Impedance
Last updated 28 December 2021
There is more incorrect information and wrong thinking about impedance and SWR than just about any other topic in ham radio. It doesn’t need to be difficult, but you have to start with the right understanding.
It isn’t what you don’t know that gets you in trouble, but what you know for sure that just ain’t so.
– Mark Twain
Let’s start with probably the biggest source of confusion:
Impedance is always a 2-dimensional complex number, not a scalar.
When we talk about the feedpoint impedance of an antenna, for example, it always has two components. Generally at RF these are R (resistance) and X (reactance), expressed in rectangular coordinates as a complex number: Z = R + jX. (Electrical engineering uses “j” instead of “i” for the square root of -1 to avoid confusion with “i” to indicate current.) Sometimes it is more convenient to use admittence (the reciprical of impedance), which has two components, conductance and susceptance. Occasionally you see impedance as a magnitude and phase angle in polar coordinates: all these forms are equivalent.
The important thing, however, is that you cannot express impedance as a single (scalar) value with out losing information. Yes, we often see a reference to a “50 ohm antenna” or “50 ohm coax”, but that is based on the assumption that the reactance is zero. (In practice, the characteristic impedance of feedlines has a small reactive term that accounts for the losses.)
However, some references and even textbooks (particularly those focused on low frequency applications, such as AC power) seem to confuse impedance with the magnitude of impedance, that is, the square root of ( R squared + X squared). That gives a scalar that is more convenient to calculate with than a complex number, but none of the math works out if you do that, and you’ll end up with unreal fantasies that will lead you to great errors in thinking.
Unfortunately, some antenna analyzers insist on displaying the magnitude of impedance and calling it “Z“. This just contributes to the confusion, as there are very few occasions in antenna work where we can use the magnitude of impedance. If you can’t shut off that function, just ignore it.
Most of us, however, do not naturally think in complex numbers, or do the math in our heads as quickly as we do with scalars. So it is often more convenient to use examples where X = 0, that is, the impedance is a pure resistance. But remember that this is a special case, and, in general, the full form should be used as a complex number in the general case.
So, what is impedance anyway?
In the same way that resistance (R) is defined as voltage divided by current ( R = E / I ) in a DC circuit, impedance is the ratio of voltage to current in an AC circuit where there may be a phase difference between the two terms. So, if we drive 1 A into the feedpoint of an antenna and measure the voltage developed across it (including the phase angle relative to the current), we can calculate the feedpoint impedance. Power can only be dissipated (or radiated) due to the resistance (R), while the reactance (X) is a factor that we have to correct for to deliver maximum power to the antenna.
Voltage is developed across the reactance based on the current through it, just as with a resistor, but the voltage is 90 degrees out of phase with the current, which is why reactance doesn’t dissipate power. However, we need to consider that voltage when considering insulators for your antennas. The voltage drop across the loading coil for an 80m mobile antenna, for example, can be thousands of volts, even at a modest power level.
The feedpoint impedance is a characteristic of an antenna that varies with shape, size, frequency, height above ground, and other factors. Sometimes we can design an antenna to have a specific feedpoint impedance to make it easier to deliver maximum power to it: most commonly, we want the impedance to match the characteristic impedance of our feedline (often 50 ohms). Other times we accept whatever the impedance is, and choose an appropriate way to match it to the transmitter.
Resonance
When the reactance is zero, we often say that the antenna is “resonant”, although there are multiple definitions of “resonant” that might or might not apply. (For example, if a coax cable is terminated in a dummy load, it might not technically be “resonant”, even if the reactance is 0. But we cannot tell the difference between that and a perfectly matched antenna by impedance alone.) And, as we will discuss in later articles, the impedance can vary along a feedline…
Some folks make a big deal about having a resonant antenna. It really doesn’t make any difference, although sometimes it is more a matter of semantics. Especially since a resonant antenna might not present a resonant load to the transmitter. So I’ll write a separate article on The Importance – or Not – of Resonance. But for now, just know that whether or not an antenna is “resonant”, in and of itself, does not affect its performance.