# Introduction to Impedance

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## What It Is, and How It Applies to Embedded Systems

### Impedance is an important concept in electronics, even though most of the time it’s like air-conditioning—we ignore it until something doesn’t work. A lot of electronics depend on impedance, such as USB cables, the cables inside a computer, semiconductors, and motors. The power supply that powers your computer or charges your phone presents an impedance to the AC line it’s plugged into. In this article, I introduce the basics of impedance in embedded systems.

• What is impedance?
• How do you measure impedance?
• How does impedance work in embedded systems?
• Time-domain reflectometer
• Printed circuit board assembly
• Resources not for print

So, what is impedance? And how is it different from resistance? And what’s an article about impedance doing in a magazine about embedded systems?

##### What is impedance?

Start with resistance. Resistance is the degree to which a component such as a resistor opposes an electric current. If you put a 1kW resistor across a 1.5V battery, the current will be 1.5ma. Higher resistance equals lower current and vice versa. For batteries and DC power supplies, resistance is just resistance—in other words it’s just a value in ohms. For more about resistors, see my article “Getting Started—Resistors” in Circuit Cellar issue #382.

Resistance is fairly simple. If you apply a battery across the two leads of a spool of a two-conductor speaker cable, and then measure the voltage at the other end with a high-impedance voltmeter, you’ll measure the battery voltage. If you short one end of the cable and measure the resistance at the other end, you’ll get the resistance of the wire, which will typically be a fraction of an ohm.

When you apply AC voltage to a two-conductor speaker cable, you get different results because of the impedance of the cable. Impedance can be described as “AC resistance,” but it is not the same as DC resistance. In the case of a resistor, impedance is the same as DC resistance—except at very high frequencies, because then the resistor’s capacitance and inductance become significant. But for other types of circuits, the AC signal interacts with other components of the device or the cable, resulting in an impedance value that is quite different from the DC resistance value.

##### Impedance and Frequency

Impedance is a complex number, meaning it consists of what are called “real numbers” and “imaginary numbers.” I will explain that a little more shortly. The imaginary number in the impedance is the inductance and/or capacitance, while the real number is the DC resistance. The sum of these two values makes up the impedance, usually denoted with the symbol Z.

Figure 1 shows some simple examples. In the top schematic, R1 is just a resistor, and the impedance consists only of the resistance, the real part of the impedance value. So, if R1 is 50Ω, then resistance=impedance=50Ω. The remaining circuits in Figure 1 are all impedance circuits. The resistance presented to an AC signal varies with frequency, as described below.

If you connect an AC signal to the input and another resistor from the output to ground, as in the example at the top of Figure 2, the result is a voltage divider. The output can be measured as either the peak-to-peak or RMS value. The graph of voltage versus frequency shows that the voltage remains the same across the frequency range for the resistive divider.

For the remaining circuits in Figure 2, the signal seen with a resistive load will vary with frequency, and it will have phase variations between voltage and current. For the R2-L1 circuit, the output voltage will decrease as frequency increases. L1 appears as a short at DC and its impedance rises with frequency. For the R3-C1 circuit, the output voltage will be zero at DC, but rises with frequency. The impedance of C1 is highest at DC, effectively an open circuit, and falls as the frequency rises.

As a complex number, impedance is written as R + jx, where R is the effective resistance and jx is the reactance. The j coefficient represents the square root of -1, which is not possible in the real numbers—precisely why it’s called the “imaginary unit.” Sometimes in other fields you will see i instead of j, but j is normally used inelectrical engineering because i indicates current. The term jx is positive if the reactance is inductive, and negative if the reactance is capacitive.

Impedance can also be written in polar form, meaning as a number and a phase angle such as 100Ω∠40°. This would be read as 100Ω at a 40° phase angle. Any impedance value can be written in either complex or polar notation, and one can be converted to the other as needed.

The impedance of an inductor in ohms can be calculated by 2ΠfL, where f is the frequency and L is the inductance in Henries. So as frequency rises, the impedance rises. For a capacitor, the formula is 1/2ΠfC, where C is capacitance in farads. For a capacitor, impedance decreases as frequency rises.

##### Transmission Lines

Any pair of wires, whether parallel, twisted together, or in the form of a coaxial cable, has inductance in the individual wires and capacitance between the two wires. This can be modeled as a series of inductor-capacitor pairs as shown in Figure 3. This is where impedance gets interesting. If you chop up a transmission line into shorter and shorter pieces, the inductance and capacitance both get smaller. But an actual transmission line consists of an infinite number of infinitely small pieces, each with an infinitesimal inductance and capacitance. These interact so that the transmission line appears as a pure resistance over a range of frequencies.

An ideal transmission line has zero resistance at DC, but obviously that’s not the real world. The inductors that model the line can be more accurately represented by an inductor in series with a resistor that has a very small value. The impedance of a transmission line is very low, just the resistance of the wire, from DC to some frequency. Above that frequency, the impedance increasingly appears reactive. In a specific range of frequencies, the transmission line appears to have a pure resistance again, but not the same as the DC resistance. Transmission lines in the form of coaxial cables—used for amateur radios and public service radios for organizations such as the police and fire departments—are usually 50Ω. Television cables are normally 75Ω. Over-the-air television antennas, although not very common now, are 300Ω. Even the transmission lines that bring power to your home have impedance, and since those lines can be tens or hundreds of miles long, impedance becomes an important calculation in power distribution.

A transmission line presents a resistive impedance to an AC signal within a certain (usually wide) band of frequencies. The impedance of the cable is approximated by the formula zo=√(L/C), where L is the inductance and C is the capacitance. We are ignoring resistance here as it is usually small. The derivation of that formula is quite complicated, so I won’t delve into it here. A standard RG58 50Ω cable has a distributed capacitance of about 25pF per foot. The impedance of the cable will be right around 50Ω over a wide range of frequencies, but the cable losses increase as the frequency increases. RG58 loss in decibels is typically specified for a given length at specific frequencies, starting at 1 MHz.

##### Velocity Factor

Transmission lines also have a “velocity factor,” which can be thought of as the speed of light in the line. Because of the inductance and capacitance in the line, and the materials from which it is made, it takes more time for a signal to travel from one end of the line to the other than light takes to travel the same distance in free space. The velocity factor of a transmission line is the time it takes a signal to travel through the line for a given distance divided by the time it takes light to travel that same distance in free space, expressed as percent or a decimal.

##### TDR

Time-domain reflectometers (TDRs) are devices used to look for breaks in a transmission line, to verify the impedance of traces on a PCB, and for other, similar features [1]. A TDR sends a pulse down one end of a transmission line and displays the resulting waveform. An ideal square wave consists of the fundamental frequency and all the odd harmonics. So, sending a pulse down a transmission line is equivalent to simultaneously sending a wide band of frequencies at the same time (I’m simplifying a little bit here).

##### TDR test circuit

Figure 4 is a schematic for a simple circuit that allows a scope to be used as a (somewhat imprecise) TDR. Q1 and Q2 form a two-transistor multivibrator circuit. D1 and D2 are steering diodes to improve the rise time of the transistors, and RV1 adjusts the frequency. U1 is a high-speed CMOS buffer with the outputs each having a 330Ω resistor in series. With the outputs connected, the six 330Ω resistors make an output impedance of about 55Ω.

Below the schematic is a diagram of how the circuit is connected, using a scope to capture the result. The ground lead on the scope probe is connected to the circuit ground. The circuit used here doesn’t produce waveforms as clean as a real TDR, but this set up is good enough for this discussion.

##### Reflections

If you send a pulse down a transmission line with the other end unterminated, the energy must go somewhere because energy cannot be destroyed. Since the transmission line isn’t using it and there is no load, it is reflected back down the line.

I used a 100’ length of speaker wire as a test transmission line. That’s long enough to measure the velocity factor, and to see the effects of driving it with a pulse. Plus, it was readily available at a local discount store. Figure 5 shows what happens when it is driven with the TDR pulse circuit, and with the far end, opposite the TDR circuit and oscilloscope, left unconnected. You can see the initial pulse, and after a delay that corresponds to the speed of light in the transmission line you can see the voltage step which is the reflection of the original pulse. The step doesn’t reach double the original input because there are losses in this setup, but the propagation delay followed by the large return step is clearly visible.

That voltage step represents reflected power. The oscilloscope shows voltage, but most of the power sent down the cable is reflected back. The rounded rising edge on the reflection occurs because the transmission line has more attenuation at higher frequencies. For this simple circuit the reflected power is negligible. But for commercial or amateur radio installations which operate at hundreds or thousands of watts, that reflected voltage represents a lot of power, and it has to go somewhere. That is why radio installations want to match the antenna impedance to the transmitter and cable impedance—to minimize reflected power and maximize transmitted power.

The voltage step in Figure 5 occurs after approximately 300ns. The propagation time of a pulse over 200’ of free space (100’ out, 100’ back) is approximately 203ns (200’ x 1.017 ns/foot). So, the velocity factor of the speaker cable is about 0.67 (67%). This means that light’s travel time is 67% that of a signal’s through the line—or, inversely, that a signal through the line is approximately 149% slower than light through free space. Service personnel can use a TDR to locate a break or other anomaly in a transmission line—the time of the reflection represents the distance of the break.

If a transmission line isn’t left open but is terminated in a resistance that is higher than the characteristic impedance of the line, you will see a positive reflected pulse, but it will be smaller than the open-line pulse. An open line has an infinite resistance, so the pulse is ideally the same voltage as the drive voltage, as shown earlier.

If a transmission line is terminated in its characteristic impedance, the pulse is absorbed by the termination and no reflection occurs. The speaker cable I used has an impedance around 90Ω. To estimate that figure, I terminated the line with a potentiometer, adjusted it so that there was no pulse reflection, and then measured the potentiometer. Figure 6 shows the resulting waveform.

If the line is terminated in a resistance that’s smaller than the characteristic impedance, the return pulse will be negative as shown in Figure 7. Here, the line is terminated in a 47Ω resistor.

The TDR circuit provides a square wave that is composed of multiple frequencies. If the transmission line is driven with a sine wave and is a quarter of a wavelength or half of a wavelength long, then the interaction of the line with the load changes. But that is a topic for a different article.

##### So what?

If you’re not an amateur radio operator or an RF engineer, then what does this have to do with you? Why does it matter? In a word—cables.

Well, not just cables. It’s cables, printed circuit board assemblies (PCBAs), connectors and any wired connection that is involved in sending signals from one point to another. If you are sending audio signals down a cable, unless they are going a long distance or using very high power, this is probably not important. But for other, higher-speed signals, it is.

There are many kinds of standardized interfaces with specified impedance. USB cables have a specified impedance of 90Ω. Serial ATA (SATA) and SAS cables have a specified impedance of 100Ω differential, 50Ω common mode. The PCIe bus is 85Ω. A Cat-6 Ethernet cable is 100Ω. All of these are defined to allow designers to match cables, connectors, PCBA traces, and terminations to the same impedance so as to minimize reflections, data errors, and loss of signal power. If you’re using a cable to connect two systems or two boards, that cable has an impedance. This is true whether the cable has two conductors or 20.

Many years ago, a circuit board PCB trace was unlikely to be long enough to create problems in most digital designs. But today’s high-speed circuits mean even a few inches of PCB trace have the characteristics of a transmission line and must be managed. In addition, higher speed circuits need more power to drive them into the capacitance of transmission lines in any form. This is why the test circuit earlier paralleled all six of the CMOS driver’s outputs—to obtain a sharp rising edge.

Despite the higher speeds and higher currents, there is also a desire to control the rise time (edge speed) of signals. A fast rise time has more high-frequency components, so it generates more electro-magnetic interference (EMI) and is more likely to create crosstalk, where signals on one conductor couple into another. Coupling between two wires can be inductive or capacitive. Slowing down the rise time eliminates some of the higher frequency components of a signal (Figure 8), reducing the EMI and coupling. So, in addition to specifying impedance, many standards also specify a minimum acceptable rise time to control extremely sharp edges. For SATA, this limit is 150 picoseconds (ps), or 10-12 seconds). USB 2.0 specifies a rise time lower limit of 300ps. The idea of limiting rise time isn’t new. The old RS-232 specification, originally released in 1960, specified a minimum rise time of 30v/µS to prevent crosstalk. This works out to about 250ns for a 12V system. RS232 specifies the rise time as a range in volts per µS because unlike the other interfaces mentioned, RS232 can have different signalling voltages.

##### PCBA

If you’re working with high-speed circuits such as USB or SATA, then you’ll have a connector on a PCBA with some cable plugged in, which has some specified impedance. You have to get your signals to and from that connector. This means that your PCBA needs to have controlled impedance traces that match the impedance of the cabling. Controlled impedance traces are configured to match a specified impedance.

There are three basic kinds of controlled impedance traces on a PCBA. The first is a microstrip, which is a trace on one layer over a ground plane on another layer. The second is a stripline, which is a microstrip on an inner layer, sandwiched between two planes. The third is a differential pair, which is a pair of traces over a ground plane that are spaced to have a specific differential impedance. For all three types, the width of the traces, distance from the ground plane, and dielectric constant of the PCB material all affect the impedance. For differential traces, the spacing between the traces also matters. Figure 9 is the described placement and layering of the three types of controlled impedance traces, as viewed from the edge of the PCBA.

I’ve used both microstrips and differential impedance lines. I’ve never needed a stripline. The steps involved in creating these different types of controlled impedance traces are outside the scope of this article, although I plan to cover it in a future article. There are also resources online that describe them, as well as calculators to help you design them. The more sophisticated computer-aided design (CAD) layout packages such as Xpedition and OrCAD will let you set up controlled impedance traces on your design. Altium has an article about differential pair traces, and Sierra Circuits, a PCB vendor, has an article about the difference between microstrips and striplines. Links to these articles are available in the resources on the Circuit Cellar website.

In some cases, if you keep the traces very short, you don’t have to use controlled impedance traces. The reality is that in most cases, every time a signal transitions between two connectors, a connector and a PCBA, or a PCBA trace and the integrated circuit to which it connects, there will be a small discontinuity in the impedance. In a properly designed system, these are too small to affect the functionality. If they weren’t, none of those cables would do what they are supposed to do. You want to minimize these discontinuities, but as long as the reflections don’t interfere with data transfer, everything still works. I created a board once where the impedance was wrong (as I recall, the dielectric thickness was incorrect) and there were all kinds of problems. The traces weren’t that long, but it didn’t work as intended. So, impedance is not something to ignore.

Without controlled impedance traces, an error-correcting interface such as SATA may work, but poorly. Data transfer rates might be extremely slow due to excessive error-correcting and retransmission. So, if you cut corners on the PCBA in any high-speed interface with a specified impedance, test thoroughly to be sure you didn’t degrade things. To do high-speed design on a PCBA, you really need CAD software that supports it. Otherwise, you are asking for trouble.

You may also use controlled impedance traces when working with very high-speed or RF circuits. Some integrated circuit amplifiers have a specified input impedance—typically 50Ω for RF amplifiers, and 75Ω for video amplifiers. At high frequencies, you want to design the PCB to match the device impedance.

##### Conclusion

We can often ignore impedance because it’s built into whatever we are using. But at some point, you may have to design something in which your circuit must match the impedance of something else. Hopefully, if you are in that situation, you will understand impedance enough to make everything work as it should.

REFERENCES
[1] Wikipedia article about time-domain reflectometers:
https://en.wikipedia.org/wiki/Time-domain_reflectometer
[2] Altium article on differential pair impedance:
https://resources.altium.com/p/differential-pair-impedance-using-calculator-design-your-pcb
[3] Sierra Circuits article on microstrips and striplines: https://www.protoexpress.com/blog/difference-between-microstrip-stripline-pcb/

RESOURCES
Aavid | boydcorp.com/aavid.html
Linear Technology (Acquired by Analog Devices) | www.analog.com
STMicroelectronics | www.st.com
Texas Instruments | www.ti.com

PUBLISHED IN CIRCUIT CELLAR MAGAZINE • DECEMBER 2022 #389 – Get a PDF of the issue

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