When we fail to match the impedance of an antenna to its input transmission lineleading from the transmitter or to the receiver, the system degrades due to reflectedpower. The input impedance is measured with respect to some transmission line orsource characteristic impedance. When the two are not the same, a voltage wave isreflected, ρV, where ρ is the voltage reflection coefficient:

 ----(1)

ZA is the antenna impedance and Z0 is the measurement characteristic impedance.On a transmission line the two traveling waves, incident and reflected, produce astanding wave:

 ----(2)

 ----(3)

VSWR is the voltage standing-wave ratio. We use the magnitude of ρ, a complexphasor, since all the terms in Eq. (1) are complex numbers. The reflected power isgiven by V_i²/Z₀. The incident power is V_i²|ρ|²/Z₀. The ratio of the reflected power tothe incident power is |ρ|². It is the returned power ratio. Scale 1 gives the conversionbetween return loss and VSWR:

return loss = −20 log |ρ|    ----(4)

SCALE 1 Relationship between return loss and VSWR.

 

SCALE 2 Reflected power loss due to antenna impedance mismatch.

The power delivered to the antenna is the difference between the incident and thereflected power. Normalized, it is expressed as

1 − |ρ|² or reflected power loss(dB) = 10 log(1 − |ρ|²)    ----(5)

The source impedance to achieve maximum power transfer is the complex conjugateof the antenna impedance. Scale 2 computes the power loss due to antennaimpedance mismatch.

If we open-circuit the antenna terminals, the reflected voltage equals the incidentvoltage. The standing wave doubles the voltage along the transmission line comparedto the voltage present when the antenna is loaded with a matched load. We considerthe effective height of an antenna, the ratio of the open-circuit voltage to the inputfield strength. The open-circuit voltage is twice that which appears across a matchedload for a given received power. We can either think of this as a transmission line witha mismatch that doubled the incident voltage or as a Théveninequivalent circuit withan open-circuit voltage source that splits equally between the internal resistor and theload when it is matched to the internal resistor. Path loss analysis predicts the powerdelivered to a matched load. The mathematical Thévenin equivalent circuit containingthe internal resistor does not say that half the power received by the antenna is eitherabsorbed or reradiated; it only predicts the circuit characteristics of the antenna loadunder all conditions.

Possible impedance mismatch of the antenna requires that we derate the feed cables.The analysis above shows that the maximum voltage that occurs on the cable is twicethat present when the cable impedance is matched to the antenna. We compute themaximum voltage given the VSWR using Eq. (2) for the maximum voltage:

 ----(6)

 

The polarization of a wave is the direction of the electric field. We handle all polarizationproblems by using vector operations on a two-dimensional space using the far-fieldradial vector as the normal to the plane. This method is systematic and reduces chanceof error. The spherical wave in the far field has only θ and φ components of the electricfield:  are phasor components in the direction of the unitvectors and . We can also express the direction of the electric field in terms ofa plane wave propagating along the z-axis: . The direction of propagationconfines the electric field to a plane. Polarization is concerned with methodsof describing this two-dimensional space. Both of the above are linear polarizationexpansions. We can rewrite them as

 ----(1)

where is the linear polarization ratio, a complex constant. If time is inserted intothe expansions, and the tip of the electric field traced in space over time, it appears asan ellipse with the electric field rotating either clockwise (CW) or counter clockwise(CCW) (Figure 1). τ is the tilt of the polarization ellipse measured from the x-axis(φ = 0) and the angle of maximum response. The ratio of the maximum to minimumlinearly polarized responses on the ellipse is the axial ratio.

FIGURE 1 Polarization ellipse.

If , the ellipse expands to a circle and gives the special case of circularpolarization. The electric field is constant in magnitude but rotates either CW (lefthand) or CCW (right hand) at the rate ωt for propagation perpendicular to the page.