Robert Lacoste's The Darker Side: Practical Applications for Electronic Design Concepts from Circuit Cellar

By Robert Lacoste

Robert Lacoste's The Darker Side column has quickly become a must read among Circuit Cellar devotees. His column provides readers with succinct theoretical concepts and practical applications on topics as far reaching as digital modulation to antenna basics. Difficult concepts are demystified as Robert shines a light on complex topics within electronic design.

This book collects sixteen Darker Side articles that have been enriched with new, exclusive content from the author. An intro into The Darker Side will give examples of material that can enhance and optimize the way you design. A Scilab tutorial along with Scilab software and all project material will be included with this package so that all projects can be tackled hands-on. It's time to stop being afraid of the dark, let this book easily guide you through the time-draining, problematic elements of your application design.

• Tips and tricks to enhance design performance
• Practical advice on topics from digital signal design to electromagnetic interference

• Language: English
• Publisher: Elsevier Science
• Release date: Nov 27, 2009
• ISBN: 9780080962054

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Impedance Matching Basics

Note: This chapter is partially based on the column The darker side: Antenna basics, Circuit Cellar, no. 211, February 2008.

Let’s start our exploration of the darker side with a quite simple but often misunderstood topic: impedance matching. As we will see, this concept is fundamental for all radio and small signal designs, basically because radio signal power is expensive and must be well used. Just try to replace your 50-m 75-ohm TV antenna cable with a 50-ohm cable to understand what I mean: You may be disappointed with the image quality. However, matching is now also fundamental in nearly all high-speed digital designs. Our standard PC motherboard, with its GHz clock, will simply not work at all if impedance matching was not managed properly by its designers. In that case, the problem would not be one of efficient energy use but one of signal reflection linked to impedance mismatches. We will see an actual demonstration of this phenomenon in Chapter 3 on time domain reflectometry. Antennas are also fundamentally related to matching. Basically, an antenna is roughly a matching network between an electrical signal and space, but more on that in Chapter 14.

Let’s start our exploration of the darker side with a quite simple but often misunderstood topic: impedance matching. As we will see, this concept is fundamental for all radio and small signal designs, basically because radio signal power is expensive and must be well used. Just try to replace your 50-m 75-ohm TV antenna cable with a 50-ohm cable to understand what I mean: You may be disappointed with the image quality. However, matching is now also fundamental in nearly all high-speed digital designs. Our standard PC motherboard, with its GHz clock, will simply not work at all if impedance matching was not managed properly by its designers. In that case, the problem would not be one of efficient energy use but one of signal reflection linked to impedance mismatches. We will see an actual demonstration of this phenomenon in Chapter 3 on time domain reflectometry. Antennas are also fundamentally related to matching. Basically, an antenna is roughly a matching network between an electrical signal and space, but more on that in Chapter 14.

Optimal Heating for the Winter

Okay, let’s take the most basic example first. Suppose that you are in the middle of a deep forest during a cold winter’s night. You still have some coffee, but it is, unfortunately, really cold. Because you are an electrical engineer, you always have a 12-V battery pack in your bag, as well as plenty of power resistors and extra wires. Bingo, you could easily build a heater for your coffee just by connecting a resistor R across the terminals of the battery and putting this resistor in your coffee. The resistor will heat up, and so will your coffee. Assuming there aren’t any thermal losses, the power dissipated by the resistor will be fully transformed into a coffee temperature increase. But what would be the optimal value for this resistor R in order to use your battery’s energy in the most efficient way—that is, to get as much energy as possible from it? This is exactly an impedance-matching problem (Figure 1.1)

Figure 1.1: The most basic impedance matching issue.

Since no perfect voltage sources exist, any so-called fixed voltage supply has at least an internal parasitic serial resistance Rint. Suppose that your battery pack has an internal resistance of 1 ohm. This means that the output voltage of the battery will be U0 = 12 V with no load; 11 V with a 1A load, as the voltage drop on the 1-ohm internal resistance will be 1 V when the current is 1A; 10 V with 2A; and so on. And of course let’s assume that you can’t change the internal resistor of the battery. The application of Ohm’s law is then straightforward.

The battery is loaded with two resistors in series, Rint and R, so the current going through the circuit loop is

The power dissipated in your coffee will then be

What are your choices? If you use a high value for the resistor R, then the current going through it will be small, and therefore the dissipated power will be low—see the preceding equation. If R increases, then the denominator will increase faster than the numerator and the power will go down to zero if R is very high. Conversely, if you use a very-low-value resistor (close to a short circuit), then the current through the circuit will be very high; however, the power dissipated in your resistor will also be very low because the voltage across a short circuit will be nearly zero. The preceding equation also shows that P will be zero if R is null. In fact, in this last case, a lot of power will be dissipated, but only in the battery’s internal resistance and not in your heating resistor. So there must be an intermediate optimal resistor value for maximum power transfer from the battery to your resistor.

Let’s plot the power P, defined by the previous equation, when the value of the loading resistor R varies. This is also a simple but good opportunity to present one of the simulation tools I will use throughout this book, Scilab, which is a French numerical calculation tool similar to software such as MathLab (© The Mathworks) but available freely on an open-source license basis. You will find a tutorial on Scilab in Appendix A, but its syntax is quite straightforward for such a simple plotting example:

Rint=1; // internal resistor = 1 ohm

U0=12; // Battery open voltage = 12V

R=[0:0.05:10]; // R range from 0 ohm to 10 ohm with 0.05 ohm steps

P=R./(Rint+R)∧2*U0∧2; // Calculate the corresponding power

plot2d(R,P); // Plot it

If you run Scilab, you will find that there is actually an optimal value for R. This value is 1 ohm, which is the same as the source resistance. For those who remember how to calculate a derivative, you can also derive the equation given P, and you will find that this derivative is null for R = Rint. The maximum power dissipated in R is then 36 W, and exactly the same power of 36 W is dissipated in the battery’s internal resistor Rint (Figure 1.2).

Figure 1.2: The power P dissipated in the resistor is maximum for R = 1 ohm = Rint.

So we have found that the optimal value of the loading resistor, meaning the value that allows as much power as possible to be obtained from the source, is the same resistance as the internal resistance of the source. This result is the fundamental theorem of DC impedance matching. Keep in mind that it is applicable any time that a power transfer must be optimized. That doesn’t mean that you must always load a battery with a matching load, but you should if your goal is to draw as much power as possible from the source to the outside world. Be careful—a common misinterpretation of impedance matching is trying to see it as a way to save energy or power. It isn’t. Impedance matching is a way to get as much power as possible from a source that has a given, not-null, internal resistance.

What About AC?

If we move from DC to AC, the situation is exactly the same: replacing resistance with impedance. If you have a source with a given impedance ZS and connect it to a load with a given impedance ZL, then the power transfer will be maximum if both impedances are matched. There are just two differences in AC. An impedance Z is a complex value—the sum of a real resistance R and an imaginary reactance X. These impedances can be written respectively:

and

Do not be intimidated by this complex notation; just consider it an easy way of managing a pair of different values, R and X, as a single complex number written as R + jX. If you wish, refer to the short introduction on complex numbers provided in Appendix B.

With these notations, it can then be easily shown that matching is achieved when the load impedance is the complex conjugate of the source impedance. This simply means that both resistances should be equal, as in the DC case; however, both reactances should be equal in value but with opposite signs. Recall that a reactance is positive if the circuit is inductive at the working frequency, and negative if it is capacitive. In other words, a slightly capacitive source must be matched with a slightly inductive load and vice versa. The AC matching condition can then be written as

The other difference between DC and AC is that the reactance of a load changes with the frequency. For simple capacitors and inductors the well-known formulae are the