03.Impedance matching

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),

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:

50IMPEDANCE MATCHING

(c)Design a three-element matching circuit with a bandwidth of f D 50 kHz, and check the results with SPICE to determine the actual bandwidth.

(d)Design a double tuned transformer matching circuit with a bandwidth off D 50 kHz, and check the results with SPICE to determine the actual bandwidth.

(e)Repeat part (d) for a 3 dB bandwidth of 2 MHz. Again, check the results using SPICE.

3.2The matching circuit shown in Fig. 3.4 is used to match the load R D 1000 " to R00 D 80 ". If the intermediate resistance level is R0 D 20 ", determine the following:

(a)What is Q1?

(b)What is Q2?

(c)What is B1, the first susceptance nearest R?

(d)What is the estimated 3 dB bandwidth for this circuit in terms of the center frequency, f0?

3.3The tapped capacitor transformer is to be used in a narrowband of frequencies around ω D 4 Ð 109 rad/s. In designing the matching circuit, the tapped C circuit is converted to an ‘L’ matching circuit. If R2 in Fig. 3.10 is 50 ", C2 D 8 pF, and C1 D 5.0 pF, then what is the total capacitance for the ‘L’ matching circuit?

REFERENCES

1.P. L. D. Abrie, The Design of Impedance-Matching Networks for Radio-Frequency and Microwave Amplifiers, Norwood, MA: Artech House, 1985.