# 4.3: Resonant Antennas

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With a resonant antenna the current on the antenna is directly related to the amplitude of the radiated EM field. Resonance ensures that the standing wave current on the antenna is high.

## 4.3.1 Radiation from a Current Filament

The fields radiated by a resonant antenna are most conveniently calculated by considering the distribution of current on the antenna. The analysis begins by considering a short filament of current, see Figure 4.2.1(a). Considering the sinusoidal steady state at radian frequency \(\omega\), the current on the filament with phase \(χ\) is \(I(t) = |I_{0}| \cos(\omega t + χ)\), so that \(I_{0} = |I_{0}|e^{-\jmath χ}\) is the phasor of the current on the filament. The length of the filament is \(h\), but it has no other dimensions, that is, it is considered to be infinitely thin.

Resonant antennas are conveniently modeled as being made up of an array of current filaments with spacings and lengths being a tiny fraction of a wavelength. Wire antennas are even simpler and can be considered to be a line of current filaments. Ramo, Whinnery, and Van Duzer [1] calculated the spherical EM fields at the point \(P\) with the spherical coordinates \((\phi ,\theta , r)\) generated by the \(z\)-directed current filament centered at the origin in Figure 4.2.1. The total EM field components in phasor form are

\[\begin{align}\label{eq:1} H_{\phi}&=\frac{I_{0}h}{4\pi}e^{-\jmath kr}\left(\frac{\jmath k}{r}+\frac{1}{r^{2}}\right)\sin\theta ,\quad\overline{H_{\phi}}=H_{\phi}\hat{\phi}\\ \label{eq:2}E_{r}& =\frac{I_{0}h}{r\pi}e^{-\jmath kr}\left(\frac{2\eta }{r^{2}}+\frac{2}{\jmath\omega\varepsilon_{0}r^{3}}\right)\cos\theta ,\quad \overline{E_{r}}=E_{r}\hat{\mathbf{r}} \\ \label{eq:3}E_{\theta}&=\frac{I_{0}h}{4\pi}e^{-\jmath kr}\left(\frac{\jmath\omega\mu_{0}}{r}+\frac{1}{\jmath\omega\varepsilon_{0} r^{3}}+\frac{\eta}{r^{2}}\right)\sin\theta ,\quad\overline{E_{\theta}}=E_{\theta}\hat{\theta}\end{align} \]

where \(\eta\) is the free-space characteristic impedance, \(\varepsilon_{0}\) is the permittivity of free space, and \(\mu_{0}\) is the permeability of free space. The variable \(k\) is called the **wavenumber **and \(k = 2π/\lambda = \omega\sqrt{\mu_{0}\varepsilon_{0}}\). The \(e^{-\jmath kr}\) terms describe the variation of the phase of the field as the field propagates away from the filament. Equations \(\eqref{eq:1}\)-\(\eqref{eq:3}\) are the complete fields with the \(1/r^{2}\) and \(1/r^{3}\) dependence describing the near-field components. In the far field, i.e. \(r ≫ \lambda\), the components with \(1/r^{2}\) and \(1/r^{3}\) dependence become negligible and the field components left are the propagating components \(H_{\phi}\) and \(E_{θ}\):

\[\label{eq:4}H_{\phi}=\frac{I_{0}h}{4\pi }e^{-\jmath kr}\left(\frac{\jmath k}{r}\right)\sin\theta,\quad E_{r}=0,\quad\text{and}\quad E_{\theta}=\frac{I_{0}h}{4\pi}e^{-\jmath kr}\left(\frac{\jmath\omega\mu_{0}}{r}\right)\sin\theta \]

Now consider the fields in the plane normal to the filament, that is, with \(θ = π/2\text{ radians}\) so that \(\sin θ = 1\). The fields are now

\[\label{eq:5}H_{\phi}=\frac{I_{0}h}{4\pi}e^{-\jmath kr}\left(\frac{\jmath k}{r}\right)\quad\text{and}\quad E_{\theta}=\frac{I_{0}h}{4\pi}e^{-\jmath kr}\left(\frac{\jmath\omega\mu_{0}}{r}\right) \]

and the wave impedance is

\[\label{eq:6}\eta =\frac{E_{\theta}}{H_{\phi}}=\frac{I_{0}h}{4\pi}e^{-\jmath kr}\frac{\jmath\omega\mu_{0}}{r}\left(\frac{I_{0}h}{4\pi}e^{-\jmath kr}\frac{\jmath k}{r}\right)^{-1}=\frac{\omega\mu_{0}}{k} \]

Note that the strength of the fields is directly proportional to the magnitude of the current. This proves to be very useful in understanding spurious radiation from microwave structures. Now \(k=\omega\sqrt{\mu_{0}\epsilon_{0}}\), so

\[\label{eq:7}\eta =\frac{\omega\mu_{0}}{\omega\sqrt{\mu_{0}\epsilon_{0}}}=\sqrt{\frac{\mu_{0}}{\epsilon_{0}}}=377\:\Omega \]

as expected. Thus an antenna can be viewed as having the inherent function of an impedance transformer converting from the lower characteristic impedance of a transmission line (often \(50\:\Omega\)) to the \(377\:\Omega\) characteristic impedance of free space. Further comments can be made about the propagating fields (Equation \(\eqref{eq:4}\)). The EM field propagates in all directions except not directly in line with the filament. For fixed \(r\), the amplitude of the propagating field increases sinusoidally with respect to \(\theta\) until it is maximum in the direction normal to the filament.

The power radiated is obtained using the **Poynting vector**, which is the cross-product of the propagating electric and magnetic fields. From this the time-average propagating power density is (with the SI units of \(\text{W/m}^{2}\))

\[\label{eq:8}P_{R}=\frac{1}{2}\Re (E_{\theta}H^{\ast}_{\phi}=\frac{\eta k^{2}|I_{0}|^{2}h^{2}}{32\pi^{2}r^{2}}\sin^{2}\theta \]

and the power density is proportional to \(1/r^{2}\). In Equation \(\eqref{eq:8}\) \(\Re(\cdots )\) indicates that the real part is taken.

## 4.3.2 Finite-Length Wire Antennas

The EM wave launched by a wire antenna of finite length is obtained by considering the wire as being made up of many filaments and the field is then the superposition of the fields from each filament. As an example, consider the antenna in Figure 4.2.1(b) where the wire is half a wavelength long. As a good approximation the current on the wire is a standing wave and the current on the wire is in phase so that the current phasor is

\[\label{eq:9}I(z)=I_{0}\cos (kz) \]

From Equation \(\eqref{eq:4}\) and referring to Figure 4.2.1 the fields in the far field are

\[\label{eq:10}H_{\phi}=\int^{\lambda /4}_{-\lambda /4}\frac{I_{0}\cos (kz)}{4\pi}e^{-\jmath kr'}\left(\frac{\jmath k}{r'}\right)\sin\theta 'dz \]

\[\label{eq:11}E_{\theta}=\int^{\lambda /4}_{-\lambda /4}\frac{I_{0}\cos (kz)}{4\pi}e^{-\jmath kr'}\left(\frac{\jmath\omega\mu_{0}}{r'}\right)\sin\theta 'dz \]

where \(θ′\) is the angle from the filament to the point \(P\). Now \(k = 2π/\lambda\) and at the ends of the wire \(z = ±\lambda /4\) where \(\cos (kz) = \cos (±π/2) = 0\). Evaluating the equations is analytically involved and will not be done here. The net result is that the fields are further concentrated in the plane normal to the wire. At large \(r\), of at least several wavelengths distant from the antenna, only the field components decreasing as \(1/r\) are significant. At large \(r\) the phase differences of the contributions from the filaments is significant and results in shaping of the fields. The geometry to be used in calculating the far field is shown in Figure \(\PageIndex{1}\)(a). The phase contribution of each filament, relative to that at \(z = 0\), is \((kz \sin θ)/\lambda\) and Equations \(\eqref{eq:10}\) and \(\eqref{eq:11}\) become

\[\label{eq:12}H_{\phi}=I_{0}\left(\frac{\jmath j}{4\pi r}\right)\sin (\theta )e^{-\jmath kr}\int^{\lambda /4}_{-\lambda /4}\frac{\sin (kz)}{4\pi}\sin (z\sin (\theta ))dz \]

\[\label{eq:13} E_{\theta}=I_{0}\left(\frac{\jmath\omega\mu_{0}}{4\pi r}\right)\sin (\theta ))e^{-\jmath kr}\int^{\lambda /4}_{-\lambda /4}\sin (kz)\sin (z\sin (\theta ))dz \]

Figure \(\PageIndex{1}\)(b) is a plot of the near-field electric field in the \(y-z\) plane calculating \(E_{r}\) and \(E_{θ}\) (recall that \(E_{\phi} = 0\)) every \(90^{\circ}\). Further from the antenna the \(E_{r}\) component rapidly reduces in size, and \(E_{θ}\) dominates.

A summary of the implications of the above equations are, first, that the strength of the radiated electric and magnetic fields are proportional to the

Figure \(\PageIndex{1}\): Wire antenna: (a) geometry for calculating contributions from current filaments of length \(dz\) with coordinate \(z\) \((d = −z \sin θ)\); and (b) instantaneous electric field in the \(y-z\) plane due to a \(\lambda /2\) long current element. There is also a magnetic field.

Figure \(\PageIndex{2}\): Monopole antenna showing total current and forwardand backward-traveling currents: (a) a \(\frac{1}{2}\lambda\)-long antenna; and (b) a relatively long antenna.

current on the wire antenna. So establishing a standing current wave and hence magnifying the current is important to the efficiency of a wire antenna. A second result is that the power density of freely propagating EM fields in the far field is proportional to \(1/r^{2}\), where \(r\) is the distance from the antenna. A third interpretation is that the longer the antenna, the flatter the radiated transmission profile; that is, the radiated energy is more tightly confined to the \(x-y\) (i.e. \(\Theta =0\)) plane. For the wire antenna the peak radiated field is in the plane normal to the antenna, and thus the wire antenna is generally oriented vertically so that transmission is in the plane of the earth and power is not radiated unnecessarily into the ground or into the sky.

To obtain an efficient resonant antenna, all of the current should be pointed in the same direction at a particular time. One way of achieving this is to establish a standing wave, as shown in Figure \(\PageIndex{2}\)(a). At the open-circuited end, the current reflects so that the total current at the end of the wire is zero. The initial and reflected current waves combine to create a standing wave. Provided that the antenna is sufficiently short, all of the total current— the standing wave—is pointed in the same direction. The optimum length is about a half wavelength. If the wire is longer, the contributions to the field from the oppositely directed current segments cancel (see Figure \(\PageIndex{2}\)(b)).

In Figure \(\PageIndex{2}\)(a) a coaxial cable is attached to the monopole antenna below the ground plane and often a series capacitor between the cable and the antenna provides a low level of coupling leading to a larger standing wave. The capacitor also approximately matches the characteristic impedance of the cable to the input impedance \(Z_{in}\) of the antenna. If the length of the monopole is reduced to one-quarter wavelength long it is again resonant, and the input impedance, \(Z_{in}\), is found to be \(36\:\Omega\). Then a \(50\:\Omega\) cable can

Figure \(\PageIndex{3}\): Mobile antenna with phasing coil extending the effective length of the antenna.

Figure \(\PageIndex{4}\): Dipole antenna: (a) current distribution; (b) stacked dipole antenna; and (c) detail of the connection in a stacked dipole antenna.

be directly connected to the antenna and there is only a small mismatch and nearly all the power is transferred to the antenna and then radiated.

Another variation on the monopole is shown in Figure \(\PageIndex{3}\), where the key component is the phasing coil. The phasing coil (with a wire length of \(\lambda /2\)) rotates the electrical angle of the current phasor on the line so that the current on the \(\lambda /4\) segment is in the same direction as on the \(\lambda /2\) segment. The result is that the two straight segments of the loaded monopole radiate a more tightly confined EM field. The phasing coil itself does not radiate (much).

Another ingenious solution to obtaining a longer effective wire antenna with same-directed current (and hence a more tightly confined RF beam) is the stacked dipole antenna \(\PageIndex{4}\). The basis of the antenna is a dipole as shown in Figure \(\PageIndex{4}\)(a). The cable has two conductors that have equal amplitude currents, \(I\), but flowing as shown. The wire section is coupled to the cable so that the currents on the two conductors realize a single effective current \(I_{\text{dipole}}\) on the dipole antenna. The stacked dipole shown in Figure \(\PageIndex{4}\)(b) takes this geometric arrangement further. Now the radiating element is hollow and a coaxial cable is passed through the antenna elements and the half-wavelength sections are fed separately to effectively create a wire antenna that is several wavelengths long with the current pointing in one direction. Most cellular antennas using wire antennas are stacked dipole antennas.

Figure \(\PageIndex{5}\): Vivaldi antenna showing design procedure: (a) the antenna; (b) a stepped approximation; and (c) transmission line approximation. In (a) the black region is a metal sheet.

### Summary

Standing waves of current can be realized by resonant structures other than wires. A microstrip patch antenna, see Figure 4.1.2(b), is an example, but the underlying principle is that an array of current filaments generates EM components that combine to create a propagating field. Resonant antennas are inherently narrowband because of the reliance on the establishment of a standing wave. A relative bandwidth of \(5\%–10\%\) is typical.