SNAA434 March 2025 LMX2820

B Appendix: Calculations for Reactive Matching Network

Introduce the following terms:

Equation 29.

X_{L} = 2 π \times f \times j \times L

Equation 30.

X_{C} = - j \times \frac{1}{2 π \times f \times C}

Equation 31.

z = X_{L} + \frac{R_{S o u r c e}}{N}

The impedance as looking into and out of the load needs to be the same.

Equation 32.

R_{L o a d} = z | | X_{C} = \frac{z \times X_{C}}{z {+ X}_{C}}

This equation can be rearranged as:

Equation 33.

z = \frac{X_{C} \times R_{L o a d}}{X_{C} {- R}_{L o a d}}

The impedance as looking out from inductor needs to be equal to the impedance looking in

Equation 34.

2 \times \frac{R_{S o u r c e}}{N} = z + R_{L o a d} | | X_{C} = z + \frac{R_{L o a d} \times X_{C}}{R_{L o a d} {+ X}_{C}}

Combine these equations to get

Equation 35.

2 \times \frac{R_{S o u r c e}}{N} = \frac{R_{L o a d} \times X_{C}}{X_{C} - R_{L o a d}} + \frac{R_{L} \times X_{C}}{X_{C} {+ R}_{L o a d}} = \frac{2 \times R_{L o a d} \times {X_{C}}^{2}}{{X_{C}}^{2} - {R_{L o a d}}^{2}}

Defile the following term:

Equation 36.

Q = \sqrt{\frac{{N \times R}_{L o a d}}{R_{S}} - 1}

Being very careful to get the correct root and get the correct branch. For example, realize that these equations can be combined as

Equation 37. i.e.

\sqrt{\frac{1}{- 1}} = - j \neq j

Keeping this in mind, these equations can be combined as

Equation 38.

X_{C} = - j \times \frac{R_{L o a d}}{\sqrt{\frac{{N \times R}_{L o a d}}{R_{S}} - 1}} = - j \times \frac{R_{L o a d}}{Q}

Substituting this back into the values for x and y yield:

Equation 39.

\frac{(- j \times \frac{R_{L o a d}}{Q}) \times R_{L o a d}}{(- j \times \frac{R_{L o a d}}{Q}) {- R}_{L o a d}} = X_{L o a d} + \frac{R_{S o u r c e}}{N}

This can be simplified to say:

Equation 40.

X_{L} = \frac{(- j \times \frac{R_{L o a d}}{Q}) \times R_{L o a d}}{(- j \times \frac{R_{L o a d}}{Q}) {- R}_{L o a d}} - \frac{R_{S o u r c e}}{N} = \frac{j \times Q \times R_{L o a d} + R_{L o a d}}{Q^{2} + 1} - \frac{R_{S o u r c e}}{N} = \frac{j \times Q \times R_{L o a d} + R_{L o a d}}{\frac{{N \times R}_{L o a d}}{R_{S o u r c e}}} - \frac{R_{S o u r c e}}{N} = j \times \frac{Q \times R_{S o u r c e}}{N}

Finally, the values for the inductance and capacitance can be found:

Equation 41.

L = \frac{j \times \frac{Q \times R_{S o u r c e}}{N}}{2 π \times f \times j} = \frac{R_{S o u r c e} \times Q}{2 π \times f \times N}

Equation 42.

C = \frac{- j}{2 π \times f \times X_{C}} = \frac{Q}{R_{L o a d} \times 2 π \times f}