10-12] STANDING WAVES AND IMPEDANCE MATCHING 541 



traveling in opposite directions. It is the periodic constructive and destruc- 

 tive interference between the incident and reflected waves which creates 

 the standing wave pattern. 



Minima occur every half wavelength where the incident and reflected 

 waves are out of phase and subtract {Ei — Er), while maxima occur where 

 the two waves are in phase and add (Ei + Er). The voltage standing wave 

 ratio (VSWR) is the ratio of these two quantities: 



VSWR = ^' "^ f = ^ (10-25) 



where Ei and Er are the voltage amplitudes of the incident and reflected 

 waves respectively. The relative position of a voltage minimum is a 

 function of the phase angle of the reflected wave, which is determined by 

 the complex impedance of the mismatch. When the transmission line is 

 perfectly matched and terminated, there is no reflected wave and hence no 

 standing wave, and the VSWR reduces to unity. For the case of fofa/ 

 reflection^ Er = Ei, and VSWR is infinite, meaning zero voltage at the 

 standing wave minima, while the voltage maximum is 2Ei. This condition 

 is approximated in a short-circuited transmission line. 



It is often more convenient to measure and express the condition of 

 mismatch in terms of a reflection coeficient. The magnitude of the voltage 

 reflection coefficient is simply the ratio of the reflected voltage to the 



incident voltage, or p = ^. The voltage standing wave ratio and the 

 Ei 



voltage reflection coefficient are thus related by the following equations: 

 VQWD 1 + P VSWR - 1 ,^ ... 



Unless kept to a very low magnitude, standing waves can significantly 

 aff"ect proper operation of a microwave radar. The reflected signal respon- 

 sible for the standing wave represents RF power flowing back toward the 

 source. If the voltage reflection coefficient is known, the relative power 

 reflected is simply p^, as the reflected power is proportional to the square of 

 the reflected voltage. When transmitting, this is power which does not 

 reach the antenna. Fig. 10-16 shows the amount of power lost as a function 

 of the voltage reflection coefficient. Good system design practice calls for 

 an overall VSWR not to exceed 2 : 1 (p = 0.33). Even this can represent 

 an 11 per cent reflected power loss, both on transmitting and receiving. 

 Power handling capacity of a transmission line is reduced under standing 

 wave conditions because of the increase in voltage at the standing wave 

 maxima. The voltage maximum is increased by the factor (1 + p). 



A more frequently encountered eff"ect of standing waves in microwave 

 systems is their reaction upon the RF generator. Magnetrons in particular 



