**The matching impedance process for RF circuits is very important. Impedance matching is also called the tuning process.** Physically it represents the circuit between the load and transmission line that can perform the following functions: it maximises the delivered power when the load is matched with the transmission line and minimises power loss; it improves the signal-to-noise ratio of the system; it also minimises the amplitude and phase errors. When considering the matching process it is important to keep in mind the following factors: complexity, bandwidth, adjustability and implementation aspects.

The simplest way of approaching the matching process is by using lumped components. Depending on the position of the normalised impedance $\frac{{Z}_{L}}{{Z}_{0}}$ on the Smith Chart, it can be used in one of two matching circuits: if the normalised impedance is inside the $1+jx$ circle on the Smith Chart (Figure 1), then we must use the matching scheme depicted in Figure 2a, and if the normalised impedance is outside the $1+jx$ circle of the Smith Chart, then the scheme in Figure 2b must be used.

Figure 1. The Smith Chart (used in impedance matching).

a

b

Figure 2. L-section impedance matching circuits.

Let’s consider the load resistance as ${Z}_{L}={R}_{L}+j{Z}_{L}$.

If the normalised impedance is out of the $1+jx$ circle, then ${R}_{L}<{Z}_{0}$, then $X=\pm \sqrt{{R}_{L}({Z}_{0}\u2013{R}_{L})}\u2013{X}_{L}$ and $B=\pm \frac{\sqrt{({Z}_{0}\u2013{R}_{L})/{R}_{L}}}{{Z}_{0}}$ .

If the normalised impedance is in the $1+jx$ circle , then ${R}_{L}>{Z}_{0}$, then $X=\frac{1}{B}+\frac{{X}_{L}{Z}_{0}}{{R}_{L}}\u2013\frac{{Z}_{0}}{B{R}_{L}}$, and $B=\frac{{X}_{L}\pm \sqrt{{\displaystyle \frac{{R}_{L}}{{Z}_{0}}}}\sqrt{{{R}_{L}}^{2}+{{X}_{L}}^{2}\u2013{Z}_{0}{R}_{L}}}{{{R}_{L}}^{2}+{{X}_{L}}^{2}}$.

In practice it is always easier to calculate the matching impedance using the Smith Chart.[1]

[1] “Microwave engineering”, 4th edition, David M. Pozar.