#### Standing Wave Ratio

_{i}

_{i }=Z

_{L}

_{L}

_{1}

_{2}

Figure 1: Replacement circuit diagram of a transmission line driven by a generator

_{i}

_{i }=Z

_{L}

_{L}

_{1}

_{2}

Figure 1: Replacement circuit diagram of a transmission line driven by a generator

#### Standing Wave Ratio

Standing waves occur in transmission lines when they are used mismatched or maladjusted.

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 Z_{L}, 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.

Figure 2: Temporal sequence of the voltage on a transmission line

Figure 2: Temporal sequence of the voltage on a transmission line

##### 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.

**Example given:**A 5 m long transmission line (Z_{L}= 75 Ω) is fed by a generator
(R_{i}= 75 Ω) and it is terminated with a resistance of R_{a}= 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 | = | 3·10^{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 |

(Remember: contrary to this upper result using the speed of light, the wave length in a conductor differs from the wave length in free space!)

_{i }=Z

_{L}

_{reflected}

_{incident}

_{a }≠ Z

_{L}

Figure 3: An un-customised transmission line

_{i }=Z

_{L}

_{reflected}

_{incident}

_{a }≠ Z

_{L}

Figure 3: An un-customised transmission line

##### Mismatched Impedance

What happens with this wave, if there isn't a customization, while a load of e.g. 50 Ω (instead of 75 Ω) is in use?

The generator delivers the power P_{gen}. This power splits up after the following equation:

P_{Ri} = P_{ZL} = ½ P_{gen} |

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.
It is independently of this, whether R_{a} > Z_{L}
or R_{a} < Z_{L}.
In this case one speaks about maladjustment.

##### 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: Risung of a standing wave

Figure 4: Risung of a standing wave

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-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.