cause higher harmonic levels and thus can cause noise problems.
The ringing of the measurement result in Fig. 2-4-12 is relatively smaller than the measurement result in Fig. 2-4-11. This is considered to be because of attenuation in a short period of time since the actual circuits more or less have some losses in the IC and wring. The voltage magnitude is also smaller, which is less than 3 V in Fig. 2-4-12.
In addition, the measurement uses an FET probe with the frequency band of 2.5GHz as a voltage probe, which has a voltage ratio of 10:1. Therefore, the values of the spectrum shown in Fig. 2-4-12 are 20dB smaller than the actual values.
(4) Inductance in wiring causes resonance and in turn causes ringing
The ringing shown in Fig. 2-4-11 is considered to be a result of a resonant circuit being created inside the signal circuit due to the inductance of the wiring. Fig. 2-4-13(a) shows a model diagram.
In Fig. 2-4-13(a), the inductance and electrostatic capacity of the wiring are described in lumped parameters. This arrangement allows us to understand that an RLC series-resonant circuit has been created in the signal circuit.
When magnifying the ringing generated in the rising section of the signal of Fig. 2-4-11, you can see the damped oscillating waveform with a cycle of approx. 7ns as shown in Fig. 2-4-13(b). The cycle of 7ns is equal to 143MHz in frequency, which almost corresponds to the frequency (150MHz) of the rising harmonics observed in Fig. 2-4-11.
(5) How much inductance in wiring?
The inductance and electrostatic capacity for the 20 cm wire that has been assumed in Fig. 2-4-11 are calculated to be approx. 140nH and 10pF respectively based on the unit-length parameters indicated by the transmission theory. If these values are applied to the RLC series-resonant circuit shown in Fig. 2-4-13(a), the resonance frequency is estimated to be approx. 110MHz. Although this result is 30% smaller than 150MHz observed in Fig. 2-4-11, it is generally accordant, and the simplified model in Fig. 2-4-13(a) is thus considered to be relevant to understand the mechanism of ringing.
If you need to estimate the resonance frequency more accurately, the wiring needs to be considered as a transmission line instead of using lumped parameters such as inductance and electrostatic capacity. (Please refer to technical books for how to calculate the unit-length parameters of wiring and how to handle wiring as a transmission line
(6) Absorb ringing by a ferrite bead
Generally, in order to suppress resonance, dumping resistors are used. If you would like to implement noise reduction at the same time, it is effective to use a ferrite bead instead of damping resistors. Fig. 2-4-14 shows a calculation result wherein a ferrite bead is used in the previous model. In addition, Fig. 2-4-15 shows a calculation result wherein a ferrite bead is used in the test circuit used in Fig. 2-4-12.
Since a ferrite bead has been attached in Fig. 2-4-14 and Fig. 2-4-15, the ringing has been eased and the rise in the harmonics around 150MHz has disappeared, as well as lowering the harmonic levels in the entire frequency range up to 500MHz at the same time. In this manner, ferrite beads can efficiently suppress not only resonance but also unwanted harmonics. Ferrite beads have been widely used for eliminating noise that has been caused by digital signal harmonics.
2-4-8. Elimination of harmonics by EMI suppression filter
(1) EMI suppression filter eliminates harmonics that can cause noise
Using an EMI suppression filter such as a ferrite bead allows thorough elimination of unwanted harmonics in a digital circuit and thus suppresses noise from harmonics. Although EMI suppression filter and how it is used will be further described in a separate chapter, this section will describe an example of its effects.
Although harmonic suppression can be achieved to some extent by slowing down the rise time with use of a slow-speed IC (as described above) or general-purpose parts such as a resistor, more effects can be gained by using EMI suppression filters. Even if it looks like the same signal waveform, there could be a difference of 10dB or more in terms of the noise suppression effect.
(2) Use 50MHz cut-off filter for a 20MHz clock signal
Fig. 2-4-16 shows an example of experiment that has eliminated noise of 20MHz clock generator with use of an EMI suppression filter. Here, the case of using a three-terminal capacitor and the case of using a
-type filter with a cut-off frequency of 50MHz (steeper frequency characteristic) are compared with each other. Although the noise reduction effect is excellent in both cases, you can see that the changes in the signal waveform and rise time do not necessarily correspond to the noise suppression effect. The
-type filter seems to be capable of eliminating noise while maintaining the pulse-shape signal waveform and rise time.
(3) Noise looks different on oscilloscope or spectrum analyzer
This is because of the trend that relatively lower frequency components seem to stand out in the signal waveform, while relatively higher frequency components seem to stand out in noise measurement. Since the observation of signal waveform shows a waveform in which all frequencies are added, the lower-order harmonics with large amplitude exert stronger influence. In contrast, noise measurement discretely observes each frequency and is more influenced by high (higher order) frequency components due to its disposition of being easily emitted from a smaller antenna.
(4) EMI suppression filter for signals
If you use a filer with a steep frequency characteristic just like the
-type EMI suppression filter shown in Fig. 2-4-16, noise may be efficiently suppressed while the signal quality is maintained. This type of EMI suppression filter will be further described in later sections.
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“2-4. Harmonics in digital signal” - Key points
- Digital signal is composed of harmonics.
Signal waveform can be maintained by the lower-order harmonics. Unwanted higher-order harmonics are likely to cause noise.
Rise time significantly affects the levels of the higher-order harmonics.
EMI suppression filter allows effective elimination of unwanted harmonics.