### Section 6: Wave Bounce Complex Structures

In Section 5, we took a look at how the voltage waves bounce along relatively simple structures, generally a single transmission line between a source and load impedance. In this section, we will take a look at how waves bounce between multiple transmission line structures. This produces more complicated return loss structures.#### Simple Return Loss

The return loss is the amount of energy that returns back towards the source... it is loss in the sense that it not transmitted to the load. Loss can occur in two ways- it can be reflected back to the source instead of going to the load, or it can be lost by being burned up in the resistance of the lines. Here, we are only considering the energy that returns back towards the source.Typically, the return loss is expressed in decibels (dB), as follows:

Equation 1.6.1 |

In the previous section, we showed voltage waves being transmitted towards open circuits and short circuits. Because no energy can be transmitted to either, and no energy is lost in the transmission of the waves, all of the energy is reflected back to the source. The ratio of reflected energy to incident energy is one, therefore:

Equation 1.6.2 |

A perfectly matched load will have no reflection:

Equation 1.6.3 |

In practice, there are no perfectly matched loads. An exceptionally good match might be within a few percent of the source.

Equation 1.6.4 |

In practice, the match will typically be much worse; in practice, we're content to have a return loss on the order of -15dB or better.

#### Frequency Dependent Return Loss

The return loss equations () described above related the impedance between a source impedance and load impedance, with no transmission line in between. However, Equation 1.6.5 shows the impedance looking into a transmission line of impedance Z0, which is terminated with a resistor of ZL. The important thing to note is that the impedance looking into the line (which relates to the return loss, Equation 1.6.6) is related to the line length and the frequency. More explicitly, the return loss for a complex structure will vary over frequency.The return loss measures how well the circuit impedance matches the source impedance. Based on equation 1.6.6, if the circuit impedance is perfectly matched (a 50-ohm source into a 50-ohm transmission line with a 50-ohm load), there will be no reflections. If it is well matched (a 51-ohm source into 49-ohms), there will be very little reflection (-34dB, or 2% of the voltage wave)

#### Reflections with complex structures

The first structure we will examine has multiple transmission lines, each is two wavelengths long at 1GHz. The first transmission line is 50 ohms, the second is 70.7 ohms, and the third is 100 ohms. The input impedance is 50 ohms and the output impedance is 100 ohms.

Where,

Equation 1.6.5 |

Equation 1.6.6 |

#### Reflections with complex structures

The first structure we will examine has multiple transmission lines, each is two wavelengths long at 1GHz. The first transmission line is 50 ohms, the second is 70.7 ohms, and the third is 100 ohms. The input impedance is 50 ohms and the output impedance is 100 ohms.Z0 = 50 ohms, Length0 = 2 lambda

Z1 = 70.7 ohms, Length1 = 2 lambda

Z2 = 100 ohms, Length2 = 2 lambda

Z_load = 100

Z1 = 70.7 ohms, Length1 = 2 lambda

Z2 = 100 ohms, Length2 = 2 lambda

Z_load = 100

Video of Voltage Waves Bouncing Around Structure

The return loss of this structure varies between around -9.5dB (an okay match) and below -20dB (a very good match). The match of -9.5dB is the return loss for a 100ohm structure into a 50ohm structure. The impedance transformation making the 50-ohm source matched to the circuit occurs because the reflection off of the load (100-ohms) cancels with the reflection off of the transmission line (70.7-ohms). The step in impedance is the same amplitude going from the 50-ohm source to the 70.7-ohm transmission line, as it is going from the 70.7-ohm transmission line to the 100-ohm load. But, in the time it takes that second bounce to travel 90-degrees (a quarter wavelength), the incoming wave has traveled 90-degrees, and therefore the two waves are 180-degrees out of phase and cancel. Waves sum when in phase (0-degrees relative to each other) , and waves cancel when out of phase (180-degrees relative to each other).

Figure 6.1: Return Loss over Frequency

The second structure we
will examine has multiple transmission lines, but the middle line is a quarter wavelength long at 1GHz. The first transmission line is 100 ohms, the second is 70.7 ohms,
and the third is 50ohms. The input impedance is 100 ohms and the
output impedance is 50 ohms.

Where,

Where,

Z0 = 100ohms, Length0 = 2 lambda

Z1 = 70.7 ohms, Length1 = 0.25 lambda

Z2 = 50ohms, Length2 = 2 lambda

Z_load = 50

Z1 = 70.7 ohms, Length1 = 0.25 lambda

Z2 = 50ohms, Length2 = 2 lambda

Z_load = 50

Video of Voltage Waves showing a Quarter-Wave Transformer

Here, we can see that the quarter-wave transformer provides a perfect match at the center frequency of 1GHz.