Introduction

Driven by the growing demand for safe nuclear fuel post-treatment processes, the China initiative Accelerator Driven System (CiADS) is being constructed as a clean solution for nuclear fission power sources [1-3]. To showcase the potential of a high-power continuous wave (CW) proton beam for this project, the China ADS Front-End Demo Linac (CAFe) was built. This Linac is a 162.5 MHz superconducting (SC) radio-frequency (RF) machine operating in the CW mode and consists of both normal conducting (NC) and SC sections (Fig. 1). The NC section includes an ion source, low-energy beam transport line, RF quadrupole accelerator, and medium-energy beam transport line. Conversely, the SC section comprises SC accelerating units, including 23 SC half-wave resonator cavities assembled into four cryomodules (CM1–CM4) [4-7]. The commissioning tests conducted on CAFe in the CW mode with a current of 10 mA and energy of 20 MeV successfully demonstrated its ability to accelerate and transmit high-intensity beams.

Fig. 1Full-screen View

(Color online) Layout of the CAFe facility. Two types of half-wave resonator superconducting cavities (HWR010 and HWR015) are implemented. The cavity CM_3-3 is marked by a red triangle. Note that for a cavity CM_m-n, the subscripts m and n represent the m^th cryomodule and n^th cavity, respectively.

For an SC cavity, the loaded quality factor (Q_L) reflects the consumption of the stored electromagnetic energy inside the cavity. In an ideal situation, in the absence of a beam passing through the cavity, Q_L indicates the power dissipation from the cavity wall owing to the surface resistance (termed Q₀ [8, 9]) and the power leakage from the coupler ports (termed Q_e) [10]. Thus, Q_L is a critical parameter that must be carefully selected to match the impedance of the RF generator with the particle beam load during operation [11]. Furthermore, dark current loading can negatively affect Q_L, making it an important figure of merit for identifying such effects [12, 13]. In addition, Q_L (or the cavity half-bandwidth (f_0.5)) plays a crucial role in the design of model-based controllers [14-16]. To satisfy the aforementioned application requirements, the measurement error for Q_L should not exceed 5%. The value of Q_L can be calibrated using the cavity resonant frequency (f₀) and f_0.5, where $f_{0.5}=\frac{f_0}{2Q_{\text{L}}}$. Therefore, the precise measurement of f_0.5 is a prerequisite for calibrating Q_L. The decay curves of the cavity amplitude and phase (after the RF power is turned off) obtained from the cavity differential equation contain information on f_0.5 and the cavity detuning parameter (Δ f), respectively. Many laboratories, including DESY, KEK, and CSNS, employ the “field decay method" to measure the aforementioned physical quantities [17-21].

We tracked the long-term changing regularity of f_0.5 based on the data obtained when the RF power was turned off and accumulated while the CAFe facility was operated. Occasionally, we found that the cavity half-bandwidth calculated using the amplitude decay curve (i.e., f_{0.5, decay}) and the cavity detuning parameter calculated using the phase decay curve (i.e., Δ f_decay) appeared to be correlated. However, in principle, they should be independent. To better understand the above issue, we thoroughly examined the “field decay method.” Our findings revealed that this method is based on the zero-input response of the cavity differential equation, which indicates that the RF system must satisfy the “zero-input” condition. Thus, the cavity incident power must drop to zero after the RF power is turned off. However, if this condition is not met (i.e., owing to impedance mismatch), the remaining incident power may influence the decay process and render the “field decay method” ineffective. We constructed an equivalent circuit that included RF power sources, transmission lines, input couplers, and SRF cavities and derived a solution for the cavity differential equation when a source impedance mismatch occurred. Finally, we modified the formula in the “field decay method” to explain the aforementioned correlation.

The “field decay method” is always employed to calibrate Q_L. If the aforementioned “zero-input” condition is not satisfied owing to impedance mismatch, considerable errors may occur in the measurement of Q_L. To improve measurement accuracy, this study focuses on a modified calibration algorithm based on an equivalent circuit.

Phenomena and Possible Interpretation

The conventional “field decay method" is briefly reviewed in this section. The naming rules for the cavity forward voltage signal (V_f) and cavity voltage (V_c) in polar coordinates are illustrated in Fig. 2. In polar coordinates, V_f and V_c can be expressed as $V_{\text{f}}=\frac{1}{2}\rho e^{i\theta }$ and V_c=reiφ, respectively, where ρ, r, θ, and φ represent the amplitudes of 2 V_f and V_c and the phases of V_f and V_c, respectively. When the RF power is turned off, ρ immediately decreases to zero in the ideal case. Under this condition, according to the SC cavity differential equation without the beam in polar coordinates [22-24], f_0.5 and Δ f can be expressed as $｛\begin{array}{l}{f}_{0.5\text{,decay}=-}\left(\frac{r^{\text{'}}}{r}\right)\cdot \frac{1}{2\pi }\hfill \\ \Delta f_{\text{decay}}=\frac{{\phi }^{\text{'}}}{2\pi }\hfill \end{array}\text{.}$ (1)

Fig. 2Full-screen View

Schematic interpreting the notations of the cavity differential equation in polar coordinates.

We refer to the various methods used in this study to calculate f_0.5 and Δ f. For clarity, we first provide their detailed definitions in Table 1.

For the cavity CM_3-3 (marked by a red triangle in Fig. 1) at CAFe, we measured the cavity-field decay curves for different values of Δ f_decay (Fig. 3). f₀ was tuned using a frequency tuner. After the RF power was turned off, the slope of the cavity phase varied with the detuning parameter (Fig. 3(c)) because the phase decay curves are directly associated with the detuning parameter according to Eq. (1). Because the cavity Q_L is independent of the detuning parameter, the field decay curves of the cavity are expected to overlap under different detuning conditions; however, they appear to be affected by the detuning parameter (Fig. 3(a) and (b)). We conducted several studies to address this perplexing phenomenon.

Fig. 3Full-screen View

(Color online) Cavity amplitude (a and b) and phase (c) decay curves before and after the RF power is turned off for various Δ f on the cavity CM_3-3. The parameter f_{0.5, decay} is obtained by calculating the slope of the decay curve between 0.95 and 0.75 of the steady-state V_c (see the middle plot).

First, we calibrated f_{0.5, decay} and Δ f_decay with four different values of V_c for the cavity CM_3-3 (Fig. 4(a)). The value of V_c is less than the onset gradient of the field emission (approximately 1.2 MV). All four curves show the dependencies between f_{0.5, decay} and Δ f_decay. A similar dependence appears in another cavity (CM_3-4) (Fig. 4(b)).

Fig. 4Full-screen View

(a) f_{0.5, decay} and Δ f_decay have dependency relations for different cavity voltages on CM_3-3, particularly when V_c is greater than 0.4 MV. (b) The dependency relations can also be observed in CM_3-4 (red triangles). (c) Comparison of the dependency relationships at CM_3-3 when Δ f_decay is scanned by the signal source (red triangles) and tuner (gray dots). The two curves exhibit the same tendency.

Initially, we suspected that the frequency tuner might have disturbed the input coupler, causing variations in the coupling coefficients (β) and Q_L (or f_{0.5, decay}). Consequently, we turn off the tuner and achieve cavity detuning by scanning the frequency of the signal generator at the same CM_3-3. However, a similar dependence was observed in both cases (Fig. 4(c). The deviation in Fig. 4(c) is primarily because of the slight differences in the cavity field levels.

Assuming that $V_{\text{f}}^{*}$ and $V_{\text{r}}^{*}$ represent the forward signal and reflected signal measured by the directional coupler, respectively, a further calibration of $V_{\text{f}}^{*}$ and $V_{\text{r}}^{*}$ is necessary to obtain the true forward and reflected signals (V_f and V_r) using $V_{\text{f}}=X{V}_{\text{f}}^{*}$ and $V_{\text{r}}=Y{V}_{\text{r}}^{*}$, respectively (assuming that we neglect the channel crosstalk). The complex coefficients (X and Y) can be obtained by solving a linear regression equation [22]. Fig. 5 compares the V_f, V_r and V_c signals, and a residual attenuation signal of V_f (i.e., $X{V}_{\text{f}}^{*}$) is observed after turning off the RF power. There are two possible reasons for this (Fig. 6).

Fig. 5Full-screen View

Voltage and phase of V_c, $X{V}_{\text{f}}^{*}$, and $Y{V}_{\text{r}}^{*}$ before and after turning off the RF power.

Fig. 6Full-screen View

Two causes of $V_{\text{f}}^{*}\ne 0$ (RF off): crosstalk between the measurement channels (green line) and source impedance mismatch (red line).

1. The first reason is the crosstalk between the measurement channels, that is, the residual signal of V_f is coupled with the signal V_r [22]. In this case, the residual signal is a false measurement signal.

2. The second reason is the source impedance mismatch, where the V_r signal is reflected from the generator side and mixed with the V_f signal. In this case, the residual signal is a true signal.

Before 2021, directional couplers with poor directivity (approximately 20 dB) were commonly used in our RF system, at CAFe facility. Previous studies have suggested that the limited directivity of these couplers was primarily responsible for the residual signals [24]. In 2022, we replaced all the old directional couplers with new ones exhibiting high directivity (40 dB). This resulted in almost negligible channel crosstalk; however, we decided not to install a high-power circulator in CAFe because of cost constraints. Based on these factors, we conclude that the residual V_f signal in Fig. 5 could be attributed to impedance mismatch rather than crosstalk.

The algorithm in Eq. (1) must be modified because the “zero-input" condition is not satisfied. The specific calibration algorithms are described in Sec. 3.

Theory and Algorithm

In this section, we establish cavity differential equations for the mismatched source impedance condition and use them to derive new formulas for calibrating f_0.5 and Δ f.

3.1

Radio-frequency and cavity circuit under the mismatched source impedance condition

Fig. 7(a) presents a simplified model of an RF cavity coupled to an RF generator using a rigid coaxial line and an RF input power coupler [19]. In this model, the coaxial line is represented by a transmission line with a characteristic impedance (Z₀) and complex propagation constant (α+iβ). If the source impedance (Z_g) (on the generator side) is not equal to Z₀, a portion of the cavity-reflected signal is measured by the direction coupler as the cavity forward signal after turning off the RF power. This process is described using an equivalent circuit (Fig. 7(b)). Assuming that the cavity input coupler has a transformation ratio of 1:N, the voltage signals V_c, V_f, and V_r, can be transformed into $V_{\text{c}}^{♯}$, $V_{\text{f}}^{♯}$, and $V_{\text{r}}^{♯}$, respectively, using the following transformation equation: $V^{♯}=\frac{1}{N}\cdot V$.

Fig. 7Full-screen View

(a) Simplified model of a cavity coupled to an RF generator by a rigid coaxial line and an RF input power coupler. (b) Equivalent circuit diagram of (a) after the RF power is turned off. The cavity voltage (V_c) is transformed into $V_{\text{c}}^{♯}$ on the left side of the input coupler.

In Fig. 7(b), the reflection coefficient (Γ_g) for the forward generator side at z=0 can be expressed as [25] ${\Gamma }_{\text{g}}=\frac{V^{-}}{V^{+}}=\frac{Z_{\text{g}}-Z_0}{Z_{\text{g}}+Z_0}\mathrm{,}$ (2) where V⁺ and V^- represent the voltages of the incident and reflected waves at z=0, respectively. After the RF power is turned off, the reflection coefficient at z=-L (L is the length of the transmission line) is given by ${\Gamma }_{\text{L}}=\frac{V_{\text{f}}^{♯}}{V_{\text{r}}^{♯}}=\frac{V^{-}{e}^{-\left(\alpha +i\beta \right)L}}{V^{+}{e}^{\left(\alpha +i\beta \right)L}}={\Gamma }_{\text{g}}{e}^{-2\left(\alpha +i\beta \right)L}\mathrm{.}$ (3) Therefore, after turning off the RF power, the transformed cavity voltage ($V_{\text{c}}^{♯}$) can be expressed as $V_{\text{c}}^{♯}=V_{\text{f}}^{♯}+V_{\text{r}}^{♯}\mathrm{,}$ (4) Inserting Eq. (3) into Eq. (4) and eliminating $V_{\text{r}}^{♯}$, we obtain $V_{\text{f}}^{♯}=\frac{{\Gamma }_{\text{L}}}{1+{\Gamma }_{\text{L}}}{V}_{\text{c}}^{♯}\mathrm{.}$ (5) Accordingly, signal V_f is associated with signal V_c by $V_{\text{f}}=\frac{{\Gamma }_{\text{L}}}{1+{\Gamma }_{\text{L}}}{V}_{\text{c}}\mathrm{.}$ (6)

In the absence of the beam current, the cavity differential equation can be expressed as: $\frac{\text{d}{V}_{\text{c}}}{\text{d}t}+\left({\omega }_{0.5}-i\Delta \omega \right)V_{\text{c}}={\omega }_{0.5}u\mathrm{,}$ (7) and $u=\frac{2{\beta }_{\text{c}}}{{\beta }_{\text{c}}+1}{V}_{\text{f}}\mathrm{.}$ (8) Here, the parameter β_c is the coupling factor, which is generally considerably larger than 1 (i.e., β_c>>1). Thus, u can be simplified as u=2V_f. By substituting V_f with Eq. (6) into Eq. (7), we obtain $\frac{\text{d}{V}_{\text{c}}}{\text{d}t}+\left({\omega }_{0.5}-i\Delta \omega \right)V_{\text{c}}={\omega }_{0.5}\cdot \alpha \cdot V_{\text{c}}\mathrm{,}$ (9) where α=2Γ_L/(1+Γ_L) denotes a complex factor. Eq. (9) does not satisfy the “zero-input" condition. However, by rearranging the terms, we obtain an equation that satisfies the following condition: $\frac{\text{d}{V}_{\text{c}}(t)}{\text{d}t}+\left[{\omega }_{0.5}\left(1-\alpha \right)-i\Delta \omega (t)\right]V_{\text{c}}(t)=\mathrm{0,}$ (10)

3.2

New calibration algorithm for f_0.5 and Δ f

It is more convenient to normalize the steady state (V_c(t)) to one, that is, V_c(0)=1 if we assume that the RF power is turned off at t=0 and the signal V_c(t) is in steady state at t≤0. Under the aforementioned restrictions, the solution to Eq. (10) at t≥0 is given by: $\begin{array}{c}{V}_{\text{c}}(t)=V_{\text{c}}(0)e^{{\displaystyle {\int }_0^t}\left[{\omega }_{0.5}\left(\alpha -1\right)+i\Delta \omega (\tau )\right]\text{d}\tau }\\ =e^{{\displaystyle {\int }_0^t}\left[{\omega }_{0.5}\left(\alpha -1\right)+i\Delta \omega (\tau )\right]\text{d}\tau }.\end{array}$ (11) The complex factor (α) can be decomposed into real and imaginary components (α_r and α_i), i.e., $\alpha ={\alpha }_{\text{r}}+i{\alpha }_{\text{i}}$.

Separating Eq. (11) into real and imaginary parts yields the following: $V_{\text{c}}(t)=e^{\left({\alpha }_{\text{r}}-1\right){\omega }_{0.5}t+i\left({\alpha }_{\text{i}}{\omega }_{0.5}t+{\displaystyle {\int }_0^{\tau }}\Delta \omega (\tau )\text{d}\tau \right)}\mathrm{,}\left(t⩾0\right).$ (12) Consequently, the amplitude and phase of V_c are given by $｛\begin{array}{l}r=|V_{\text{c}}(t)|=e^{\left({\alpha }_{\text{r}}-1\right){\omega }_{0.5}t}\hfill \\ \phi =\angle V_{\text{c}}(t)={\alpha }_{\text{i}}{\omega }_{0.5}t+{\displaystyle {\int }_0^{\tau }}\Delta \omega (\tau )\text{d}\tau \hfill \end{array}.$ (13) Equation (13) provides some insights. If the source impedance (Z_g) perfectly matches the transmission line impedance (Z₀), then Γ_g=0, Γ_L=0, and α=0. In this scenario, according to the field decay algorithm, the cavity half bandwidth (ω_0.5) and cavity detuning parameter (Δω) can be easily obtained by fitting the slope of the cavity amplitude and phase, i.e., ω_0.5=-r’/r and Δω=φ’. However, if the source impedance is not correctly matched to the transmission line (${\Gamma }_{\text{g}}\ne 0$), the waves reflected from the cavity side will reflect again, resulting in nonzero forward power even after the RF power is turned off. This significantly affects the shape of the field curves, rendering the original algorithm inapplicable.

Using the original field decay method, the cavity half-bandwidth (ω_0.5,decay) and detuning parameter (Δω_decay) were calibrated using $｛\begin{array}{l}{\omega }_{\mathrm{0.5,}\text{decay}}=\frac{-r^{\text{'}}}{r}=\left(1-{\alpha }_{\text{r}}\right){\omega }_{0.5}\hfill \\ \Delta {\omega }_{\text{decay}}={\phi }^{\text{'}}={\alpha }_{\text{i}}{\omega }_{0.5}+\Delta \omega \hfill \end{array}\mathrm{.}$ (14) To calibrate the actual cavity half-bandwidth and detuning parameters, the original algorithm must be modified as follows: $｛\begin{array}{l}{f}_{\text{0}\mathrm{.5,}\text{cali}}=\frac{1}{2\pi }\cdot \frac{-r^{\text{'}}/r}{1-{\alpha }_{\text{r}}}=\frac{f_{\text{0}\text{.5,decay}}}{1-{\alpha }_r}\hfill \\ \Delta f_{\text{cali}}=\frac{1}{2\pi }{\phi }^{\text{'}}-{\alpha }_{\text{i}}{f}_{\text{0}\text{.5,cali}}=\Delta f_{\text{decay}}-{\alpha }_{\text{i}}{f}_{\text{0}\mathrm{.5,}\text{cali}}\hfill \end{array}.$ (15) The estimation error of the original field-decay algorithm can be easily evaluated for a specified Γ_L. According to Eq. (15), the accuracies of f_0.5,decay and Δ f_decay can be expressed as: $｛\begin{array}{l}\frac{f_{\text{0}\mathrm{.5,}\text{decay}}-f_{\text{0}\mathrm{.5,}\text{cali}}}{f_{\text{0}\mathrm{.5,}\text{cali}}}=-{\alpha }_r=-\Re \left(\frac{2{\Gamma }_{\text{L}}}{1+{\Gamma }_{\text{L}}}\right)\hfill \\ \frac{\Delta f_{\text{decay}}-\Delta f_{\text{cali}}}{f_{\text{0}\mathrm{.5,}\text{cali}}}={\alpha }_i=\Im \left(\frac{2{\Gamma }_{\text{L}}}{1+{\Gamma }_{\text{L}}}\right)\hfill \end{array}\mathrm{.}$ (16) The accuracy of the original field-decay algorithm depends on the parameter α, where the real (α_r) and imaginary (α_i) parts of α determine the accuracies of f_{0.5, decay} and Δ f_decay, respectively. Fig. 8(a) and (b) illustrate the calibration errors of f_0.5,decay and Δ f_decay, respectively, as functions of Γ_L. To ensure that the accuracies of f_{0.5, decay} and Δf_decay/f_0.5,cali lie within the ± 5% band, the coefficient Γ_L must be located inside the red circle (that is, Γ_L|<0.024 ≈ -32 dB) in Fig. 8(c). Fig. 8(d) illustrates the specific relationship between the accuracies of f_{0.5, decay}, Δf_decay/f_0.5,cali and Γ_L.

Fig. 8Full-screen View

(Color online) Calibration error of the cavity half bandwidth (a) and detuning parameter (b) as a function of the reflection coefficient (Γ_L), whereΓ_L ranges from 0 – 0.18. The cavity bandwidth (f_{0.5, decay}) and Δ f_decay are calibrated based on the field decay curves. To ensure that the accuracies of f_{0.5, decay} and $\Delta f_{\text{decay}}/f_{\text{0}\mathrm{.5,}\text{cali}}$ lie within the ± 5% band, Γ_L should be located inside the red circle (i.e., Γ_L|<0.024 ≈ -32 dB), as illustrated in (c). The results in (d) show that the calibration error of f_{0.5, decay} and Δ f_decay increases with Γ_L|, particularly when Γ_L| exceeds -32 dB.

Modeling and Simulation

The state–space formalism in Eq. (7) is given by [19, 26, 27]: $\begin{array}{c}\frac{\text{d}}{\text{d}t}\left(\begin{array}{l}{V}_{\text{c,r}}\hfill \\ V_{\text{c,i}}\hfill \end{array}\right)=\left(\begin{array}{ll}-{\omega }_{0.5}\hfill & -\Delta \omega \hfill \\ \Delta \omega \hfill & -{\omega }_{0.5}\hfill \end{array}\right)\left(\begin{array}{c}{V}_{\text{c,r}}\\ V_{\text{c,i}}\end{array}\right)\\ +\left(\begin{array}{cc}{\omega }_{0.5}& 0\\ 0& {\omega }_{0.5}\end{array}\right)\left(\begin{array}{c}{u}_{\text{r}}\\ u_{\text{i}}\end{array}\right)\mathrm{.}\end{array}$ (17) Here, V_c,r and V_c,i represent the real and imaginary parts of the complex quantity (V_c), while u_r and u_i represent the real and imaginary parts of the complex quantity (u), respectively. Eq. (17) can easily be transformed into its discrete-time form as follows [26, 27]: $\begin{array}{c}\left[\begin{array}{c}{V}_{\text{c,r}}(n+1)\\ V_{\text{c,i}}(n+1)\end{array}\right]=\left[\begin{array}{cc}1-T_{\text{s}}{\omega }_{0.5}& -T_{\text{s}}\Delta \omega (n+1)\\ T_{\text{s}}\Delta \omega (n+1)& 1-T_{\text{s}}{\omega }_{0.5}\end{array}\right]\left[\begin{array}{c}{V}_{\text{c,r}}(n)\\ V_{\text{c,i}}(n)\end{array}\right]\\ +\left[\begin{array}{cc}{T}_{\text{s}}{\omega }_{0.5}& 0\\ 0& T_{\text{s}}{\omega }_{0.5}\end{array}\right]\left[\begin{array}{c}{u}_{\text{r}}(n)\\ u_{\text{i}}(n)\end{array}\right]\mathrm{,}\end{array}$ (18) where T_s represents the sampling period. According to Eq. (9), the drive signal u can be expressed as $\left[\begin{array}{c}{u}_{\text{r}}(n+1)\\ u_{\text{i}}(n+1)\end{array}\right]=\left[\begin{array}{cc}{\alpha }_{\text{r}}& -{\alpha }_{\text{i}}\\ {\alpha }_{\text{i}}& {\alpha }_{\text{r}}\end{array}\right]\left[\begin{array}{c}{V}_{\text{c,r}}(n+1)\\ V_{\text{c,i}}(n+1)\end{array}\right].$ (19) Other equations related to the dynamic behavior of the cavity are derived in Appendix A. The simulation parameters are presented in Table 2. After steady-state operation is achieved for 0.4 ms, the RF power is turned off. The signal reflected from the cavity is also reflected as Γ_g is not zero. Fig. 9 compares the signals V_c, V_f, and V_r based on the cavity model (red dash line) and real SC cavity (thick solid gray line). The red dash lines and thick solid gray lines showed good consistency. In addition, the simulation results for the perfect impedance matching case (Γ_g=0, indicated in green) are included for comparison.

Fig. 9Full-screen View

(Color online) Comparison of the cavity voltage (a), cavity forward (b), and reflected signal (c) measurements based on the cavity model (red) and real SC cavity (gray). A stable operation is maintained for 0.4 ms before the RF power is turned off to create a field decay event. In the cavity model, the LFD dynamics is assumed to be determined by a first-order differential equation, resulting in curved trajectories in the cavity phase signal (red dashed lines) instead of linear ones. When the LFD effect is neglected, the cavity phase curve is linear (blue dotted lines), as shown in (a). In addition, the simulation results for the perfect impedance match case (Γ_g=0, indicated by green solid line) are also included for comparison.

According to Appendix A, the Lorentz force detuning (LFD) dynamics are assumed to be determined by a first-order differential equation. Consequently, the cavity phase signal presents curved trajectories (red dashed lines in Fig. 9) rather than linear trajectories. However, without the LFD dynamics, the cavity phase curve is linear, as indicated by the blue dotted lines in Fig. 9(a). The slope of the linear phase curve represents the cavity detuning parameter (Δω_decay). The cavity phase curves overlap in the first 80 μs after the RF power is turned off, regardless of whether the LFD is included (Fig. 9(a)). For clarity, the LFD dynamics during the field decay process were examined (Fig. 10). For the subsequent 80 μs after the RF power is turned off, the LFD-induced phase error and maximum LFD value are less than 0.01 deg and approximately 0.8 Hz, respectively. Consequently, it is reasonable to fit the 80 μs cavity phase data to obtain Δ f.

Fig. 10Full-screen View

Simulation results showing the effect of the LFD on the measurement accuracy of the cavity phase and cavity detuning. For the initial 80 μs period after turning off the RF power, the LFD-induced phase error is less than 0.01 deg, which can be ignored.

Experimental Verification

Based on the algorithm in (15), we recalibrated the measurement results shown in Fig. 4(a). The details of the process are described below.

1. Calibrating the actual forward and reflected signals using $V_{\text{f}}=X{V}_{\text{f}}^{*}$ and $V_{\text{r}}=X{V}_{\text{r}}^{*}$, respectively: The complex factors (X and Y) were determined by solving a linear regression equation [22].

2. Determining the factor α: According to (6), we calculated α as twice the ratio of V_f to V_c after turning off the RF power. Here, we averaged α over a time interval of 50 μs (e.g., from 0.45 ms to 0.5 ms in Fig. 11) to reduce uncertainty.

Fig. 11Full-screen View

(a) Cavity forward signal (V_f) for different detuning values on cavity CM_3-3(the corresponding cavity voltage signal is given in Fig. 4). (b) Method for estimating α from the V_c and V_f signals after the RF power is turned off.

3. Calibrating f_{0.5, decay} and Δ f_decay using the traditional “field decay method": To ensure a sufficient signal-to-noise ratio, we determined f_{0.5, decay} by calculating the slope of the decay curve between 0.95 and 0.75 in the steady-state (V_c)(Fig. 3(b)). To avoid the LFD effect, we determined the derivative of the cavity phase within an 80 μs interval after the RF power was turned off to obtain Δ f_decay.

4. Calibrating f_{0.5, cali} and Δ f_cali: We used the formula (15) with the known values of α, f_{0.5, decay}, and Δ f_decay to calibrate f_{0.5, cali} and Δ f_cali.

We used the aforementioned procedure to calibrate f_{0.5, cali} and Δ f_cali (Fig. 12(a)). In contrast to the strong correlation observed between f_{0.5, decay} and Δ f_decay measured by the cavity amplitude and phase decay curve, the new calibrated value of f_0.5,cali was not independent of Δ f_cali at different V_c levels.

Fig. 12Full-screen View

(Color online) (a) Relationship between the cavity half bandwidth (f_0.5) and cavity detuning parameter (Δ f_cali) in the calibrated and uncalibrated algorithms for various values of V_c. (b) Comparison of the results of Δ f_decay and Δ f_cali measurements with the cavity detuning value (Δ f_ss) obtained in the steady state.

We also compared Δ f_cali and Δ f_decay with the steady-state cavity detuning parameters (Δ f_ss), which were calculated from the steady-state values of V_f and V_c (Fig. 12(b)). The following expression was used to calibrate Δ f_ss: $\Delta f_{\text{ss}}=\tan \left(\phi -\theta \right)\cdot f_{\mathrm{0.5,}\text{cali}}\mathrm{.}$ (20) Figure 12(b) indicates that the discrepancy between Δ f_cali and Δ f_ss is less than ± 5 Hz, whereas there is an offset of approximately 40 Hz between Δ f_decay and Δ f_ss.

To further validate the proposed algorithm, we used a network analyzer to measure CM_3-3 and perform a comparison. Figure 13(a) and (b) show the measurement setup, and the frequency-response measurement and corresponding fitted curve, respectively. The measurement results were fitted using the following formula $A_{\text{fit}}=\frac{A_{\max }\cdot f_{0.5}}{\sqrt{{\left(f-f_{\text{offset}}\right)}^2+f_{0.5}^2}}\mathrm{.}$ (21) where A_max is the maximum value of the amplitude–frequency-response curve. The parameters f and f_offset represent the stimulus frequency and frequency offset between the cavity resonant frequency and RF, respectively. In Fig. 13(b), the estimated value of f_0.5 for the fitted curve is 183.7 Hz.

Fig. 13Full-screen View

(a) Measurement block diagram based on the network analyzer. (b) Frequency response function of cavity CM_3-3, as measured by the network analyzer. The f_offset parameter is excluded for better comparison.

Next, we utilized a network analyzer to scan the cavities in CM₁-CM₃ and estimated f_{0.5, scan} by curve fitting. We then compared the derivation between f_0.5,cali and f_0.5,scan for the 16 cavities (excluding CM_2-6 and CM_3-1 because of cavity faults). In addition, we plot the deviation between f_0.5,decay and f_0.5,scan for comparison purposes. The results are shown in Fig. 14. The deviation between f_{0.5, scan} and f_{0.5, cali} was maintained roughly within ±2%, indicating that the proposed calibration algorithm accurately estimated the values of f_0.5.

Fig. 14Full-screen View

Comparison of the deviation between f_0.5,decay and f_0.5,cali for the 16 cavities and the network analyzer measurement results, f_0.5,cali.

Conclusion

After the RF power was turned off, the residual signal (V_f) caused by source impedance mismatch could affect the field decay process, resulting in measurement errors for f_0.5 and Δ f. The simulation results indicate that, to ensure that the measurement errors of f_0.5 and Δ f were less than 5%, the reflection coefficient at the generator side must be less than -32 dB. To improve the measurement accuracy, we derived a calibration algorithm based on the cavity differential equation for cases in which there was a source impedance mismatch. Using this algorithm, we recalibrated f_0.5 and Δ f and compared the results with those obtained from network analyzer measurements. The maximum error between the calculated and measured values was within 2%.

α is a crucial factor in the calibration algorithm that ultimately depends on the impedance Z_g on the generator side. Unfortunately, we do not have a comprehensive understanding of the factors that determine Z_g. One possible factor is the power output of the RF generator. For instance, different Vc and detuning parameters require different RF powers, which lead to different Z_g values. Nevertheless, it is important to stress that Z_g is not determined solely by the generator power, and the same power may result in different values of Z_g. In our future work, we shall focus on identifying the mechanisms and physical quantities that affect Z_g.