Davis W.A.Radio frequency circuit design.2001

X '2 = X2 – R 'Q1

R '

FIGURE 3.6 Equivalent series reactance after first transformation.

TABLE 3.2 p Matching Circuit Design Formulas

The X2, B3 combination is a series L section with “load” of R0. Consequently

A summary for the design process is shown below. To make R00 < R, make Q1 > Q2 and Q1 D Qmax, and follow the design steps in the first column of Table 3.2. For R00 > R, use the second column.

3.7T TRANSFORMATION CIRCUIT

The T transformation circuit is the dual to the transformation circuit and is shown in Fig. 3.7. In this circuit, however, the series reactance X1 first raises the resistance level to R0, and the remaining shunt susceptance lowers the resistance level as indicated in Fig. 3.8. The design formulas are derived in the same way as the circuit formulas and are summarized in Table 3.3.

FIGURE 3.7 T transformation circuit.

R '

R

R "

FIGURE 3.8 Diagram showing impedance transformation for the T circuit.

TABLE 3.3 T Matching Circuit Design Formulas

3.8TAPPED CAPACITOR TRANSFORMER

The tapped capacitor circuit is another approximate method for obtaining impedance level transformation. The description of this design process will begin with a parallel RC to series RC conversion. Then the tapped C circuit will be converted to an L-shaped matching circuit. The Q1 for an equivalent load resistance Reqv will be found. Finally, a summary of the circuit synthesis procedure will be given.

3.8.1Parallel to Series Conversion

Shown in Fig. 3.9 is a parallel RC circuit that will be forced to have the same impedance as the series RC circuit, at least at one frequency. The conversion is of

44 IMPEDANCE MATCHING

C 1

FIGURE 3.10 Tapped C transformation circuit.

FIGURE 3.11 Intermediate equivalent transformation circuit.

This is the transformed resistance looking through C1 toward the left. Looking toward the right through Cseqv and again using the parallel to series conversion, Eq. (3.36),

These two expressions for Rseqv can be equated and solved for Qp:

3.8.3Calculation of Circuit Q

An approximate value for Q can be found by equating the impedances of the two circuits in Fig. 3.12:

If the Q of the right-hand circuit is approximately that of the left-hand circuit in Fig. 3.12, then

FIGURE 3.12 Equate the leftand right-hand circuits.

TABLE 3.4 Tapped C Matching Circuit Design Formulas

The variable C represents the total capacitance of C1 and Cseqv in series, as implied in Fig. 3.11 and represented in Fig. 3.12. For a high Q circuit, circuit analysis gives the resonant frequency:

Here f is the bandwidth in Hz and f0 is the resonant frequency.

3.8.4Tapped C Design Procedure

The above ideas are summarized in Table 3.4, which provides a design procedure for the tapped C matching circuit. Similar expressions could be found for a tapped inductor transforming circuit, but such a circuit is typically less useful because inductors are more difficult to obtain than capacitors.

3.9PARALLEL DOUBLE-TUNED TRANSFORMER

Each of the above described T, , or tapped C matching circuits provide some control over the bandwidth. Where precise control over the bandwidth is required, a double tuned circuit allows controlling bandwidth by specifying two different frequencies where maximum transmission occurs. For a small pass band, the

46 IMPEDANCE MATCHING

G T

f m1 f m2

FIGURE 3.13 Double tuned transformer response.

M

FIGURE 3.14 Real transformer with resonating capacitances.

midband dip in the transmission can be made small. Furthermore the double-tuned circuit is especially useful when a large difference in impedance levels is desired although its high end frequency range is limited. The filter transmission gain is shown in Fig. 3.13.

The double-tuned circuit consists of a coupled coil transformer with resonating capacitances on the primary and secondary side. This circuit is shown in Fig. 3.14. The transformer is described by its input and output inductance as well as the coupling coefficient k. The turns ratio for the transformer is

The circuit in Fig. 3.14 can be replaced by an equivalent circuit using an ideal transformer (Fig. 3.15a). Since an ideal transformer has no inductance, the inductances and coupling factor, k, must be added to the ideal transformer. The final circuit topology is shown in Fig. 3.15b. Looking toward the right through the ideal transformer, Fig. 3.15b shows the circuit values are:

(a )

L '2

(b )

FIGURE 3.15 (a) Alternate equivalent circuit with an ideal transformer, and (b) final equivalent circuit.

FIGURE 3.16 Double tuned circuit split into two.

The circuit elements will be chosen to give exact match at the two frequencies, fm1 and fm2. The circuit in Fig. 3.15b can be conceptually split into two (Fig. 3.16). The resistance R1 with the parallel resonant circuit will never be larger than RG. The right-hand side is an L matching circuit with the reactance of the shunt element monotonically decreasing with frequency. Hence R2 monotonically decreases. Consequently, if RL is small enough, there will be two frequencies where R1 D R2. This is illustrated in Fig. 3.17.

A design procedure for the parallel double-tuned circuit has been reviewed in [1] and is summarized below. The typical synthesis problem is to design a circuit that will match RG and RL over a bandwidth, f, at a center frequency, f0, with a given pass band ripple. The bandwidth and center frequency are approximated by the following:

1. Determine fm1 and fm2: