Matching Network Tuning and Control Methods

Abstract

It is a trend that tunable matching networks are expected to play an important role in the realization of adaptive and reconfigurable radio front-end architectures. One particular example of using tunable impedance matching networks is the compensation of mobile phone antenna impedance mismatch loss caused by phone user proximity effects.

Appendices

Appendix 1: MWO Script of Single Frequency Tuning Algorithm

Appendix 2: Script of Duplex Pair Frequency Tuning Approach

Appendix 3: Script of Frequency Band Tuning Approach

Appendix 4: Formula Derivations of Sect. 6.2

6.4.1 Derivation of Formulas (6.14)–(6.17)

The derivation of equations of (6.14)–(6.17) needs to be split into two cases, i.e., R _L  < R _o and R _L  > R _o , then the derivation can be done by means of two different configurations of the equivalent circuit topologies. For (6.14) and (6.15), our derivation is based on the equivalent circuit configuration shown in Fig. 6.37. C ₂ in the pi-network tuner has been absorbed by X _e and R _e as given in (6.11a and 6.11b).

Fig. 6.37

Equivalent circuit for deriving (6.14) and (6.15)

The impedance on the left side of the dashed line is given by,

$$ {Z}_{left}=\frac{1}{G_o+j{B}_{C1}}=\frac{G_0}{G_o^2+{B}_{C1}^2}-j\frac{B_{C1}}{G_o^2+{B}_{C1}^2} $$

(A4.1)

In the conjugation match condition, we should have (A4.2) and (A4.3) satisfied.

$$ \frac{G_0}{G_o^2+{B}_{C1}^2}={R}_e $$

(A4.2)

$$ \frac{B_{C1}}{G_o^2+{B}_{C1}^2}={X}_e+2\pi f\cdot {L}_e $$

(A4.3)

where B _C1 = 2πf ⋅ C ₁. From (6.59), we obtain (6.14) and from (6.60) we derive (6.10) as,

$$ \begin{array}{c}{C}_1=\frac{1}{2\pi f}{B}_{C1}=\frac{1}{2\pi f}\sqrt{\frac{G_o}{R_e}-{G}_o^2}=\\ {}=\frac{1}{2\pi f}\sqrt{G_o\left({G}_e-{G}_o\right)}=\sqrt{\frac{1}{R_o}\left(\frac{1}{R_e}-\frac{1}{R_o}\right)}\end{array} $$

(6.14)

$$ \begin{array}{c}{L}_e=\frac{1}{2\pi f}\left(\frac{B_{C1}}{G_o^2+{B}_{C1}^2}-{X}_e\right)=\frac{1}{2\pi f}\left(\frac{R_e}{G_o}\sqrt{\frac{G_o}{R_e}-{G}_o^2}-{X}_e\right)=\\ {}=\frac{1}{2\pi f}\left(\sqrt{R_e\left({R}_o-{R}_e\right)}-{X}_e\right)\end{array} $$

(6.15)

In the derivation of (6.16) and (6.17), the equivalent circuit as depicted in Fig. 6.38 is used. For R _L  > R _o , C ₁ is set to minimum C _min. Z _left and Z _right in Fig. 6.38 can be expressed, respectively, as

Fig. 6.38

Equivalent circuit for deriving (6.16) and (6.17)

$$ {Z}_{left}=\frac{1}{G_o+j{B}_{C1 \min }}=\frac{R_o}{1+{R}_o^2{B}_{C1 \min}^2}-j\frac{B_{C1 \min }{R}_o^2}{1+{R}_o^2{B}_{C1 \min}^2} $$

(A4.4)

and

$$ \begin{array}{l}{Z}_{right}=\frac{1}{G_L+j\left({B}_{C2}+{B}_L\right)}\\ {}=\frac{R_L}{1+{R}_L^2{\left({B}_{C2}+{B}_L\right)}^2}-j\frac{R_L^2\left({B}_{C2}+{B}_L\right)}{1+{R}_L^2{\left({B}_{C2}+{B}_L\right)}^2}\end{array} $$

(A4.5)

where B _C1 min = 2πf ⋅ C _1,min and B _C2 = 2πf ⋅ C ₂

In conjugation match condition, we should have the real part of (A4.4) equal to

$$ \frac{R_o}{1+{R}_o^2{B}_{C1 \min}^2}=\frac{R_L}{1+{R}_L^2{\left({B}_{C2}+{B}_L\right)}^2} $$

(A4.6)

and the imaginary part of (A4.4) without the negative sign equal to the following expression

$$ \frac{B_{C1 \min }{R}_o^2}{1+{R}_o^2{B}_{C1 \min}^2}=2\pi f\cdot {L}_e-\frac{R_L^2\left({B}_{C2}+{B}_L\right)}{1+{R}_L^2{\left({B}_{C2}+{B}_L\right)}^2} $$

(A4.7)

From (A4.6), after manipulating we derive (6.16)

$$ {C}_2=\frac{1}{2\pi f}\left(\sqrt{\frac{G_L}{R_o}\left(1+{R}_o^2{B}_{C1, \min}^2\right)-{G}_L^2}-{B}_L\right) $$

(6.16)

and from (A4.7) and (6.16) we obtain (6.17)

$$ {L}_e=\frac{1}{2\pi f}\left(\frac{\sqrt{\frac{G_L}{R_o}\left(1+{R}_o^2{B}_{C1, \min}^2\right)-{G}_L^2}}{\frac{G_L}{R_o}\left(1+{R}_o^2{B}_{C1, \min}^2\right)}+\frac{R_o^2{B}_{C1, \min }}{1+{R}_o^2{B}_{C1, \min}^2}\right) $$

(6.17)

6.4.2 Derivation of Input Reflection Coefficient

The input reflection coefficient Γ_in is defined as

$$ {\Gamma}_{in}=\frac{Z_{in}-{R}_o}{Z_{in}+{R}_o} $$

(A4.8)

where Z _in is the input impedance as shown in Fig. 6.39, and it can be expressed as

Fig. 6.39

Equivalent circuit for deriving Γin

$$ {Z}_{in}=\frac{1}{j{B}_{C1}+1/{Z}_x}=\frac{1}{j{B}_{C1}+{Y}_x} $$

(A4.9)

where B _C1 = 2πf ⋅ C ₁ and Y_x is expressed as

$$ {Y}_x=\frac{1}{Z_x}=\frac{G_L+j\left({B}_{C2}+{B}_L\right)}{1-{X}_{Le}\left({B}_{C2}+{B}_L\right)+j{G}_L{X}_{Le}} $$

(A4.10)

where B _C2 = 2πf ⋅ C ₂ and X _Le  = 2πf ⋅ L _e . Substituting (A4.10) to (A4.9), we obtain Z_in to be

$$ {Z}_{in}=\frac{1-{X}_{Le}\left({B}_{C2}+{B}_L\right)+j{G}_L{X}_{Le}}{G_L\left(1-X{}_{Le}B_{C1}\right)+j\left[{B}_{C1}+\left({B}_{C2}+{B}_L\right)\left(1-{X}_{Le}{B}_{C1}\right)\right]} $$

(A4.11)

Finally, plugging (A4.11) into (A4.8), we derive the input reflection coefficient Γin (6.18) to be expressed as,

$$ {\Gamma}_{in}=\frac{G_o\left[1-{X}_{Le}\left({B}_L+{B}_{C2x}\right)\right]-{G}_L\left(1-{X}_{Le}{B}_{C1x}\right)+j\left[{G}_o{G}_L{X}_{Le}-{B}_{C1x}-\left({B}_{C2x}+{B}_L\right)\left(1-{X}_{Le}{B}_{C1x}\right)\right]}{G_o\left[1-{X}_{Le}\left({B}_L+{B}_{C2x}\right)\right]+{G}_L\left(1-{X}_{Le}{B}_{C1x}\right)+j\left[{G}_o{G}_L{X}_{Le}+{B}_{C1x}+\left({B}_{C2x}+{B}_L\right)\left(1-{X}_{Le}{B}_{C1x}\right)\right]} $$

(6.18)

Appendix 5: Matlab Code for Tuning Algorithm