Matching Network Yin-Yang - Part 2

The conclusion of this article covers transmission line matching networks, plus a discussion of how characteristics of the load affects matching bandwidth and the choice of network topologies 

 A well-known distributed matching network is the quarter-wavelength long transmission line transformer. I will refer to this network as a type 11. The characteristic impedance of this line is given by (41) For example, a 100 ohm load is matched to a 50 ohm source using a 90° line with characteristic impedance 70.71 ohms. The matchable space of the quarter-wavelength transformer is small, essentially only the real axis on the Smith chart. Nevertheless, it enjoys widespread use. A quarter-wavelength line is also used in filter design as an impedance inverter to convert series resonant circuits to parallel resonance, and vice versa [4]. 

The General Transmission Line Transformer 

Perhaps less well-known is that a single series transmission line can match impedances not on the axis of reals. The matchable space of this type 12 network is plotted in Figure 9.  

The Shortened Quarter-Wavelength Transformer 

Another less well-known but useful adaptation of the quarter-wavelength transformer is the shortened, doublesection transformer depicted in Figure 10. I will refer to this as type 13. Like the standard transformer, it is used to match real impedances. But the required length is shorter and it uses lines with characteristic impedance equal to the impedances being matched. These are both practical features in many applications. Notice that the transmission line with characteristic impedance equal to the load is adjacent to the source. Both transmission lines have the same length. The maximum line length is 30°, and it decreases as the load impedance is much higher or lower than the source impedance.  

The Challenge 

Since all complex loads are matchable by two element networks and sometimes a single transmission line, why is matching sometimes difficult? For loads with a large reflection coefficient, the element values may be difficult to realize. This is particularly true for distributed circuits. But more often the problem is bandwidth. A simple circuit matches at a single frequency. Obtaining a good match over an extended frequency range may require many elements and finding values is very challenging. Before I cover this subject, let me introduce another fundamental concept.  

Q of the Load 

The term Q is used for several properties. Mastery of each is critical to understanding oscillators, filters, matching networks and other circuits [5]. One definition of loaded Q is the center frequency divided by the 3 dB bandwidth of a resonant circuit response. It is a finite value even if the circuit is built using components with infinite Q. Component Q, or unloaded Q, is a measure of component quality; the ratio of stored energy to dissipated energy in the component. It is as high as 200 for excellent inductors. But unloaded Q increases with physical size, so modern miniature inductors have much lower Q. Q of the load described in this section is yet a third definition of Q. I often feel engineers would be less confused if these properties were labeled Q, R and S. However, their definitions have similar roots.  

Element Impedance Transforms Plotted on the Smith Chart The solid red arc that begins at 300 ohms (6 normalized) on the real axis, right of center, is the resulting action at 100 MHz of a shunt 11.84 pF capacitor that transforms the 300 ohm load resistance to 50–j110 ohms. The arc from this point to 50+j0 is the result of a series inductor of 178.4 nH, or +j110 ohms. These arcs are not responses plotted versus frequency but rather the length of these arcs correspond to increasing values of the capacitor and inductor. These concepts are the basis of network design using the Smith chart [1, 2]. The red dashed Q arcs were drawn so they intersect the maximum extent of the solid L-network arcs, so the one-section L-network has a Q of 2.3. You can see that the arcs of the 3-section L-network remain closer to the real axis. The Q of the 3-section L-network is only 0.9. Plotted in brown and green in Figure 12 are the transmission and return loss responses of the one section L-network. Plotted in red and blue are the transmission and return loss responses of the three section L-network. The 15 dB return loss has a bandwidth of about 17% for the one-section L-network and about 59% for the 3-section L-network. The ratio of the bandwidths is 59/17 = 3.5 and the ratio of the Q values is 3/0.9 = 3.3. The exact relationship between the bandwidth and Q arcs depend on the return loss used to define the bandwidth. However, the relation is clear: matching networks with impedance arcs that remain closer to the real axis have better bandwidth. Using impedance arcs is insightful when designing both lumped and distributed matching networks. But this process is increasingly ineffective when attempting to match multiple frequencies over a wide bandwidth. The blue arcs in Figure 11 were drawn for a single frequency, 100 MHz. The solid blue return loss response in Figure 12 reveals a near perfect match has been achieved at 100 MHz as expected since the last blue arc in Figure 11 ends at 50 ohms. But further examination of the blue trace in Figure 12 reveals that the response is not centered on 100 MHz, the design frequency. The dashed magenta and brown plots in Figure 12 are the result of optimizing all 6 element values to center the response on 100 MHz. When broad bandwidth is required, drawing arcs on a Smith chart, either with pencil and paper charts or by computer program, is cumbersome at best. Synthesis routines such as those in the Impedance Matching module of GENESYS are effective. Alternatively, optimization of an initial Smith chart design via computer simulation is effective for problems with well-behaved loads. This will be discussed in more detail later.