CHAPTER 1
Antenna Circuit Model. Antenna Matching. Antenna Bandwidth

SECTION 1 LUMPED CIRCUIT MODEL OF AN ANTENNA. ANTENNA INPUT IMPEDANCE

• 1.1 Antenna Circuit Model. Antenna Loss
• 1.2 Maximum Power Transfer to (and from) Antenna
• 1.3 Antenna Efficiency
• 1.4 Antenna Input Impedance and Impedance Matching
• 1.5 Point of Interest: Input Impedance of a Dipole Antenna and Its Dependence on Dipole Length
• 1.6 Beyond the First Resonance
• 1.7 Numerical Modeling
• References
• Problems

1.1 ANTENNA CIRCUIT MODEL. ANTENNA LOSS

The generic transmitter (TX) circuit with an antenna is shown in Figure 1.1. The generator (g) is modeled as an ideal (sinusoidal or pulse) voltage source Vg in series with the generator resistance Rg, connected to a TX antenna. The typical generator resistance is 50 O. This model is known as Thévenin equivalent of the generator circuit. The Norton equivalent may also be used when necessary.

Figure 1.1 A generator (its Thévenin equivalent) connected to an antenna.

The portion depicted in the shaded box is an antenna. The antenna in Figure 1.1 is assumed to be resonant, which means that its equivalent impedance, Za, is purely real, i.e.

In order words, the resonant antenna is simply modeled by a resistor Ra.

The antenna resistance Ra includes two parts:

1. Radiation resistance of the antenna Rr that describes the circuit power loss due to radiation by the antenna into free space.
2. Loss resistance of the antenna RL that describes the circuit power loss in the antenna itself. Case in point: a long thin wire with a significant ohmic resistance or a helical antenna with a ferrite lossy core.

One thus has

Parasitic antenna resistance RL has the following features:

1. it is zero for ideal antennas (a metal antenna made of perfect electric conductors);
2. it is usually relatively small for metal antennas covering the band 0.3-3 GHz (UHF, L-band, S-band) where it may be often ignored;
3. it may be very significant for printed antennas on lossy dielectric substrates and in the vicinity of lossy dielectric (such as FR4, ABS, human body, etc.);
4. it is vital for very small antennas whose size is much less than the wavelength.

Example 1.1

A small antenna operating at f = 10 MHz uses a thin copper wire with the diameter D of 0.25 mm and with the wire length l of 1 m. Calculate antenna loss resistance RL.

Solution: The DC resistance of the wire is given by

where s is the material conductivity and A is the wire cross section. However, we cannot use this formula since most of the high-frequency current flows in a thin skin layer around the wire. The correct result has the form:

where P is wire perimeter and µ0 is vacuum permeability. A short MATLAB script given below accomplishes the task and gives RL approximately equal to 1 O. This value may be comparable to the radiation resistance of a small antenna, and may even exceed it.

1.2 MAXIMUM POWER TRANSFER TO (AND FROM) ANTENNA

One question you have to ask yourself is this: for a fixed resistance Rg, can the electric power delivered to the antenna be maximized, and at which value of Ra does the maximum occur? In other words, we would like to know what parameters the antenna should have in order to acquire and radiate maximum electric power from the given RF generator (an RF amplifier). In an electric circuit, the passive load - the antenna - may have only one such parameter - the antenna resistance. All other antenna parameters (geometrical, material, etc.) are implicitly included into antenna's resistance.

The answer is given by the maximum power transfer theorem and found by solving the circuit in Figure 1.1. We assume that voltages and current are all functions of time and solve the circuit for an arbitrary time moment. First, the current is determined from the given voltage source ?g(t) and the total resistance using the series equivalent,

This allows us to compute the (instantaneous) power delivered to the antenna based on

For a generator with fixed resistance Rg, the load resistance determines the power Pa(t) at any time instant. Eq. (1.5b) is identical to the corresponding result at DC.

Example 1.2

Calculate and plot the average acquired antenna power when the generator with a periodic waveform ?g(t) = ?g(t + T) is characterized by the rms voltage and generator resistance given by

Solution: We use Eq. (1.5b) and average it over period T to obtain average power . The result has a form:

Then, we plot the average antenna power as a function of the load resistance. The short MATLAB script given below accomplishes the task:

This important result is given in Figure 1.2. We see that the load power does have a maximum at a particular value of the load resistance. Our next step will be to find this maximum.

It is instructive to find the maximum of the average antenna power analytically since it gives us insight into the optimization process. We treat Pavg in Eq. (1.5d) as a function of Ra, i.e. Pavg = Pavg(Ra). From basic calculus it is known that a function has a maximum where its first derivative is zero. Consequently, differentiating Pavg with respect to Ra gives

Figure 1.2 Average antenna power as a function of the antenna resistance for fixed Vrms = 9 V, Rg = 50 O.

The necessary and sufficient condition for Eq. (1.6) to hold, and thus maximizing the antenna power, is simply

This result has a great practical value despite, or maybe thanks to, its simplicity. The maximum output radiated power is achieved when the antenna resistance is exactly equal to the internal resistance of the generator. In other words, the antenna is matched to the generated; it is called the matched antenna. The design of such an antenna over a frequency band of interest is called antenna matching. Such a design is a critical step, and it may be a great challenge for an RF engineer. It does not matter if the antenna radiates a sinusoidal or other periodic signal or a pulse; Eq. (1.7) holds in either case since it also maximizes power at any time instant.

However, it must be clearly stated that no more than 50% of the total generator power can be extracted even in this best case. This statement makes sense if we again examine the circuit in Figure 1.1 with two equal resistors. We see that the power delivered to each resistor is obviously the same. Since Rg is internal resistance, half of the total power is spent to heat up the generator.

Note:

The power maximum in Figure 1.2 is relatively flat over the domain Ra > Rg; however, the power drops sharply when Ra < < Rg. This last condition should be avoided if at all possible. The corresponding example is given below.

Example 1.3

A transmitting antenna in a radio handset features a monopole antenna. It is connected to a sine wave generator that has the same basic form as Figure 1.1 with an internal (generator) resistance of 50 O. The antenna has the radiation resistance of 50 O (which generates power loss in terms of electromagnetic radiation); its loss...