Know the Bandwidth: How to Find Bandwidth of a Signal

This article relates to both printed electronics and PCBs. I acquired my bandwidth design techniques working in PCB design, and later applied the same principles to my printed electronics design. In this article, I’ll explain my knowledge of bandwidth, and how I was able to apply it to both PCBs and printed electronics.

Before going into details on bandwidth, I think it’s worth revisiting what we in electronics mean by the term “signal”. For me, at least, it’s easiest to understand “signal” as follows: a signal is an electrical quantity containing information. Typically, the electrical quantity in question is ‘voltage’, and the information contained is ‘voltage level as a function of time’. We are interested in knowing the exact voltage level at a specific moment in time. For example, digital signal information is a queue of logic 0 and 1 states, and we get the right information only if the correct logic levels are detected at the correct time. From an electronics perspective, information is 0 or 1, and if we lose any pulses, then we lose information. Analog signals do not make a difference: a certain voltage at a certain time generates information. If our electronics cause distortion of the analog signal—for example, by clipping—then we lose information. Signal information is necessary for reliable product operation, and sometimes it may be needed to supply information to the end-user, perhaps in the form of audio or visual feedback.

Signals, when calculated by Fourier transform from the time domain to the frequency domain, may contain several frequency components. A time domain signal is the sum of all included frequency components, and the shape of the signal depends on the power level of each individual frequency. Digital signals contain a DC component followed by a lot of lower intensity AC components whose intensities decrease as frequency increases. And faster signals mean higher frequency components. Each of these AC frequencies is a very narrow band—being single-frequency sine wave signals. Therefore, a digital signal is the sum of DC plus a huge amount of sine wave signals; thus its bandwidth becomes wider.. Pure AC signals can be narrow band (like a sine wave), because these do not contain a DC component.

Signal information lies somewhere in the frequency range, and all the frequency components required for that information determine the bandwidth. Frequencies outside of the bandwidth are unnecessary and can be rejected, for example by filtering, because these do not carry additional information for the signal. Bandwidth can be thought of as the operational region for the electrical signal, in which it doesn’t lose information, and is also required for the electrical path (i.e. trace) or load of the signal. Electronics are then designed accordingly, and in the optimal case, when a signal is fed to the trace, remains unchanged. If the signal speed is higher than the bandwidth of the trace or filter, then the signal is modified, which typically means that some frequency components are filtered away. The trace itself will have bandwidth limits, especially for high speed signals, but a trace can contain designed bandwidth limiting components like passive filters.

Bandwidth for a signal is determined by the signal rise time (10% to 90%) requirement, and can be expressed by the following thumb rule:

Bandwidth = 0.35/tr (1)

Signal frequency is not as critical as rise-time requirement, simply because signals are different. Digital signals (50% duty cycle) have different rise and fall time requirements than PWM signals (10% to 90% duty cycles), even if the signal frequencies are exactly the same. In PWM signals, when the signal On state is short (duty cycle 10%) compared to the Off state (90%), it means that the rise time must be much faster compared to longer On state pulses. Of course, signal frequency matters as well, because the higher the frequency, the faster its rise time needs to be. This bandwidth rule of thumb is my number one tool for signal bandwidth-related design tasks. I first learned this a long time ago from electronics design lecturers at my university, and I’ve since utilized it many times in my design.

Figure1. Relation between signal rise time and bandwidth ⁽²⁾

I used to limit the speed of signals, both analog and digital, to the minimum required bandwidth, and this can be done easily with filters. One of the simplest filters is the RC-filter, which is easy to implement and easy to design. For RC filters, the -3dB cut-off frequency calculation is:

The cut-off frequency of an RC filter ⁽³⁾  

Where R is the series resistance and C is the capacitance connected to ground after the resistor. The schematic implementation is presented in Figure 2 below.

Figure 2. Schematic implementation of RC-filter

You must remember that if your selected resistance of RC filter is approximately the same ohm level as the output resistance of the signal driver, then you have to take the output resistance into account as well when calculating -3dB cut-off frequency. The calculation formula for RC filter becomes:

The cut-off frequency of an RC filter on a signal driver with an output resistance

Bandwidth can be thought to be the same as -3dB cut-off frequency. Cut-off frequency means that the frequency at this point has been attenuated to half of its original power level. Other filters can be used as well, and for example, many MCUs include a slew rate limiter for GPIO rise and fall times. This feature creates the same result as filters, but can be controlled by software and does not need additional components. There are basically two reasons why I filter signals: 1) I want to minimize the overall noise level of this particular signal, and 2) I want to minimize the coupled disturbance due to crosstalk. Of course, it makes sense to minimize crosstalk by optimal PCB stackup design, but filters give us another tool to minimize it.

The figures below show an oscilloscope measurement of a 3MHz digital signal pulse, with and without filtering. Figure 3 is an “unfiltered” signal, and its speed is limited by signal driver output resistance Rs and load capacitance only, and the measured rise time is 7.8ns. We don’t know the exact load capacitance, but according to the datasheet of the component which is driven by this signal, there is a protection diode reverse biased to ground, and typically the capacitances of these are ~10pF. However, in this case, the exact capacitance value is not important, because according to the thumb rule the bandwidth is now 0.35/7.8ns = 44.9MHz, and we can filter it quite a bit without losing signal information. Added filter capacitance will be much higher than diode capacitance.

Figure 3. 3MHz signal without filtering

Figure 4 shows the same signal, but filtered by an RC filter. I selected the 100Ω resistor and the 100pF capacitor, and in addition we have measured a 38Ω output resistance of signal driver and ~10pF IC load capacitance, which must be taken into account. The RC filter calculator shows the cut-off frequency to be:

f-3dB = 1/2π(100Ω + 38Ω)*(100pF + 10pF) = 10.484MHz

According to bandwidth calculations, the fastest rise time for this bandwidth is then 0.35/10.484MHz = 33.4ns. From figure 4 we see the measured rise time was 37.4ns, which in this case is close enough to the calculated value. Typically I don’t expect a perfect match between calculated and measured values, but usually I check the result by quick calculation to determine it’s close enough for the application requirement. Most often, I design filters which are not on the edge, but which are effective enough and have enough margins.

Figure 4. 3MHz signal with RC filter

This signal was digital, and from the shape we can see that we didn’t lose information after filtering. We can still reliably detect the pulse to be logic 1, and the signal still gets to the low state fast enough before the next period starts. In addition, it’s much less noisy because the high frequency harmonics have been attenuated. With this method, I have successfully decreased the crosstalk between the digital bus trace and a sensitive sensor trace, and got the sensor working properly without re-routing traces. This was accomplished by filtering only the signal that generates disturbances and not touching the analog signal at all because the sensor bandwidth requirement was higher than digital bus.

Limiting bandwidth to a proper level is even more important in printed electronics than in PCBs. The main reason for limiting bandwidth in printed electronics is to reduce disturbances due to crosstalk. Printed electronics are much more limited by building optimal stack-ups in terms of impedance and crosstalk, and I need to use filters or slew rate limited signals. When we consider the stack-up of printed electronics we can see that the traces crossing each other are separated only by a local thin printed dielectric layer. Its thickness is only tens of micrometers, which means capacitive coupling between crossing traces is quite strong. Capacitance between traces depends on the area of cross-over and thickness of the dielectric layer in between. In printed electronics, traces tend to be wider than in PCB,s and the dielectric layer much thinner than in PCBs, which leads to the capacitance between traces also being bigger. Bigger capacitance means that even lower frequencies are coupled through this “capacitor”. In addition to this, layout areas may be nearly the same size as product size, which means the lengths of traces are long, which thereby increases the inductances of traces. And like higher capacitances, higher inductances affect lower frequencies.

Because of the various materials and stack-ups involved, printed electronics bring low-frequency bandwidth challenges, but solving these problems is possible through known principles and methods widely used in PCB design. Additionally, an understanding of bandwidth is important in printed electronics design, and needs to consider carefully. Because of material differences, signal speed related challenges in printed electronics are similar to those in PCBs, but in printed electronics, we may face them far less often. Personally, because of this, I write down critical signal frequencies in my schematics to remind myself that they’re important, and if I notice potential issues, I limit bandwidths to the minimum required level.

Would you like to find out more about how Altium can help you with your next PCB design? Talk to an expert at Altium or discover more about adopting signal integrity in your high-speed design process.

References:

• Bandwidth = 0.35/tr
• Rule of Thumb #1: Bandwidth of a signal from its rise time
• Lecture material of University of Oulu: Analog Circuits 1, Kostamovaara Juha
• f-3dB = 1/2πRC
• RC circuit