#### Standing Wave Ratio

generator
Ri
R=ZL
two-wire line
ZL
U1
U2

Figure 1: Equivalent circuit diagram of a line connected to a generator

generator
Ri
R=ZL
two-wire line
ZL
U1
U2

Figure 1: Equivalent circuit diagram of a line connected to a generator

#### Standing Wave Ratio

The standing wave ratio (or voltage standing wave ratio, VSWR) is a measure that describes how well the load is impedance matched to the transmission line to which it is connected.

Standing waves occur in lines when these operate in a mismatched manner. If there is matching everywhere in an RF system, the entire power from the source to the receiver is transmitted without loss. However, since absolute matching can never be achieved in practice, the technology deals with the problems that arise when mismatching. For simplicity’s sake, it uses two extreme cases of mismatch:

• open-ended line, and other cases as a
• short-circuit at the end.

A high-frequency generator feeds a two-wire line stylized as symmetrical antenna cable, below which the diagram Voltage as a function of line length is indicated. Animation: On the ordinate of the diagram, a sine wave continuously wanders. The distance between two wave crests is equal to the wavelength λ. At any point on the ordinate, the measured voltage, therefore, changes as a function of time.

Before these special cases are considered in more detail, it is necessary to clarify what theoretically happens in an infinitely long line when an HF oscillation is fed in. There should be power matching (Ri = ZL).

At the moment of switching on, the generator begins to transmit its power to the line (see Figure 2). At time t = 0 the voltage should have its minimum value. This voltage value moves along the line with the propagation speed of the wave. This wave is called a traveling wave. It is characterized by the fact that the same signal can be measured qualitatively at any point on the line.

Fig. 2: Temporal sequence of the voltage on an RF line (so-called “travelling wave”)

Fig. 2: Temporal sequence of the voltage on an RF line (so-called “travelling wave”)

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 of 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.

If a transmission line is terminated in its characteristic impedance (ZL = Ra) then all power is transformed in heat inside the resistance Ra. The signal on this line behaves exactly as if it were infinitely long. The result is a traveling wave. It is characterized in that the signal is qualitatively measurable at each point of the transmission line.

##### Matched transmission line

If the line is terminated with a resistor Ra which is as large as the characteristic impedance ZL of the line, the entire power is converted in the resistor Ra.

If the line is short-circuited, then the entire incoming power is reflected. If the end of the line is open, i.e. the terminating resistor goes towards infinity, then the power is also completely reflected. In addition, a phase jump of 180° occurs in this case.

R=ZL
Preflected
Pincident
R≠ ZL

Figure 3: A mismatched transmission line

R=ZL
Preflected
Pincident
R≠ ZL

Figure 3: A mismatched transmission line

##### Mismatched Impedance

What happens with a wave if there is no matching but also no direct short circuit or open circuit, e.g., with a terminating resistor of 50 Ω on a line system of 75 Ω ?

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.

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

##### Interference

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.

Figure 4: Rising of a standing wave
Note: The maximum voltage of the standing wave is twice as high as the fed wave. This can even destroy the source!

Figure 4: Rising of a standing wave
Note: The maximum voltage of the standing wave is twice as high as the fed wave. This can even destroy the source!

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| = Ureflected = |Ra-ZL| Uincident |Ra+ZL|

 s = Umax = Uincident · (1+r) = (1+r) Umin Uincident · (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-power value is better measurable instead of 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.