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Microwave an RF Supplementary Project
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Equivalent Impedance of an Antenna

Introduction:
Antenna impedance is a measure of the resistance to an electrical signal in an antenna. Many factors have an impact on an antenna’s ability to transmit a signal, including the environment that the antenna is in and the design and composition of the antenna. Understanding antenna impedance is important when designing components that connect an antenna to a receiver or transmitter. The ratio to voltage current, which is equal to antenna impedance, is expressed in ohms. The antenna impedance represents the power that is absorbed by the antenna as well as the power that is dispersed by it as it comes into contact with electromagnetic waves.
Voltage and Current Relationship:
Antenna impedance relates the voltage to the current at the input to the antenna.
Let us assume that the antenna has an impedance of 50 ohms. And if a sinusoidal voltage is applied at the antenna terminals with amplitude of 1 volt, then the current will have amplitude of I=VR=150=0.02 A. Since the impedance is a real number, the voltage is in phase with the current.
Alternatively suppose the impedance is given by complex number say Z=50+j50 ohms.
If Z=50+j50, then the impedance has the magnitude equal to:
502+502=70.71
The phase will be equal to: tan-1ImZReZ=45° This means the phase of the current will lag the voltage by 45°. That is the current waveform is delayed relative to the voltage waveform. To spell it out, if the voltage with frequency f at the antenna terminals is given by:
Vt=cos⁡(2πft)
The electric current will then be equal to:
It=170.71cos⁡(2πft-π180×45)
Hence antenna impedance is a simple concept. Impedance relates the voltage and current at the input to the antenna. The real part of the antenna impedance represent power that is either radiated away or absorbed within the antenna .the imaginary part of the impedance represent power that is stored in the near field of the antenna. This is non-radiated power. An antenna with real input impedance is said to be resonant. As we can see that the impedance of an antenna will vary with frequency.
Before we go ahead with finding the total impedance of an antenna we should explain why this is important considering both 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 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. Antenna circuit model
From circuit theory, we know that P=I×V. The power that is delivered to the antenna is:
PPA=V2×ZA(ZA+Zs)2
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:
ZA=ZS*
The * in the above equation represents complex conjugate. So if Zs=30+j30 ohms, then for maximum power transfer the antenna should impedance ZA=30-j30 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 - particularly over a wide frequency range.
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 Zo) and Length L. The end of the transmission line is hooked to an antenna with impedance ZA.

Figure 2. high frequency antenna circuit
It turns out (after studying transmission line theory for a while), that the input impedance Zin is given by:
Zin=ZoZA+jZotan⁡(2πflc)Zo+jZAtan⁡(2πflc)
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). This loss of power is known as impedance mismatch. Hence, we see that having a tuned impedance for an antenna is extremely important.
VSWR:
We see that an antenna's impedance is important for minimizing impedance-mismatch loss. A poorly matched antenna will not radiate power. This can be somewhat alleviated via impedance matching, although this doesn't always work over a sufficient bandwidth (bandwidth is the next topic).
A common measure of how well matched the antenna is to the transmission line or receiver is known as the Voltage Standing Wave Ratio (VSWR). VSWR is a real number that is always greater than or equal to 1. A VSWR of 1 indicates no mismatch loss (the antenna is perfectly matched to the tx line). Higher values of VSWR indicate more mismatch loss.
As an example of common VSWR values, a VSWR of 3.0 indicates about 75% of the power is delivered to the antenna (1.25 dB of mismatch loss); a VSWR of 7.0 indicates 44% of the power is delivered to the antenna (3.6 dB of mismatch loss). A VSWR of 6 or more is pretty high and will generally need to be improved.
The parameter VSWR sounds like an overly complicated concept; however, power reflected by an antenna on a transmission line interferes with the forward travelling power - and this creates a standing voltage wave - which can be numerically evaluated by the quantity Voltage Standing Wave Ratio (VSWR).

Refrences
"Antenna Impedance." Impedance of an Antenna. N.p., 1 Jan. 2008. Web. 1 Dec. 2013. <http://www.antenna-theory.com/basics/impedance.php>.
Cline, J.. "An Antenna Impedance-Measuring Instrument." Proceedings of the IRE 40.12 (1952): 1686-1689. Print.
Fox, C.B., and Susan Barwick. "what is an antenna impedance." WiseGeek. Conjecture, 18 Nov. 2013. Web. 1 Dec. 2013. <http://www.wisegeek.com/what-is-antenna-impedance.htm>.
Stiles, Jim. "Antenna Impedance Notes." notes. N.p., 18 Nov. 2007. Web. 1 Dec. 2013. <http://www.ittc.ku.edu/~jstiles/622/handouts/section_6D_Antenna_Impedance_package.pdf>.