Introduction

Resonant cavities have many applications in accelerator fields, such as pulse compressors^[1,2] standing wave deflectors^[3,4], and standing wave linacs^[5,6]. In these structures, the resonant cavities are all one port devices. The unloaded quality factor (unloaded Q factor), coupling coefficient, and resonant frequency are the main parameters. In particular, the unloaded Q factor can be used to predict the performance of the structures. In linacs and deflectors, the unloaded Q factor and input power can be used to estimate the field in the structures. In pulse compressors, the Q factor and coupling coefficient determine the peak power of the output pulse and efficiency of the system^[1].

Tsinghua University has conducted research on radio frequency (RF) pulse compressors, wherein high Q factor resonant cavities are key parts in storing the energy of an input RF pulse. Precise measurement of the Q factor of resonant cavities is necessary before powering them with a high-powered microwave. Many methods, such as energy decay in the cavity, the Smith chart, and reflection coefficient, can be utilized to measure the unloaded Q factor of a resonant cavity with one port^[7-14]. In the decay method, the power leaving a resonant cavity decreases exponentially and is measured to determine the Q factor^[14]. In the Smith chart and reflection coefficient methods, the unloaded quality factor can be obtained by measuring the half-power points on the Smith chart or the reflection and coupling coefficients of the resonant cavity^[15,16]. For devices with two ports, the Q factor can be derived from the transmission coefficient data^[17-20].

The Smith chart and reflection coefficient of a resonant cavity can be measured using a vector network analyzer (VNA) that is used to characterize passive and active devices by measuring their network parameters, called scatter parameters (S-parameters), as functions of frequency. Using VNA enables the Smith chart reflection coefficient methods to be the two common methods for measuring the Q factor of a one-port resonant cavity.

Many factors influence the measurements of the Smith chart and reflection coefficient through VNA. One of the factors is VNA calibration that reduces systematic errors and corrects the imperfections of cables and connectors^[21]. Other factors include the ambient temperature and mechanical vibration from the measurement circumstances. The noises from these factors influence the half-power points in the Smith chart and reflection coefficient curves and may lead to significant errors when measuring the unloaded Q factor.

We demonstrated a new method to measure the unloaded Q factor of resonant cavities for pulse compressors. The method is based on the reflection-type measurement by VNA. The data required to calculate the Q factor are near the resonant frequency of the reflection coefficient curve.

Equivalent circuit model

An equivalent model of a parallel resonant circuit driven by a current consists of a RF-resonant cavity coupled with the waveguide, as shown in Fig. 1. In this model, the resonant cavity consists of inductance (L), capacitance (C), and circuit shunt resistance (R). This resonant cavity couples with the waveguide through a coupler, which is the transformer in the equivalent model.

Fig. 1.Full-screen View

Equivalent circuit of a resonant cavity coupled with a waveguide.

From the circuit model, the impedance of the resonant cavity is^[15]:

In Eq. (1), Z₀ is the characteristic impedance of the input transfer line and Q₀ is the unloaded Q factor given by:

where f₀ is the resonance frequency. The coupling coefficient is defined as:

where Q_eis the external Q factor.

The input reflection coefficient, which can be directly measured by VNA, is:

We can rewrite the reflection coefficient from the previous relations as follows:

where $\xi =\omega /{\omega }_0-{\omega }_0/\omega$. The reflection curve and Smith chart with different coupling coefficients and the same unloaded Q factors are shown in Fig. 2.

Fig. 2.Full-screen View

Reflection coefficient at the input port with Q₀=6000 for three different coupling coefficients: (a) Reflection coefficient curve and (b) Smith chart.

Equation (5) can be rewritten as:

We define:

where a and b are dimensionless. Then, Eq. (6) becomes:

In Eq. (6) the values of $\xi$ and $\Gamma$ are obtained by measuring the resonant cavity, so x and y in Eq. (7) are obtained from measurement. Equation (11) represents a line and the quantities a and b are the slope and intercept of the line, respectively. After obtaining a and b, Q₀ and $\beta$ can be calculated as follows:

Error analysis of the measurement

Two kinds of errors may occur when performing the measurements. One is the systematic error (${\delta }_{\text{s}}$) that exists even after instrument calibration. The other is the noise from the measurement circumstances. The quantities directly obtained from the measurements were the frequency and reflection coefficient. Figure 3 shows the measurement scheme. The directional couplers 1 (DC1) and 2 (DC2) were used to sample the incident and reflected signals, respectively. Then, the sampled signals were downconverted to signals at low frequencies. After the analog-to-digital conversion, the digital signals were processed by the computer to obtain a₁ and b₁. The reflection coefficient of the resonant cavity was derived from the ratio of b₁ and a₁, as follows:

Fig. 3.Full-screen View

Scheme of the measurement of high Q cavity. The RF signals a₁ and b₁ were sampled by directional couplers and then downconverted to intermediate frequency (IF).

The reflection coefficient with noise error is:

The noise error of the reflection coefficient is:

Where ${\delta }_{a_1}$ and ${\delta }_{b_1}$ are the errors of a₁ and b₁, and are less than a₁.

Equation (16) can be rewritten as three terms as follows:

The third term is the product of two small values and is negligible. The first and second terms are the errors contributed by the errors of a₁ and b₁. Typically, the second term is related to the value of b₁. Moreover, a small b₁ implies a small error. At the resonant frequency, the amplitude of the reflection coefficient and b₁ were the minimum values. Therefore, the error of the reflection coefficient was minimal near the resonant frequency. After including the systematic error, the reflection coefficient with errors can be written as:

Figure 4 shows the influence of the systematic error. In Eq. (11), x and y represent a linear relation without errors. The systematic error destroys the linearity resulting in an x and y of the quadratic curve. The noise error was a random number with a small value. Figure 5 shows how the noise error affected the relationship of x and y. Without the noise error, x and y form a line or a quadratic curve. When the noise error was added, a simple curve cannot be used to represent x and y. The data with errors deviated from the curve without the noise error. A large noise error implied that the data deviated from the curve. Even the systematic and noise errors destroyed the linearity of x and y, and the linearity existed for small x values. From Eq. (8) and the reflection coefficient curve of a resonant cavity, the value of x near the resonant frequency was small. This condition implies that the data obtained near the resonant frequency contained minimal errors.

Fig. 4.Full-screen View

(Color online) Relationship of x and y without noise errors and with different systematic errors.

Fig. 5.Full-screen View

(Color online) Relationship of x and y with a fixed systematic error of 0.003 and different noise errors.

High-Q cavity and experimental setup

The geometry of the resonant cavity used in the SLED pulse compressor at Tsinghua University is a cylinder. The operation mode of the resonant cavity is TE₀₁₅ with a low loss feature. The ratio of diameter d, to length l, of the resonant cavity was selected to maximize the frequency separation from other modes. The cavity coupled with the waveguide through two coupling holes. Using two coupling holes can reduce the effect of the TM₁₁₅ mode in the resonant cavity. Changing the hole diameters can be used to tune the coupling coefficient of the cavity. The E and H fields of the resonant cavity are shown in Figs. 6 and 7. The electric and magnetic fields on the cylindrical surface were both minuscule compared with those inside the cylinder. From the Computer Simulation Technology (CST) simulation, the unloaded Q factor of the resonant cavity without coupling holes was 108,789. Moreover, the resonant frequency was 2857.05 MHz. After adding coupling holes and the waveguide to the resonant cavity, the frequency was only minimally changed to 2855.74 MHz. The unloaded Q factor was 105,273, slightly smaller than that without coupling holes. The coupling coefficient was then tuned to 6.72.

Fig. 6.Full-screen View

(Color online) E field of high-Q cavity

Fig. 7.Full-screen View

(Color online) H field of high-Q cavity.

The cavity was made of copper and the mechanical drawings are shown in the Fig. 8. The cylinder was between two copper plates with a waveguide attached to one of them, and an SLAC type flange.

Fig. 8.Full-screen View

(Color online) High Q cavity assembled with the waveguide. The resonant cavities had radii of 102.55 mm and heights of 335.7 mm.

Figure 9 shows the experiment setup. The measurement was performed using VNA, that generated, measured, and processed the RF signals at high speed. The cavity was connected to the VNA from a microwave coaxial cable and coaxial waveguide adapter, with frequency responses calibrated for precise measurement. The VNA was fully calibrated with a short open, and matched before the measurement of the reflection coefficient. In the measurement, the start and stop frequencies were set to 2853.7 and 2857.74 MHz, respectively. The number was 2001 within a span of 4 MHz. The frequency difference between two nearby points was 2 kHz. Therefore, the error of frequency measurement was less than 2 kHz, that was ignored, compared with the 2.8577 GHz frequency of the cavity.

Fig. 9.Full-screen View

(Color online) Experiment setup with the high Q resonant cavity and a VNA.

Results and discussion

The reflection coefficients of the resonant cavity from the simulation and measurement are shown in Fig. 10. The results were very similar.

Fig. 10.Full-screen View

(Color online) Results from CST and the experiment.

The experimental results of Fig. 10 are represented as a graph of x and y in Fig. 11. The distribution of the points in Fig. 11 are similar to that in Fig. 5. This finding indicates that the measurements contained both systematic and noise errors. The systemic and noise errors of the reflection coefficient were estimated by overlapping the experimental and simulation results with the systemic and noise errors. Using this method, the systemic error was in the range 0.0022 to 0.0026 and noise error amplitude between 0.0039 to 0.0041 in this case. The error of the reflection coefficient led to the error of the unloaded quality factor of approximately ±200, with the systemic error included in the calculation.

Fig. 11.Full-screen View

Graph of x and y from the experimental date.

To minimize the noise error, the data near the resonant frequency was selected. A line was fitted to the x and y data. The goodness of the fit was evaluated by the R-square:

where, $\widehat{y}$ is obtained from the fitting line and $\overline{y}$ is the mean value of y. Figure 12 shows the unloaded Q factor of the frequency span. When the span was small, the data used to fit a line was limited. In this case, the calculated unloaded Q factor was larger than that obtained in the simulation. Moreover, the R-square values of the x and y lines were less than 1 in Fig. 13. This result indicated that the linearity was poor when the span was small. When the span was between 0.224–0.292 MHz, the R-square values for x and y were greater than 0.999, indicating good linearity in this region. Therefore, the span was sufficient to fit a line. Figure 14 shows the lines of x and y with the largest R-square value of 0.9991774. Table 1 summarizes the results from the simulation and experiments.

Fig. 12.Full-screen View

Unloaded Q factor with different frequency spans.

Fig. 13.Full-screen View

R-square values with different frequency spans.

Fig. 14.Full-screen View

(Color online) Graph of x and y with span of best R-square value.

Conclusion

A novel method was demonstrated to measure the unloaded Q factor of high-Q resonant cavities. This method can efficiently eliminate data with large errors. The frequency span with useful data should be small but large enough to fit a line. We obtained good agreement between the measured and theoretical results. This method is useful for measuring a low unloaded Q factor.