Synthesized Inductor Delivers Maximum Power Transfer

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To transfer maximum power from a circuit to a load, the load’s resistance, R_L, must match the source resistance, R_o (Fig. 1). In the case of a complex load, Z_L, and a resistive source, this transfer can require the use of a bulky inductor. This idea shows how to replace the physical inductor with a smaller synthesized inductor.

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1. The simplest matching circuit consists of a resistive load, R_L, equal to the source resistance, R_o.

A good example of a complex load is a piezoelectric transducer represented by the Butterworth-Van Dyke model as C_o, R_L, C_s, and L (Fig. 2a). At series resonance, the load impedance reduces to the equivalent circuit shown in Figure 2b. The load is still complex, with R_L in parallel with C_o. This makes matching to R_o impossible without a complex conjugate match of –C_o at the source—that is, an inductor in parallel with C_o and having the same magnitude of reactance (Fig. 2c).

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2. An example of a complex load is a Butterworth-Van Dyke model of a piezoelectric transducer (a), which can be reduced to an equivalent circuit (b). To match the load to the source, an appropriate inductor, L, must be added (c).

To match R_o to R_L, L must be chosen to “resonate out” C_o:

–XC_o = X_L

– {–j[1/(ωC_o)]} = jωL

Therefore:

L = 1/(ω² × C_o)

where:

ω = 2πf

To replace the physical inductor, L, with a synthesized inductor, consider the gyrator circuit, which Bernard Tellegen invented in 1948 (Fig. 3). The circuit is basically a differentiator in which the operation of capacitor C is reversed. The simulated inductance becomes:

L = (R1)(R)(C) Henries

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3. Instead of a bulky physical inductor, the designer can use this gyrator circuit, which provides a simulated inductance equal to (R1)(R)(C).

Typically, R1 is kept to 100 Ω or less so the circuit’s Q can be at least 10. (Details of the gyrator circuit are available on the Internet. For example, see www.beigebag.com/case_gyrator.htm. Figure 4 shows the final implementation of the matched circuit. With this circuit, R can be dynamically varied to change the tuning on the fly for different complex loads, provided that the voltage across the load is measured so optimum tuning can be achieved.

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4. In a typical implementation of the gyrator matching circuit, dynamically varying R can change the tuning of the circuit for different complex loads, provided that the voltage across the load is measured to ensure optimum tuning.

Figure 5 shows the simulation results of a tuned circuit using the described gyrator circuit. The top trace is the voltage across the load, which shows a minima at the resonant frequency. The bottom trace shows the phase of the voltage across the load, which is 0° at resonance. These results indicate that the selected synthesized inductance successfully matches the load to the source.

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5. In these simulation results, the voltage across the load (top trace) shows a minima at the resonant frequency. The phase of the voltage across the load (bottom trace) is 0° at resonance. These results indicate the selected inductance created a successful match.