#### Standing Wave Ratio

_{i}

_{i }=Z

_{L}

_{L}

_{1}

_{2}

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

_{i}

_{i }=Z

_{L}

_{L}

_{1}

_{2}

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

#### Standing Wave Ratio

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 a 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 (R_{i} = Z_{L}).

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 travelling 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”)

##### Adjusted Inpedance

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 (R_{i} = Z_{L}).

Since R_{i} equals Z_{L},
one-half the applied voltage will appear across
the internal generator impedance, R_{i}, and one-half across the impedance of the line,
Z_{L}. 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.
This wave is called traveling wave.

If a transmission line is terminated in its characteristic impedance
(Z_{L} = R_{a}) then all power is transformed in heat inside the resistance
R_{a}.
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 R_{a} which is as large as the characteristic impedance Z_{L} of the line,
the entire power is converted in the resistor R_{a}.

A 5 m long transmission line (Z_{L}= 75 Ω ) is fed by a generator
(R_{i}= 75 Ω ) and got a terminating resistance of R_{a}= 75 Ω, i.e. there is matching.
The generator supplies an oscillation with a frequency of 30 GHz.
How many oscillations fit on the line?

Solution:

λ = | c | = | 3·10^{8} ^{m}/_{s} |
= 0,01 m |

f | 3·10^{10} ^{1}/_{s} |

Number of oscillations = | Length of the transmission line | = | 5 m | = 500 oscillations |

wave length | 0.01 m |

(Attention! If a cable is used as a line, the propagation speed is reduced: a velocity of light reduced by a cable-typical shortening factor is then used. In contrast, in waveguides the propagation speed is the so-called phase speed - this can be higher than the speed of light).

_{i }=Z

_{L}

_{reflected}

_{incident}

_{a }≠ Z

_{L}

Figure 3: A mismatched transmission line

_{i }=Z

_{L}

_{reflected}

_{incident}

_{a }≠ Z

_{L}

Figure 3: A mismatched transmission line

##### Mismatched Impedance

What happens with a wave if there is no matching, e.g. with a terminating resistor of 50 Ω on a line system of 75 Ω ?

The power P_{ZL} = P_{incident}
goes along the transmission line and reaches end terminator R_{a}.
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 P_{ZL} is left.
This part is reflected and goes back to the generator as P_{reflected}.

If Z_{L} is dissimilarly R_{a},
a part of the to running wave is reflected always then.
In this case, one speaks of mismatching.
It is independently of this, whether R_{a} > Z_{L}
or R_{a} < Z_{L}.

##### 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 Z_{L}, 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 Z_{L} of the line,
some of the energy is reflected. The amount of voltage reflected may be found by using the equation:

|r| = | U_{reflected} |
= | |R_{a}-Z_{L}| |

U_{incident} |
|R_{a}+Z_{L}| |

s = | U_{max} |
= | U_{incident} · (1+r) |
= | (1+r) |

U_{min} |
U_{incident} · (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.