Excitation of a Resonator by a Coupling Loop

N. A. Lashuk, V. I. Tolmachev

(Presented by the seminar of the High-Energy Laboratory of the Scientific Research Institute of Nuclear Physics, Electronics and Automation)

In order to excite a cavity resonator, i.e., to compensate the losses present in it, it is necessary to couple it to a source of high-frequency oscillations. The coupling of the source to the resonator may be inductive, conductive, capacitive, or aperture coupling. The last method is used in the microwave range when exciting a resonator by a waveguide. From the energy point of view, the first three methods are equivalent, and the use of each of them is determined by the type of oscillations that must be obtained in the resonator, by the convenience of supplying energy, and by the placement of the coupling elements.

In the case of energy transmission by a coaxial cable (feeder), it is expedient to use the inductive method of excitation, since it provides simple fastening of the cable and access to the coupling elements. The equivalent circuit for excitation of a resonator, in which the distributed inductance and capacitance may be regarded as lumped, is shown in Fig. 1. The circuit consisting of a high-frequency generator, feeder, coupling elements, and resonator is a system of two coupled circuits, the exciting and the excited.

In the circuit of Fig. 1, b,

\[ R_1 = W_{\mathrm{f}} + r_1 \quad \text{and} \quad R_2 = r_2 + r_{\mathrm{p}}, \]

where \(W_{\mathrm{f}}\) and \(r_1\) are the characteristic impedance of the feeder and the loss resistance of the loop;

\(r_2\) and \(r_{\mathrm{p}}\) are the loss resistance of the resonator and the resistance due to beam loading;

\(L_1, C_1, L_2, C_2\) are the equivalent inductances and capacitances of the excitation circuit and of the resonator;

\(e(t)\) is the voltage developed by the generator at the input of the feeder. Writing the differential equations for the system of circuits and considering the input resistance on the resonator side for the currents \(I_1\) and \(I_2\), we obtain the following expressions:

\[ \dot{i}_1 = \frac{j\omega M I_2 - e(t)}{Z_1} = -\frac{e(t)}{Z_1} \left( 1+\frac{\omega^2 M^2}{Z_1 Z_{23}} \right), \tag{1} \]

\[ \dot I_2= \frac{j\omega M e(t)} {Z_1\left[\left(R_2+\frac{\omega^2M^2}{Z_1^2}R_1\right)-j\left(X_2-\frac{\omega^2M^2}{Z_1^2}X_1\right)\right]} =\frac{j\omega M e(t)}{Z_1Z_{2э}}, \tag{2} \]

where \(Z_1=\sqrt{R_1^2+X_1^2}\), \(X_1=\omega L_1-\dfrac{1}{\omega C_1}\), \(X_2=\omega L_2-\dfrac{1}{\omega C_2}\),

\(M\) is the magnitude of the coupling between the circuits.

In solving the problem of resonator excitation, it is of interest to obtain the maximum voltage across the accelerating gap or, equivalently, the maximum circuit current. This condition is satisfied for a definite coupling between the circuits and definite equivalent and total resistances of the excitation circuit and the resonator.

Fig. 1. Equivalent circuit for excitation of a resonator heavily loaded by capacitance.

Analyzing expressions (1) and (2), it can be shown that at full resonance, i.e., when the frequencies of the generator, the resonator, and the excitation circuit are equal, the currents are determined by the expressions:

\[ \dot I_{1\max}= \frac{e(t)}{R_1+\dfrac{\omega^2M^2}{R_2}} =\frac{e(t)}{R_{1э}}, \tag{3} \]

\[ \dot I_{2\max}= \frac{j\omega M e(t)}{R_{1э}R_2} =\dot I_{1\max}\frac{j\omega M}{R_2}. \tag{4} \]

In the case of partial resonance, when the frequency of the excitation circuit is constant, while the resonator frequency is tuned at \(X_2=X_1\dfrac{\omega^2M^2}{Z_1^2}\), the expressions for the currents take the form

\[ \dot I_{1\max}=\frac{e(t)Z_2}{R_{2э}Z_1}, \tag{5} \]

\[ \dot I_{2\max}=\frac{j\omega M e(t)}{R_{2э}Z_1} =\dot I_{1\max}\frac{j\omega M}{Z_2}. \tag{6} \]

Analogous expressions can also be obtained for another partial resonance, when the resonator frequency is constant and the frequency of the excitation circuit is adjusted.

The currents in equations (5) and (6) are not limiting; they depend on the coupling between the circuits. Taking the derivative \(dI_2/d\omega M=0\) and solving with respect to the coupling, we find, for complete and partial resonances,

\[ M_{1\operatorname{opt}}=\frac{\sqrt{R_1R_2}}{\omega} \quad \text{and} \quad M_{2\operatorname{opt}}=\frac{Z_1}{\omega}\sqrt{\frac{R_2}{R_1}}. \tag{7} \]

Substituting \(M_{\operatorname{opt}}\) into the corresponding values of the currents \(I_{2\max}\), we obtain that, for each of the cases considered, the limiting currents are identical and equal to

\[ I_{2\max}=\frac{e(t)}{\sqrt{R_1R_2}}. \tag{8} \]

In the case of partial resonance, in order to obtain the limiting current it is necessary to increase the coupling in comparison with complete resonance. It can be shown that for small detunings this increase is insignificant and can easily be realized. For large detunings the increase in coupling becomes substantial and ultimately may serve as a limitation on obtaining the limiting current.

The dependence of the currents on the coupling is determined by the expression

\[ \frac{I_2}{I_{2\max}} = \frac{2\cdot\dfrac{M}{M_{\operatorname{opt}}}} {1+\dfrac{M^2}{M_{\operatorname{opt}}^2}} \cdot \frac{\sqrt{1+\dfrac{X^2}{R_1^2}}} {2+\dfrac{X^2}{R_1^2}}, \tag{9} \]

the theoretical curves of which for \(X=0\) and \(X=R\) are shown in Fig. 2. From comparison of the curves it follows that for \(X=R\) an increase of the coupling by a factor of \(\sqrt{2}\) is required in comparison with the optimum, and the maximum of the curve becomes less sharp. Naturally, the most advantageous case in exciting a resonator is that of complete resonance. However, the use of the capacitor \(C_1\) in the excitation circuit is undesirable for the following reasons. First, because of voltage resonance in the series circuit, a large rf voltage will develop across its elements, \(Q_1\) times greater than the applied voltage, where \(Q_1\) is the quality factor of the excitation circuit. Second, the use of \(C_1\) leads to a more complicated design and to reduced operating reliability. The use of a parallel circuit is also not free from drawbacks: complexity of construction, large circuit currents, etc. Therefore inductive systems for exciting resonators without a capacitor are usually used.

Let us consider this case. Setting in (2) \(Z_1=R_1+j\omega L_1\) and \(X_1=j\omega L_1\), the expression for the current in the resonator will be determined by

\[ I_2= \frac{j\omega M e(t)} {\left[ \left( R_2+\frac{\omega^2M^2}{R_1^2+\omega^2L_1^2}R_1 \right) +j\left( X_2-\frac{\omega^2M^2}{R_1^2+\omega^2L_1^2}\omega L_1 \right) \right](R_1+j\omega L_1)} . \tag{10} \]

The maximum value of the current will occur at

\[ X_2=-\frac{\omega^2 M^2}{R_1^2+\omega^2 L_1^2}\,\omega L_1 . \]

The reactive resistance introduced into the resonator will be capacitive; it can be compensated by detuning the resonator toward an increase in frequency relative to the excitation frequency. The magnitude of the optimum coupling of the excitation loop to the resonator will be

\[ M_{\mathrm{opt}}=\frac{Z_1}{\omega}\sqrt{\frac{R_2}{R_1}} =\frac{(R_1+j\omega L_1)}{\omega}\sqrt{\frac{R_2}{R_1}} . \tag{11} \]

With optimum coupling, the resistance introduced into the resonator is equal to the loss resistance, and the current reaches a maximum value equal to the current at complete resonance.

Fig. 2. Relative dependences of the current and the resonator quality factor on the magnitude of the coupling of the resonator with the source.

Thus, with optimum coupling and an arbitrary reactive resistance of the excitation circuit, the loaded quality factor of the resonator is half its intrinsic value. As the coupling is varied relative to the optimum, the introduced resistance changes, and consequently the loaded quality factor also changes. The relative change in the quality factor is determined by the expression

\[ \frac{Q_{\mathrm{n}}}{Q_0} = \frac{1}{1+\dfrac{R_{\mathrm{in}}}{R_2}} = \frac{1}{1+\dfrac{\omega^2 M^2}{Z_1^2}X_1}, \tag{12} \]

the dependence of which is shown in Fig. 2.

For very weak coupling, when \(\dfrac{R_{\mathrm{in}}}{R_2}\ll 1\), the loaded quality factor is almost equal to the intrinsic quality factor, and this condition must be satisfied in meas—

measuring the Q-factors of resonators. In practice it is sufficient that
$M \leq 0.2 M_{\mathrm{opt}}$.

When calculating high-frequency accelerating systems it is necessary to take into account the reduction of the Q-factor of the resonators not only due to coupling with the excitation feeder, but also due to loading by the beam. The load leads to a decrease in the shunt resistance and to a shift of the resonator frequency, depending on the magnitude of the accelerated current, the voltage across the resonator gap, and the phase shift between these quantities. This will require an additional increase in the coupling between the generator and the resonator and a corresponding increase in the supplied rf power.

Fig. 3. Experimental curves of the voltage and Q-factor of the resonator versus the angle of rotation of the excitation loop.

The required coupling corresponds to a definite loop area, depending on the type of resonator and its parameters. For toroidal or cylindrical resonators, heavily loaded by capacitance, this coupling is determined by the expression

\[ S_{\mathrm{p}} = \frac{\pi(\Phi_1+\Phi_2)}{\mu_0 \omega Q_0} \sqrt{ W_{\phi} R_{\mathrm{sh}} \left(1+\frac{\omega^2 L_1^2}{W_{\phi}^2}\right) } \frac{1}{\sin \alpha}, \tag{13} \]

where $S_{\mathrm{p}}$ is the loop area in $\mathrm{cm}^2$;
$\mu_0 = 4\pi \cdot 10^{-9}\ \mathrm{H/m}$ is the relative magnetic permeability;

\(\omega_0\), \(Q_0\), and \(R_{\text{sh}}\) are the frequency, intrinsic quality factor, and shunt resistance of the resonator;

\(\Phi_1\) and \(\Phi_2\) are the dimensions of the resonator;

\(\alpha\) is the angle between the direction of the magnetic field and the plane of the loop. The coupling magnitude and the loop area are related by the relation

\[ M=\frac{\mu_0 S_{\text{l}}\sin\alpha}{\pi(\Phi_1+\Phi_2)} . \tag{14} \]

Fig. 4. Experimental dependences of the resonator voltage and quality factor on the length of the excitation feeder.

In order to verify the theoretical results obtained and to obtain the most advantageous coupling to the resonator, an experimental study was carried out of inductive excitation of the resonator without tuning the loop. The circuit parameters were: \(L_1=260\ \text{cm}\), \(C_1=74\ \text{pF}\), \(L_2=106\ \text{cm}\), \(C_2=180\ \text{pF}\), \(\rho=24.4\ \Omega\), \(\omega_0=230\ \text{s}^{-1}\); \(W_{\text{f}}=60\ \Omega\), \(S_{\text{l}}=170\ \text{cm}^2\), \(\Phi_1=130\ \text{cm}\), \(\Phi_2=34.5\ \text{cm}\).

During the experiment, the power and frequency of the generator were kept constant, and tuning of the system to resonance was accomplished by changing the resonator frequency. Smooth rotation of the loop pro-

was varied within the range from 0 to 90°. The voltage across the resonator gap was measured with a calibrated inductive probe, and the quality factor was measured from the oscillation build-up time. Figure 3 gives the experimental characteristics \(\dfrac{U}{U_{\max}}\), \(\dfrac{Q_{\text{n}}}{Q_0}\), \(U\), and \(Q\) as functions of the angle of rotation of the excitation loop. It follows from the plots that, in this particular case, optimum coupling occurs when the loop is rotated through an angle of 45°, at which the loaded quality factor is 3500 and is half the intrinsic value. The effective area of the loop is \(160\ \text{cm}^2 \cdot \sin 45^\circ = 120\ \text{cm}^2\), while the calculated value for \(Q_0 = 7000\) is \(112\ \text{cm}^2\). In the case in which the inductance of the loop is compensated by its capacitance, its area is reduced to \(80\ \text{cm}^2\). From expressions (13) and (14) it follows that the coupling magnitude depends only on the parameters of the resonator and on the area of the loop, but does not depend on its inductance. Consequently, the loop must be designed with the condition that its inductance be made minimal.

For \(\dfrac{\omega^2 L_1^2}{W_{\phi}} \ll 1\), the input resistance of the loop is practically active, and its area is minimal. From these considerations it is desirable to choose a transmission line with a small characteristic impedance, and a resonator with a high quality factor.

An important role in selecting the coupling is played by the length of the excitation feeder, i.e., by its input resistance; as this changes, the optimum coupling for obtaining the maximum voltage on the resonator must also change. This can be shown by analyzing expression (2), where \(Z_1\) is taken with allowance for the feeder length. This proposition is confirmed by the experimental data shown in Fig. 4, which gives the dependences of the voltage on the resonator and of its quality factor as functions of the feeder length for a specified coupling.

The maximum voltage and quality factor at \(W_{\phi} = \omega L_1\) occur at a feeder length equal to \((2n + 1)\dfrac{\lambda}{4}\). When the inductance of the loop is compensated by capacitance, the coupling magnitude and the quality factor of the resonator do not depend on the length of the excitation feeder.

Thus, in calculating radio-frequency accelerating systems, it is necessary to take the loaded quality factor of the resonator to be 2 times smaller than the intrinsic value obtained by calculation or measured with very weak coupling to the source. When the resonator is loaded by the beam, the shunt resistance of the loaded resonator decreases and, consequently, the coupling must be increased relative to the optimum value. The theoretical results obtained are in good agreement with the experimental data obtained in studies of resonators in the meter wavelength range of the “Sirius” synchrotron.

REFERENCES

1. N. N. Krylov. Theoretical Foundations of Radio Engineering. “Morskoi Transport” Publishing House, L., 1961.

2. V. A. Vizir, N. A. Lashuk, B. A. Solntsev. Izvestiya Vuzov, “Fizika,” No. 6, 1971.