Beginners’ Guide to Impedance
last updated 28 February 2025.
Impedance can be a rather confusing concept for beginners to antennas, and even hams who have been around for a while. Some of the main reasons for this are:
- We often use shortcuts when we talk about impedances.
- Impedance is stated in ohms, but it isn’t the same as resistance, and can’t be measured with a DC ohmmeter.
- The tools for dealing with impedance that are often taught for use at the low frequencies used for power (50/60 Hz for example) are not always useful for radio frequencies.
So let’s explore a bit more about this measurement…
impedance is complex
We often talk about a “50 ohm antenna”, or a “300 ohm feedline”, but this is a shorthand that hides the actual complex value. Yes, “complex” is the proper term, because we are dealing with complex numbers. Mathematicians express complex numbers in the form x + iy, where i is the square root of negative one. Because electrical engineers often use the letter “i” for current, we use the letter “j” instead in our complex notation.
The important thing you need to know about complex numbers is that there are two parts: the real and the imaginary. They are like longitude and latitude: they act at right angles to each other, and cannot be combined into a single number without losing information. That is probably the most important thing that you need to know about complex numbers! When doing calculations, you have to use the rules for math with complex numbers, not real numbers.
In antenna work, we express these numbers in the form R + jX, where R is the resistance, and X is the reactance (which, unlike resistance, can be either positive or negative). The complex impedance is often denoted as Z = R + jX, where Z is a complex number. When the reactance at the feedpoint of an antenna is zero, we may say that it is resonant (although the definition is not quite as simple as that).
So when we talk about a “50 ohm antenna”, we are really mean the resistance is close to 50 Ω, and the reactance is close to zero ohms. But we often forget to mention that last part, so many hams assume that impedance can be expressed as a single number. The actual impedance requires both values: we cannot always assume that the reactance is zero and ignore it.
isn’t there a formula for impedance?
There is a formula for what some people call impedance, but that doesn’t help us in RF work. Let’s explore a common source of misunderstanding.
There are two different ways of expressing the location of a point in a coordinate system: rectangular (or Cartesian) coordinates, and polar coordinates. Many of us have worked problems in X and Y on a sheet of graph paper – these are rectangular coordinates. Polar coordinates instead use a distance from the origin and an angle (like a compass bearing). Either method can describe the location of any point on a 2-dimensional plane (as long as the origin and reference direction are identified). Some problems are easier to solve when expressed in one coordinate system some in the other. For example, a radar often returns data in polar coordinates (bearing angle and distance) relative to the position of the site, but those numbers can’t be directly be used by someone at a different site to locate the same point without additional calculations.
In RF work, the two components of an impedance expressed in polar form are the magnitude and the phase angle. The magnitude is given by the square root of R2 + X2, the distance from the origin. The magnitude of impedance is often denoted as |Z| (the absolute value of the complex number Z). And in some low frequency work, this is called the “impedance”. But it really isn’t: it only the magnitude of impedance, as it doesn’t include the additional information given by the phase angle that is required to fully specify the impedance.
Otherwise, this is like a radar that tells you the distance to an object, but not the direction: that results in the loss of information that may be critical.
Another problem with this misuse of the word impedance instead of the magnitude of impedance is that it returns the same number for very different impedances. For example, the impedances 50 + j0 Ω, 30 + j40 Ω, and 0 – j50 Ω, all have a magnitude of 50 Ω. If we connect them to coax cable with a characteristic impedance of 50 Ω, the SWR would be 1 : 1 in the first case, the SWR in the second case is 3 : 1, and in the last case it is infinite. And they would require very different matching networks to transform them to 50 + j0 Ω. As a result, the magnitude of impedance by itself is rarely useful in RF work.
why the difference between low frequency AC and HF?
The same principles apply, but the practical difference is due to the wavelength involved. At 50 Hz, the wavelength is 6000 km. (At 60 Hz it is 3067 miles.)
In wiring a house, for example, if we have a a 100m (300 feet) length of power cable, that is less than 0.0002 wavelengths, and any change in impedance or phase over that distance is generally insignificant. But at HF that same wire is between 1 and 10 wavelengths, and we have to pay a lot more attention to phase shift, impedance transformation, and the stray reactance of short pieces of wire. We can see the effect of changing the length of an antenna by a few cm (inches) when looking at the SWR curve, but such a change is insignificant for AC power wiring.
so, what is reactance?
Resistance and reactance are both measured in ohms, as both are defined by the ratio of the voltage across a device to the current through it. But the critical difference is that no power is dissipated in a pure reactance. Instead, energy from one part of the AC cycle is stored, then released back to the circuit in a different part of the cycle. A coil (inductance) has positive reactance, where the energy is stored in the electromagnetic field by the current. A capacitance has negative reactance, where energy is stored in an electrostatic field by the voltage. In both cases, the current is 90 degrees out of phase with the applied voltage, but in opposite directions.
So in rectangular notation we have divided the impedance into two components: the resistance, where the voltage is in phase with the current (or 180 degrees out of phase, which just means that the power is flowing in the opposite direction), and the reactance, where the voltage and current are 90 degrees out of phase (and the sign of the reactance determines whether the voltage leads the current or vice versa).
A dipole antenna that is shorter than the resonant length will have a feedpoint impedance with a negative reactance. We can add a positive reactance (loading coil) to cancel the negative reactance and achieve resonance (zero reactance). Similarly, an antenna that is somewhat longer than resonant length will have positive reactance, which we can cancel out with a series capacitor (negative reactance).
Note that canceling the reactance does not necessarily provide a low SWR: that requires that the resistance value also be reasonably close to the impedance of your feedline. So, for example, if the feedpoint impedance is 150 – j200 Ω (SWR = 8.5 : 1), we can add a coil in series with a reactance of +200 Ω and get 150 + j0 Ω (SWR = 3 : 1). But that is as low as we can get it with a series reactance: any other size of coil (or capacitor) will result in a higher SWR. The antenna is now “resonant”, but the SWR is still higher than we might like.
(We can resolve this by using reactances connected in other ways. But we’ll save that discussion for later…)
interpreting impedance
So, when you have an impedance of, say, 120 – j200 ohms (which would be an SWR of 6.5 : 1 on a 50 ohm coax), here is how to think about it:
The resistive component is 120 ohms. That’s what is responsible for the power dissipated by the load (including radiation from an antenna).
The reactance of -j200 ohms doesn’t dissipate power, but makes it more difficult to deliver power to the 120 ohms of resistance. You can add a series coil with a reactance of +j200 ohms and cancel the reactance, or, if the measurement is taken at the feedpoint, you could increase the length of the antenna a bit to lower the reactance. (Impedance gets transformed along a mis-matched feedline, so what you measure at the shack end often isn’t the actual impedance at the antenna.) Changing the antenna length will change both the resistance and reactance at the feedpoint, but usually the reactance changes faster than the resistance.
Once you cancel the reactance, you still have an impedance around 120 + j0 ohms, which would give an SWR of 2.4 : 1 on a 50 ohm feedline.
Is that too high for your radio? Well, as you learn more about feedlines, you will discover that adding an electrical quarter wavelength of 75 ohm coax would give you a very good match to 50 ohms. But that is a lesson for another day.
summary
So the important things to know in dealing with impedance are:
- Impedance has two components, resistance and reactance (or magnitude and phase angle, if dealing in polar coordinates). You can’t combine them into a single real number and use that in your calculations and expect to get the correct result, unless you explicitly know that the reactance is zero.
- When doing calculations with impedance (when the reactance is not zero), you have to follow the rules of complex arithmetic. Fortunately, there are various calculators available that will do much of that for you.
BACK TO:
impedance – main article
standing wave ratio (SWR) – main article
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