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MathematicsExploring Radio

Imaginary numbers can give real results.

Dave Casler, KEØOGEveryone knows Ohm’s law, E = I × R, which is that voltage (in volts) equals the current (in amps) times the resistance (in ohms). And the power law is just as simple, P = E × I, in which power in watts is equal to voltage times current. After all, both laws are quite basic and are part of the Technician-class license syllabus.

We can use these to answer all sorts of questions that might come up about important concepts in ham radio, including power, heat dissipation, capacitance (expressed in farads), and inductance (expressed in henries). Capacitors and inductors exhibit something called reactance, which, unlike resistance, varies with frequency. Capacitive reactance is given by XC = 1/(2 p × f × C), where XC is the reactance in ohms, and p is the constant (3.1415927...), f is the fre-quency in hertz, and C is the capacitance in farads. We note that capacitive reactance goes down as fre-quency goes up. At zero frequency, or dc, capacitive reactance is infinite (essentially an open circuit).

Inductive reactance is the reciprocal and is meas-ured in ohms. Inductive reactance is given by XL = (2 × p × f × L), where L is the inductance in henries. At zero frequency, the inductive reactance is zero, meaning the inductor acts as a short circuit at dc.

Reactive devices don’t consume energy. Rather, they accept some energy, store it in an electric field (capacitors) or a magnetic field (inductors), and then give the energy back to the rest of the system.

Now, here’s a seemingly odd thing: XC and XL vary with frequency. The reactance cannot be found with just the capacitance or inductance; you must know the frequency. It’s almost as though you’re dealing with entirely different components if you change the frequency.

Figure 1 — The representation of real numbers (plotted on the horizontal axis) and imaginary numbers (plotted on the vertical axis).

Usually, when considering reactance, we look at res-onance. This happens when there is some capaci-tance and some inductance in a circuit, and when the capacitive reactance equals the inductive reactance, the two reactances cancel out, and we achieve reso-nance in which a circuit is purely resistive (but only at one frequency). Knowing that the key to resonance is the capacitive and inductive reactances equaling each other, given two arbitrary values of capacitance and inductance, we can discover the resonant frequency by setting the capacitive and inductive reactances equal to each other: (2 × p × f × L) = 1/(2 × p × f × C). After a couple of steps, this leads to f 2 = 1/(4 × p2 × L × C), and by taking the square root of both sides, you’re left with fresonance = 1/(2 × p × √(L × C) ).

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MathematicsNow, we must consider why XC and XL are given in ohms, just like resistance. The reason stems from the fact that the familiar E = I × R is only a special case of the more general Ohm’s law, which has more to it as we dig into ac and RF circuits. The more general version of Ohm’s law is E = I × Z, where Z is the impedance, which has both resistive and reactive components. If we understand how impedance is defined, we can combine regular resistance with capacitive and inductive reactances. When we do that, we will learn that the resistance and reactance combine as impedance to affect ac waveforms, both in amplitude and phase.

Complex and Imaginary NumbersTo combine these, we need more math. Scientists long ago discovered that what we might think of as ordinary math (adding, subtracting, multiplying, and dividing real numbers) is insufficient to describe the way nature works, particularly in physics, including the physics of electromagnetic waves. So, additional mathematical tools are needed, such as complex numbers.

Squaring a number means multiplying the number by itself, and a negative number squared is always posi-tive. Therefore, a square root is the number which when multiplied by itself yields a square. There should be no such thing as the square root of a neg-ative number because no number multiplied by itself could ever be negative. In fact, if I try to take the square root of a negative number with my calculator, I get an error.

The big conceptual leap is having to imagine that there is an abstract thing that when multiplied by itself is minus 1. Because we know that no real num-ber can be multiplied by itself to be –1, such a num-ber is sheer imagination. So, mathematicians use the letter i for this thing that is defined this way: i = √(–1). Engineers (and amateurs) can’t use i because i is used for current, so they instead use the letter j to denote that imaginary thing that when squared equals –1.

When adding imaginary numbers, it is important to remember that real and imaginary numbers are sep-arate and considered orthogonal (see Figure 1). Think of real numbers as apples and imaginary num-bers as oranges. Adding apples to oranges doesn’t give you more of either, but you have more fruit. We can do something similar with real and imaginary

Figure 2 — A set of complex numbers plotted on a complex plane. Point A is the combination of a resistor of 3 ohms and an inductive reactance of 4 ohms, written as 3 + 4j ohms. Point B is another resistor of 6 ohms and a capacitive reactance of 7 ohms, written as 6 – 7j. Point C shows Point A and Point B added together, written as 9 – 3j.

numbers by graphing them, as seen in Figure 1. The real numbers are plotted on the horizontal axis, and the imaginary numbers are plotted on the vertical axis, intersecting each other at the zero point, or the origin.

Now we are prepared to represent the combination of real and imaginary numbers, which is important for dealing with capacitive and inductive reactance. On the positive imaginary line would be inductive reac-tance (in ohms). So an inductance that at some fre-quency f has an inductive reactance of 3 ohms is represented by 3j. Capacitive reactance would go on the negative imaginary line. For instance, a capacitive reactance of 4 ohms is represented by –4 j. Regular resistance is represented on the positive real line.

Now we have the tools we need to combine inductive reactance, capacitive reactance, and resistance. To add them, let’s create a set of coordinates. Let’s sup-pose we have a resistor of 3 ohms and an inductive reactance of 4 ohms. We would represent the com-bination of the two as 3 + 4j ohms (see point A in Figure 2). A combination of real and imaginary num-bers is called a complex number, and when there are multiple complex numbers plotted, it’s called the complex plane.

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Interestingly, if we added another resistor of 6 ohms and a capacitive reactance of 7 ohms (see point B), we would add them thus: (3 + 4j ) + (6 – 7j ) = 9 – 3j in ohms. The real parts and imaginary parts are added separately (see point C). Note that inductive and capacitive reactances cancel. If they cancel exactly, and the answer lies on the real line, then the circuit is resonant.

In electronics, we seem to see multiplication much more than mere addition. Multiplication can be done in Cartesian coordinates. It’s just a bit awkward. But let’s show it can be done. Let’s take the same num-bers as in the previous example, and multiply them.

(3 + 4j)(6 – 7j)

= 3 × 6 – 3 × 7j + 4j × 6 – 4j × 7j, by distributing 3 + 4j over 6 – 7j

= 18 – 21j + 24j – 28j2, but j 2 is just –1, so

= 18 + 3j + 28

= 46 + 3j

Polar CoordinatesNow we can look at a different coordinate system to represent the complex plane. This is where the real power of this approach lies. Using the real and imagi-nary perpendicular lines, we take a polar coordinates

approach. So instead of representing a complex number as a + bj, we will represent it as a vector. The vector has a length and an angle. The length is often called the magnitude, and the angle is often called the phase angle. We can represent the number as (magnitude, phase), such as (10,50°). Note that the magnitude is always zero or positive. The phase ref-erence can end up being arbitrary (it is phase differ-ences that are important in radio) and is usually taken as the phase of the voltage source.

We can convert from coordinates to vectors using the trigonometry formulas shown in Figure 3.

The magnitude is the square root of the sum of the squares of the rise and the run, and is the hypote-nuse of the triangle. The rise is the imaginary compo-nent, and the run is the real component.

We can get the angle by using a calculator to deter-mine the angle theta (Θ) using any of the following equations:

sin Θ = rise / hypotenuse

cos Θ = run / hypotenuse

tan Θ = rise / run

(Sometimes the rise is called the opposite side, and the run is called the adjacent side. The hypotenuse comes from the Pythagorean formula that says the square of the hypotenuse is equal to the sum of the squares of the rise and the run.)

Multiplication in the polar approach is simple. Multiply the magnitudes and add the angles (taking due heed for negative angles for capacitive devices or circuits). Note that division is the inverse of multiplication, so division means to divide the magnitudes and subtract the angles.

Suppose we have an RF signal of 3 V, and we place across it a network consisting of a 4 Ω resistor and an inductive reactance XL of 4 ohms. The impe-dance of the network is 4 + 4j ohms. In the polar approach, the magnitude is the square root of 42 + 42 = 16 + 16 = 32, so the magnitude is 5.66. The angle is clearly 45°, but we can calculate this as tan Θ = rise/run = 4 / 4 = 1. To get the angle itself from the tangent, we use the arctangent (inverse tangent) function: arctan(1) = 45°. So the impedance is (5.66,45°) Ω.

Figure 3 — The representation of converting from coordinates to vectors.

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Dave Casler, KEØOG, was first licensed as a Novice in 1975. An electrical engineer, most of his career involved the federal govern-ment and the Department of Defense in some way. He’s an Air Force vet and retired from IBM in 2013 to focus on his YouTube channel at www.youtube.com/user/davecasler. Dave’s chan-nel also includes training videos for Technician-, General-, and Amateur Extra-class licensees. Dave’s training videos follow the ARRL license manuals section for section and chapter for chapter. Dave focuses on the technical material and background informa-tion, and deliberately does not teach the test, so his videos are a supplement to the ARRL license manuals. Dave has been an ARRL member for over 40 years. Dave lives with his wife, Loretta, KBØVWW, near Ridgway, Colorado. See his website at www.dcasler.com. He can be reached at ke0og@arrl.net.

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Using E = I × Z to find the current, we rewrite Ohm’s law as I = E / Z. The voltage is (3,0°) (we’re using the voltage phase as the refer-ence). We have I = (3,0°) / (5.66,45°) = (3/5.66, –45°) = (0.53,–45°) amps.

Notice what this means. At this fre-quency, the current in this circuit lags the voltage by 45 degrees and is therefore inductive (see Fig-ure 4). (Yes, the current can be out of phase with the voltage. What’s happening here is that the inductor stores some of the energy in a magnetic field and then releases it a little bit later.) If the current leads the voltage, we say that the circuit is capacitive. The dividing line is 180°. If the phase angle is between zero and minus 180°, the circuit is inductive. If the phase angle is between zero and positive 180°, the circuit is capacitive.

Power in Systems with ReactanceAs it turns out, the generalization of power in this circuit to the complex plane is in units of volt- amps. P = E × I, so P = (3,0°) times (0.53,–45°) = (1.59,–45°). The magnitude of the power, in this case 1.59, is in units not of watts but of volt-amps. Let’s convert the power back to coordinates to discover how much is real power (measured in watts) and how much is imaginary reactive power. Reactive power comes only from inductance and capacitance, and is not dissipated as heat, but rather is given back to the circuit, seen in sin(–45°) = rise / 1.59 → rise = 1.59 sin(–45°) = 1.59 × (–0.707) = –1.12 W reactive.

By similar logic, the run is +1.12. The power (in volt-amps) is 1.12 – 1.12 j. Note how reactive the power is. The real part of this, 1.12, is actual power in watts, meaning converted to heat. The imaginary part of this is reactive power, which means power con-sumed, and then later in the cycle given back. And, yes, this actually happens. This circuit is quite inductive.

As an aside, the problem for the electric utility is that it has to provide 1.59 volt-amps, but is then given back 0.47 volt-amps to deal with. Given that most utility loads are inductive (think electric motors), the utility will put capacitors here and there in its system

to soak up this reactive power and release it in such a way as to cancel it. By the way, the cosine of the angle of the current to the voltage is called the power factor. In this case, it is cos(45°) = 0.707. The higher the power factor, the closer the load is to being purely resistive.

In summary, the use of complex numbers allows us to use a more general version of Ohm’s law, E = I × Z. We can calculate the combined effect of both resistance, inductance, and capacitance for a given frequency. We can look at the magnitude of the effects and the phase, and see how much power is actually consumed versus how much is given back to the source.

Figure 4 — A graph of the applied voltage and inductive current of a circuit.