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Gematronik Weather Radar Systems
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Standing Wave Ratio

If a transmission line is terminated in its characteristic impedance then effect a maximum power transfer. A line cannot always be terminated in its characteristic impedance since it is sometimes operated as an open-ended line and other times as a short-circuit at the receiving end. If the line is open-ended, it has a terminating impedance that is infinitely large. If a line is not terminated in characteristic impedance, it is said to be finite.

When a line is not terminated in ZL, the incident energy is not absorbed but is returned along the only path available - the transmission line. Thus, the behavior of a finite line may be quite different from that of the infinite line.

Schaltbild: Ein Hochfrequenzgenerator (Symbol ist ein Kreis mit den drei die Hochfrequenz symbolisierenden Sinuswellen) mit seinem Innenwiderstand Ri in Reihe geschaltet, speist eine Zweidrahtleitung mit dem Wellenwiderstand Z. Am Speisepunkt wird die Spannung U1 gemessen, am Ende die Spannung U2.
Figure 1: Replacement circuit diagram of a transmission line driven by a generator

Before these special cases are examined has to be cleared what happens theoretically in one to long line infinitely, if an AC-Voltage is fed in. Termination shall be in its characteristic impedance (Ri = ZL).

Since Ri equals ZL, one-half the applied voltage will appear across the internal generator impedance, Ri, and one-half across the impedance of the line, ZL. The generator starts at the moment of switching on to send his power on the transmission line. At the time of t = 0 the voltage shall have her minimum value. This voltage value goes along the line with the spreading speed of the wave.

Ein Hochfrequenzgenerator speist eine als symmetrisches Antennenkabel stilisierte Zweidrahtleitung, darunter ist das Diagramm Spannung als Funktion der Leitungslänge angedeutet. Animation: Auf der Ordinate des Diagramms wandert fortlaufend eine Sinuswelle. Der Abstand zwischen zwei Wellenbergen ist gleich der Wellenlänge λ. An einem beliebigen Punkt der Ordinate ändert sich also die gemessene Spannung als Funktion der Zeit.
Figure 2: Temporal sequence of the voltage on a transmission line

If a transmission line is terminated in its characteristic impedance (ZL = Ra) then all power is transformed in heat inside the resistance Ra.

Example given:A 5 m long transmission line (ZL= 75 Ω) is fed by a generator (Ri= 75 Ω) and it is terminated with a resistance of Ra= 75 Ω i.e. is there customization. The generator delivers an oscillation with a frequency of 30 Gigahertzes. How many oscillation antinodes are on this line?

Way of solving:
λ =  c  =  3·108  = 0.01 m
f 3·1010
Number of oscillations =  Length of the transmission line  =  5 m  = 500 oscillations
wave length 0.01 m

What happens with this wave, if there isn't a customization, while an end terminator of e.g. 50 Ω is in use?

The generator delivers the power Pgen. This power splits up after the following equation:

PRi = PZL = ½ Pgen

The power PZL = Pincident goes along the transmission line and reaches end terminator Ra. However, this one is smaller than at customization. Well, it cannot raise and change into warmth the complete power. A part of the power of PZL is left. This part is reflected and goes back to the generator as Preflected.

Schaltbild: Ein Hochfrequenzgenerator mit seinem Innenwiderstand Ri in Reihe geschaltet, speist eine Zweidrahtleitung. Am Ende der Zweidrahtleitung ist ein Lastwiderstand angeklemmt mit Ra kleiner als ZL.
Figure 3: An un-customised transmission line

If ZL is dissimilarly Ra, a part of the to running wave is reflected always then. It is independently of this, whether Ra > ZL or Ra < ZL. In this case one speaks about maladjustment.

The following figure shows how two waves of the same frequency and amplitude moving in opposite directions on the same conductor will combine to form a resultant wave. The deep blue line is moving steadily from left to right and is the incident wave (from the source). The ice blue line waveform is moving from right to left and is the reflected wave. The resultant waveform, the red line, is found by algebraically adding instantaneous values of the two waveforms. The resultant waveform has an instantaneous peak amplitude that is equal to the sum of the peak amplitudes of the incident and reflected waves. Since most indicating instruments are unable to separate these voltages, they show the vector sum.

Animation von zwei Wanderwellen auf einer Leitung: die dunkelblaue Welle wandert nach rechts, die hellblaue (als Reflexion der blauen Welle) nach links. Beide Spannungen überlagern sich zu einer roten stehenden Welle, die aber wie bei einer Schwebung zwischen Null und alternierend positivem und negantivem Maximum (gleich doppelter ursprünglicher Spannung!) ihren Momentanwert ändert.
Figure 4: Risung of a standing wave

Whenever the termination is not equal to ZL, reflections occur on the line. For example, if the terminating element contains resistance, it absorbs some energy, but if the resistive element does not equal the ZL of the line, some of the energy is reflected. The amount of voltage reflected may be found by using the equation:

|r| =  Urück  =  |Ra - ZL|
Uhin |Ra + ZL|

 

s =  Umax  =  Uhin(1 + r)  =  1 + r
Umin Uhin(1 - r) 1 - r

The ratio of maximum voltage to minimum voltage on a line is called the voltage standing-wave ratio (vswr). For higher frequencies the microwave-pover value is better measurable instead the voltage value. The ratio of the square of the maximum and minimum voltages is called the power standing-wave ratio (pswr). In a sense, the name is misleading because the power along a transmission line does not vary.

Publisher: Christian Wolff
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