Impedance
| Impedance | |
|---|---|
| Symbol | Z |
| Unit | Ohm (Ω) |
| Made up of | Resistance (R) and reactance (X) |
| Written as | R + jX, for example 50 + j0 ohms |
| In radio | |
| Inductive reactance | Positive; rises with frequency |
| Capacitive reactance | Negative; falls with frequency |
| Amateur standard | 50 ohms |
| Measured with | Antenna analyzer or vector network analyzer (VNA) |
Impedance is the total opposition a circuit offers to alternating current, measured in ohms. It combines ordinary resistance, which takes power from the circuit and turns it into heat or, in an antenna, into radio waves, with reactance, which stores energy and gives it back each cycle. Radio signals are alternating currents at high frequencies, so impedance comes up everywhere in ham radio: in antennas, feed lines, transmitters and filters, and above all in matching an antenna to the radio.
Resistance and reactance
A resistor opposes current the same way at every frequency, following Ohm's law. Reactance depends on frequency and comes in two opposite kinds.
- Inductive reactance comes from coils and rises with frequency: X (ohms) = 6.28 × frequency (MHz) × inductance (µH). A 1-microhenry coil has about 44 ohms of reactance at 7 MHz and 88 ohms at 14 MHz.
- Capacitive reactance comes from capacitors and falls with frequency: X (ohms) = 1,000,000 / (6.28 × frequency (MHz) × capacitance (pF)). A 100-picofarad capacitor has about 227 ohms of reactance at 7 MHz.
By convention, inductive reactance is positive and capacitive reactance negative, and the two cancel each other. Impedance is written as resistance plus reactance, with a j marking the reactive part. An antenna analyzer reading of 50 + j25 means 50 ohms of resistance and 25 ohms of inductive reactance; 35 − j10 means 35 ohms of resistance and 10 ohms of capacitive reactance.
Resistance and reactance do not simply add up. An impedance of 30 + j40 has a total magnitude of 50 ohms, the square root of 30² + 40², yet it is a poor match for a 50-ohm system and gives an SWR of 3:1. A good match needs the right resistance and no reactance.
Resonance
When a circuit's inductive and capacitive reactances are equal, they cancel and only resistance remains. The circuit is then resonant. The resonant frequency of a coil and capacitor is:
- frequency (MHz) = 159 / square root of (inductance (µH) × capacitance (pF))
A 5-microhenry coil with a 100-picofarad capacitor gives 159 / √500 = 159 / 22.4, about 7.1 MHz. Tuned circuits like this form filters, multiband antenna traps and the tuning of a magnetic loop.
An antenna is resonant at the frequency where the reactance at its feed point is zero. A dipole that is too short for the operating frequency shows capacitive reactance, and one that is too long shows inductive reactance. Resonance does not mean 50 ohms: a dipole is about 70 ohms in free space and nearer 50 at typical heights, and a quarter-wave vertical over good ground is about 36 ohms. An antenna also does not have to be resonant to radiate well, as long as it can be matched.
50-ohm systems
Amateur transmitters are designed to deliver their rated power into 50 ohms, and coax, amplifiers, dummy loads and test equipment are built to the same standard. The figure is a compromise. Air-insulated coax has its lowest loss at an impedance of about 77 ohms and handles the most power at about 30 ohms; 50 ohms sits between the two. Cable television, where low loss matters more than power, uses 75 ohms.
A feed line's 50 ohms is its characteristic impedance, set by the size and spacing of its conductors and the insulation between them. It is the ratio of voltage to current for a wave traveling along the line, not a resistance an ohmmeter can measure: an ohmmeter across an unconnected coil of coax reads an open circuit. When a line is terminated in its characteristic impedance, the radio sees that impedance whatever the line's length. With any other load, the impedance at the radio end changes with the line's length and the frequency.
Matching and SWR
When the load differs from the line's impedance, part of the power is reflected, and the result is measured as SWR. For a load with no reactance, the SWR is simply the larger impedance divided by the smaller. A 100-ohm load on 50-ohm line gives 100 / 50 = 2:1, and so does a 25-ohm load. A 70-ohm dipole gives 1.4:1. Reactance raises the SWR further.
Hams bring a load to 50 ohms in several ways:
- Adjusting the antenna itself to resonance, choosing a design whose resistance is close to 50 ohms.
- Transformers such as baluns and ununs. A 4:1 type turns 200 ohms into 50.
- Matching networks at the feed point, such as the gamma match and the hairpin used on many Yagis.
- Quarter-wave transformers, a quarter-wavelength piece of line whose impedance is the square root of the two impedances multiplied together. To match 100 ohms to 50, the ideal is the square root of 5,000, about 71 ohms, and 75-ohm coax comes close enough.
- An antenna tuner in the shack, which makes the radio see 50 ohms but does not change the SWR on the feed line beyond it.