Question 8.4: Discuss the design of the resonator exploited in the tuned l......

In class A power amplifiers, a tuned load is used where a series or (more commonly) a parallel resonator blocks higher-order harmonics from dissipating power in the load resistance R_L. The same occurs in class A to C amplifiers (see Sec. 8.6) while in class F and E amplifiers (see Sec. 8.7.1 and 8.7.2, respectively) resonators are used to suitably shape the output waveform. While distributed solutions based on transmission lines can be in principle adopted, a lumped element resonator is the most compact solution in integrated circuits.
The series (parallel) resonators considered in the discussion of power amplifiers are ideal, i.e., they behave as short (open) circuits at the resonant frequency f_0, and as open (short) circuits for f \neq f_0. Some criteria are provided here for the practical design of such elements. For the sake of simplicity, we initially neglect the effect of losses.
Resonance at \omega_0=2 \pi f_0 can be achieved by any inductance and capacitance value satisfying the condition:

\omega_0^2 L C=1 .

Let us focus first on series resonator; a suitable choice of L_S and C_S can be derived by considering the resonator impedance far from the resonance, where the resonator should ideally behave as an open circuit; in practice, we require that the series resonator impedance Z_S(\omega) be suitably larger (by a factor K > 1 ) than the load resistance R_L at the fundamental harmonic n_h \omega_0 :

\left|Z_S\left(n_h \omega_0\right)\right|=\frac{n_h^2-1}{n_h \omega_0 C_S}>K \cdot R_L .\hspace{30 pt} \text{(8.4)}

Typically, K = 10 can be sufficient. It follows:

L_S>\frac{n_h}{\left(n_h^2-1\right) \omega_0} K \cdot R_L, \quad C_S=\frac{1}{\omega_0^2 L_S} .

In most cases, condition (8.4) has to be imposed at the second harmonic \left(n_h=2\right). A dual behavior holds for parallel resonators, where a condition on the parallel resonator admittance Y_P being suitably larger than the load admittance Y_L=1 / R_L yields:

C_P>\frac{n_h}{\left(n_h^2-1\right) \omega_0} \frac{K}{R_L}, \quad L_P=\frac{1}{\omega_0^2 C_P} .

Resonator parasitic losses, neglected so far, imply that at resonance the series resonator resistance be R_S>0 and the parallel resonator conductance be G_P>0. While no impact follows on the out of resonance behavior discussed above, the design should take into account for the constraints R_S \ll R_L, G_P \ll 1 / R_L.