Impedance

An antenna's impedance relates the voltage to the current at the input to the antenna. This is extremely important as we will see. Let's say an antenna has an impedance of 50 ohms. This means that if a sinusoidal voltage is input at the antenna terminals with amplitude 1 Volt, the current will have an amplitude of 1/50 = 0.02 Amps. Since the impedance is a real number, the voltage is in-phase with the current. Let's say the impedance is given as Z=50 + j*50 ohms (where j is the square root of -1). Then the impedance has a magnitude of and a phase given by This means the phase of the current will lag the voltage by 45 degrees. To spell it out, if the voltage (with frequency f) at the antenna terminals is given by then the current will be given by So impedance is a simple concept, which relates the voltage and current at the input to the antenna. The real part of an antenna's impedance represents power that is either radiated away or absorbed within the antenna. The imaginary part of the impedance represents power that is stored in the near field of the antenna (non-radiated power). An antenna with a real input impedance (zero imaginary part) is said to be resonant. Note that an antenna's impedance will vary with frequency. While simple, we will now explain why this is important, considering both the low frequency and high frequency cases. Low Frequency When we are dealing with low frequencies, the transmission line that connects the transmitter or receiver to the antenna is short. Short in antenna theory always means "relative to a wavelength". Hence, 5 meters could be short or very long, depending on what frequency we are operating at. At 60 Hz, the wavelength is about 3100 miles, so the transmission line can almost always be neglected. However, at 2 GHz, the wavelength is 15 cm, so the little length of line within your cell phone can often be considered a 'long line'. Basically, if the line length is less than a tenth of a wavelength, it is reasonably considered a short line. Consider an antenna (which is represented as an impedance given by ZA) hooked up to a voltage source (of magnitude V) with source impedance given by ZS. The equivalent circuit of this is shown in Figure 1. Figure 1. Circuit model of an antenna hooked to a source. The power that is delivered to the antenna can be easily found to be (recall your circuit theory, and that P=I*V): If ZA is much smaller in magnitude than ZS, then no power will be delivered to the antenna and it won't transmit or receive energy. If ZA is much larger in magnitude than ZS, then no power will be delivered as well. For maximum power to be transferred from the generator to the antenna, the ideal value for the antenna impedance is given by: The * in the above equation represents complex conjugate. So if ZS=30+j*30 ohms, then for maximum power transfer the antenna should impedance ZA=30-j*30 ohms. Typically, the source impedance is real (imaginary part equals zero), in which case maximum power transfer occurs when ZA=ZS. Hence, we now know that for an antenna to work properly, its impedance must not be too large or too small. It turns out that this is one of the fundamental design parameters for an antenna, and it isn't always easy to design an antenna with the right impedance. High Frequency This section will be a little more advanced. In low-frequency circuit theory, the wires that connect things don't matter. Once the wires become a significant fraction of a wavelength, they make things very different. For instance, a short circuit has an impedance of zero ohms. However, if the impedance is measured at the end of a quarter wavelength transmission line, the impedance appears to be infinite, even though there is a dc conduction path. In general, the transmission line will transform the impedance of an antenna, making it very difficult to deliver power, unless the antenna is matched to the transmission line. Consider the situation shown in Figure 2. The impedance is to be measured at the end of a transmission line (with characteristic impedance Z0) and Length L. The end of the transmission line is hooked to an antenna with impedance ZA. Figure 2. High Frequency Example. It turns out (after studying transmission line theory for a while), that the input impedance Zin is given by: This is a little formidable for an equation to understand at a glance. However, the happy thing is: If the antenna is matched to the transmission line (ZA=ZO), then the input impedance does not depend on the length of the transmission line. This makes things much simpler. If the antenna is not matched, the input impedance will vary widely with the length of the transmission line. And if the input impedance isn't well matched to the source impedance, not very much power will be delivered to the antenna. This power ends up being reflected back to the generator, which can be a problem in itself (especially if high power is transmitted). Hence, we see that having a tuned impedance for an antenna is extremely important. For more information on transmission lines, see the transmission line tutorial.