Obtaining the Transfer Function of a Pulse Control System Containing an Ion Converter
Abstract Generated abstract
This paper derives a transfer function for a pulse automatic control system used in high-power charged-particle accelerator power supplies, where a controlled three-phase bridge rectifier charges a capacitor bank through feedback circuits. The rectifier is represented as an equivalent generator with averaged internal resistance and inductance, and the regulator equations are formulated for stepwise variations of rectifier emf and capacitor voltage. By passing from continuous operator relations to discrete Laplace transforms, the authors obtain expressions linking the discrete control signal to the emf and capacitor voltage, then close the loop by relating the control signal to changes in the rectifier ignition angle. The resulting closed-loop equation provides a basis for assessing system stability through the denominator of the discrete transfer function, for example using amplitude-phase characteristics.
Full Text
OBTAINING THE TRANSFER FUNCTION OF A PULSE CONTROL SYSTEM CONTAINING AN ION CONVERTER
V. A. KOCHEGUROV, A. A. TERESHCHENKO
To obtain electromagnetic fields in most high-power charged-particle accelerators, pulsed current sources are widely used; their operating principle is based on the slow accumulation of energy in capacitors, followed by the rapid transfer of energy to the load.
Fig. 1
Figure 1 shows the block diagram of such an installation. The circuit contains a three-phase bridge rectifier, a bank of electrical capacitors, and feedback circuits that provide stabilization of the pre-discharge voltage on the capacitors and a charge current of constant amplitude by controlling the rectifier through its grids.
In order to analyze the dynamic characteristic and the transient process of such a closed-loop pulse automatic-control system with periodically varying parameters, it is necessary to obtain its transfer function.
Works [1, 2, 3], etc., are devoted to the theory of pulse automatic-control systems; however, this theory does not provide a ready-made method for investigating the dynamics of the system described.
To obtain the transfer function of a pulse-control system with variable parameters, the methods set forth in [3, 4, 5] have been used.
These methods make it possible to regard the rectifier as a certain equivalent generator (Fig. 1), having an emf equal to
\[ \frac{3}{\pi}\sqrt{3}\,E_m \cos\alpha, \]
the mean value of the internal active resistance
\[ R=\frac{2r_{\mathrm{tr}}\left(\frac{\pi}{2}-\gamma\right)+\frac{3}{2}r_{\mathrm{tr}}\gamma}{\frac{\pi}{3}} \]
and inductance
\[ L=\frac{2L_s\left(\frac{\pi}{3}-\gamma\right)+\frac{3}{2}L_s\gamma}{\frac{\pi}{3}}, \]
where
\(L_s\) and \(r_{\mathrm{tr}}\) are the referred inductance and active resistance of the transformer,
\(\gamma\) is the commutation angle.
In setting up the equations determining the regulation process, let us first assume that the control system is open-loop, and find the value of the control signal that arises under the influence of the charging current (the equivalent emf of the rectifier) and the voltage on the capacitance, which vary according to an arbitrary stepwise law:
\[ U_{\mathrm{ctrl}(p)} = -\frac{R_{(p)}}{D_{(p)}}\,L_b{}_{(p)} + \frac{N_{(p)}}{M_{(p)}}\,\Delta U_c{}_{(p)}, \tag{1} \]
where \(\dfrac{R_{(p)}}{D_{(p)}}\) and \(\dfrac{N_{(p)}}{M_{(p)}}\) are the operators of the current and voltage regulator.
We transform equation (1), for which purpose we substitute
\[ \Delta u_c{}_{(p)}=\frac{1}{i_b{}_{(p)}\cdot C_p}, \tag{2} \]
then
\[ u_{\mathrm{ctrl}(p)} = -i_b{}_{(p)} \left[ \frac{R_{(p)}}{D_{(p)}}-\frac{N_{(p)}}{M_{(p)}}\frac{1}{C_p} \right]. \tag{3} \]
The current \(i_b{}_{(p)}\) entering the right-hand side of the equation can be found from the equivalent circuit (Fig. 1):
\[ i_b{}_{(p)} = -\frac{C_p\left[e_{(p)}-u_c{}_{(p)}\right]}{CLp^2+rCp+1}. \tag{4} \]
Equation (3), taking equation (4) into account, can be written as
\[ u_{\mathrm{ctrl}(p)} = -\frac{C_p\left[e_{(p)}-u_c{}_{(p)}\right]}{CLp^2+rCp+1} \left[ \frac{R_{(p)}}{D_{(p)}}-\frac{N_{(p)}}{M_{(p)}}\frac{1}{C_p} \right], \tag{5} \]
where \(e_{(p)}\) and \(u_c{}_{(p)}\) are the operators of the rectifier emf and the voltage on the storage capacitor.
Let us denote \(CLp^{2}+rCp+1=A_{(p)}\) and transform equation (5):
\[ u_{\mathrm{upr}(p)} = -\left[ \frac{C_{(p)}\cdot R_{(p)}\cdot M_{(p)}-N_{(p)}D_{(p)}} {A_{(p)}\cdot M_{(p)}\cdot D_{(p)}} \right] e_{(p)} + \left[ \frac{C_{(p)}R_{(p)}\cdot M_{(p)}-N_{(p)}D_{(p)}} {A_{(p)}\cdot M_{(p)}D_{(p)}} \right] u_{c(p)} . \tag{6} \]
Fig. 2
Let us write this equation for a stepwise variation of \(e_{(p)}\) and \(u_{c(p)}\), for which it is necessary to pass from operator dependences to time dependences. The value of the control signal of the regulator at the beginning of the \(n\)-th interval (Fig. 2) is determined by the expression:
\[ \begin{aligned} u_{\mathrm{upr}(n)} ={}& \sum_{m=0}^{n} e_{(m)} \left\{ \frac{C_{(0)}\cdot R_{(0)}M_{(0)}N_{(0)}D_{(0)}} {A_{(0)}\cdot M_{(0)}\cdot D_{(0)}} + \sum_{k=1}^{l} \frac{ C_{(p_k)}R_{(p_k)}M_{(p_k)}-N_{(p_k)}\cdot D_{(p_k)} } { \left[A_{(p_k)}\cdot D_{(p_k)}\cdot M_{(p_k)}\right]'p_k } \times \right. \\ &\left. \times e^{p_k(n-m)T} \right\} - \sum_{m=0}^{n-1} e_{(m)} \left\{ \frac{C_{(0)}\cdot R_{(0)}M_{(0)}-N_{(0)}\cdot D_{(0)}} {A_{(0)}\cdot M_{(0)}D_{(0)}} + \right. \\ &\left. + \sum_{k=1}^{l} \frac{ C_{(p_k)}\cdot R_{(p_k)}M_{(p_k)}-N_{(p_k)}D_{(p_k)} } { \left[A_{(p_k)}D_{(p_k)}M_{(p_k)}\right]'p_k } e^{p_k(n-m-1)T} \right\} + \\ &+ \sum_{m=0}^{n} u_{c(m)} \left\{ \frac{C_{(0)}\cdot R_{(0)}\cdot M_{(0)}-N_{(0)}D_{(0)}} {A_{(0)}\cdot M_{(0)}\cdot D_{(0)}} + \sum_{k=1}^{l} \frac{ C_{(p_k)}\cdot R_{(p_k)}M_{(p_k)}-N_{(p_k)}D_{(p_k)} } { \left[A_{(p_k)}D_{(p_k)}M_{(p_k)}\right]'p_k } \times \right. \\ &\left. \times e^{p_k(n-m)T} \right\} - \sum_{m=0}^{n-1} u_{c(m)} \left\{ \frac{C_{(0)}\cdot R_{(0)}\cdot M_{(0)}-N_{(0)}D_{(0)}} {A_{(0)}\cdot M_{(0)}D_{(0)}} + \right. \\ &\left. + \sum_{k=1}^{l} \frac{ C_{(p_k)}\cdot R_{(p_k)}M_{(p_k)}-N_{(p_k)}D_{(p_k)} } { \left[A_{(p_k)}M_{(p_k)}D_{(p_k)}\right]'p_k } e^{p_k(n-m-1)T} \right\}, \end{aligned} \tag{7} \]
where
\[
T
\]
is the duration of one interval,
\[
p_k
\]
are the roots of the equation \(A_{(p)}M_{(p)}D_{(p)}=0\),
and \(e_{(m)}\) and \(u_{c(m)}\) are the originals of the operators \(e_{(p)}\) and \(u_{c(p)}\).
The action of the control signal of the regulating device is manifested in the acceleration or delay of ignition of the rectifier valves.
The stepwise character of \(e_{(p)}\) and \(u_c{}_{(p)}\), and the discrete action of the regulating device, make it possible to use methods for calculating the stability of pulse control systems [1].
Using the discrete Laplace transform for equation (7), we obtain
\[ u_{\text{ctrl}(q)}^{*} = - W_{1(q)}^{*} e_{(q)}^{*} + W_{2(q)}^{*} u_{c(q)}^{*}, \tag{8} \]
where
\[ q=pT; \]
\[ u_{\text{ctrl}(q)}^{*},\quad e_{(q)}^{*},\quad u_{c(q)}^{*} \text{ are the discrete operators of the control signal} \]
\[ u_{\text{ctrl}[n]}\ \text{and}\ e_{[n]},\ u_{c[n]}; \]
\[ W_{1(q)}^{*} \sum_{\kappa=1}^{l} \frac{ C_{(q_\kappa)} R_{(q_\kappa)} M_{(q_\kappa)} - N_{(q_\kappa)} D_{(q_\kappa)} }{ \left[A_{(q_\kappa)} \cdot D_{(q_\kappa)} \cdot M_{(q_\kappa)}\right]' q_\kappa } \cdot \frac{e^{q}-1}{e^{q}-e^{q_\kappa}}; \]
\[ W_{2(q)}^{*} = \sum_{\kappa=1}^{l} \frac{ C_{(q_\kappa)} \cdot R_{(q_\kappa)} \cdot M_{(q_\kappa)} - N_{(q_\kappa)} D_{(q_\kappa)} }{ \left[A_{(q_\kappa)} \cdot D_{(q_\kappa)} \cdot M_{(q_\kappa)}\right]' q_\kappa } \cdot \frac{e^{q}-1}{e^{q}-e^{q_\kappa}} \]
\[ \cdot \frac{ C_{(0)} \cdot R_{(0)} \cdot M_{(0)} - N_{(0)} D_{(0)} }{ A_{(0)} D_{(0)} M_{(0)} }; \]
\[ q_\kappa=p_\kappa T. \]
Equation (8) gives the dependence of the control signal on the stepwise emf \(e_{(q)}^{*}\) and \(u_{c(q)}^{*}\).
When the control system is closed, when the control signal acts on the system, the magnitude of the equivalent emf will be composed of external disturbances and those changes in emf that arise under the action of the regulator:
\[ \left. \begin{aligned} e_{[n]} &= u_{1\,\text{dist}[n]} + u_{p[n]} \\ u_{c[n]} &= u_{2\,\text{dist}[n]} \end{aligned} \right\}. \tag{9} \]
The control signal corresponds to a certain deviation of the angle \(\alpha[n]\) from the stationary value \(\alpha[0]\) before the start of the transient process,
\[ \Delta \alpha[n]=\kappa_1 u_{\text{ctrl}}[n], \tag{10} \]
where
\[ \Delta \alpha[n]=\alpha[n]-\alpha[0]; \tag{11} \]
\(\kappa_1\) determines the steepness of the characteristic of the grid-control system. The dependence of the control angle on the control signal is linear.
The change in the rectifier emf corresponding to the increment of the angle \(\Delta \alpha[n]\) is written as
\[ u_{p[n]} = \frac{3}{\pi}\sqrt{3}\,E_m\{\cos \alpha[n]-\cos \alpha[0]\} \approx \]
\[ \approx -\frac{3}{\pi}\sqrt{3}\,E_m \sin \alpha[0]\,\Delta \alpha[n] = -\kappa_2 \sin \alpha[0]\,\Delta \alpha[n]. \tag{12} \]
Substituting the value (12) into equation (9) and applying the Laplace transforms, we obtain
\[ e_{(q)}^{*} = u_{\text{dist}(q)}^{*} - \kappa_1 \kappa_2 \sin \alpha[0]\cdot u_{\text{ctrl}(q)}^{*}. \tag{13} \]
Solving equations (13) and (8) jointly gives the equation of the closed-loop control system:
\[ e^{*}_{(q)} = \frac{ u^{*}_{1\,\mathrm{dist}\,(q)} + u^{*}_{2\,\mathrm{dist}\,(q)} \cdot k_{1} k_{2} \sin \alpha [0]\, W^{*}_{2\,(q)} }{ 1 + k_{1} k_{2} \sin \alpha [0] \cdot W^{*}_{1\,(q)} }. \tag{14} \]
To assess the stability of the control system, it is necessary to investigate the denominator of equation (14) in accordance with the general theory. Stability analysis can be carried out with the aid of amplitude-phase characteristics [1].
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