ON ENERGY RELATIONSHIPS IN EXCITATION CIRCUITS OF ACCELERATOR ELECTROMAGNETS WITH DC BIASING

V. V. IVASHIN

(Presented by the scientific seminar of the Departments of Electrical Machines and General Electrical Engineering).

In the construction of charged-particle accelerators—for example, synchrotrons or betatrons—the feeding of the accelerator electromagnet with direct current is widely used [1, 2, 3]. It is known that, in circuits with biasing of the electromagnet, the energy losses in the magnet steel are reduced by 50–65%, and the losses in copper by 25%. A significant reduction of losses in the steel makes it possible to increase approximately twofold the supply frequency and, consequently, the average intensity of the accelerated particles while maintaining the same thermal operating conditions of the machine. If the maximum induction of the accelerator magnetic field was determined not by steel saturation but by the thermal regime, then, in a circuit with biasing, it becomes possible to increase the maximum induction and, consequently, the energy of the accelerated particles.

It is known that, when biasing is applied, the alternating component of the magnetic flux is reduced by a factor of 2, which leads to a sharp reduction in the compensating reactive power of the capacitor bank and to a reduction in the cost of the power excitation system and of the accelerator as a whole.

At the same time, the energy relationships occurring in circuits with biasing have not been given sufficiently complete treatment in the literature. This results in the fact that, in the design and manufacture of circuits with biasing, the choice of circuit elements is not optimal with respect to cost.

In the present work, the energy relationships in the supply circuits of accelerator electromagnets with biasing are considered, as is the choice of the reactor in the direct-current circuit with optimal parameters.

The excitation circuits of accelerator electromagnets with direct-current biasing are complex, especially the circuits of accelerators for high energies. In such circuits, in order to reduce the voltage on the windings, capacitor banks are divided into separate parts [1, 2]. The energy relationships in any circuits with biasing are the same, and for their consideration these circuits may be replaced by a simple equivalent circuit (Fig. 1).

The biasing current \(I_{0}\) from a constant-voltage source flows through the accelerator winding with inductance \(L_{y}\), providing the required degree of biasing, and through the winding of a reactor with inductance \(L_{p}\). The alternating magnetic flux is created in the accelerator by the current \(i_{y}\) of the oscillatory circuit \(L_{y} C_{y}\).

For the alternating voltages of the circuit with current \(I_0\), one may write

\[ U_{\mathrm p}+U_{\mathrm y}=0 \tag{1} \]

or

\[ L_{\mathrm y}\frac{di_{\mathrm y\sim}}{dt} = -\,L_{\mathrm p}\frac{di_{\mathrm p\sim}}{dt}. \]

It follows from this relation that the derivatives of the currents \(i_{\mathrm y\sim}\) and \(i_{\mathrm p\sim}\), with respect to the direction of the current \(I_0\), have different signs. Consequently, when the magnetic flux of the accelerator electromagnet increases,

Fig. 1. Equivalent circuit of the power-supply system of an accelerator electromagnet with dc biasing: \(L_{\mathrm y}\) and \(L_{\mathrm p}\) are the inductances of the accelerator and the reactor; \(C_{\mathrm y}\) and \(C_{\mathrm p}\) are compensating capacitor banks; \(I_0\) is the bias current; \(i_{\mathrm y\sim}\) and \(i_{\mathrm p\sim}\) are the alternating components of the current in the accelerator and in the reactor.

that is, the currents \(I_0\) and \(i_{\mathrm y\sim}\) are directed in the same sense, the magnetic field of the reactor decreases, that is, the currents \(I_0\) and \(i_{\mathrm p\sim}\) are directed oppositely. In order that the current through the dc voltage source not change and that the source not be loaded by reactive power, a compensating capacitance \(C_{\mathrm p}\) of such magnitude is connected in parallel with the reactor that the natural oscillation frequencies of the circuits \(L_{\mathrm p}C_{\mathrm p}\) and \(L_{\mathrm y}C_{\mathrm y}\) are equal:

\[ C_{\mathrm p}=C_{\mathrm y}\frac{L_{\mathrm y}}{L_{\mathrm p}}. \tag{2} \]

At current \(I_0\), the energy stored in the reactor is equal to

\[ A_{\mathrm{po}}=\frac{L_{\mathrm p}I_0^2}{2}, \tag{3} \]

and the energy stored in the magnetic field of the accelerator is

\[ A_{\mathrm{yo}}=\frac{L_{\mathrm y}I_0^2}{2}. \]

At current \(I_0-I_{\mathrm p\sim}\), where \(I_{\mathrm p\sim}\) is the maximum value of the alternating component of the reactor current, the energy stored in the magnetic field of the reactor is

\[ A_{\mathrm{pk}}=\frac{L_{\mathrm p}\left(I_0-I_{\mathrm p\sim}\right)^2}{2}. \tag{5} \]

When the current through the reactor winding decreases, the energy of its magnetic field changes by the amount \(A_{\mathrm{pn}}\):

\[ A_{\mathrm{pn}}=A_{\mathrm{po}}-A_{\mathrm{pk}} = L_{\mathrm p}I_0I_{\mathrm p\sim} - \frac{L_{\mathrm p}I_{\mathrm p\sim}^{2}}{2}. \tag{6} \]

Let us take as the transfer coefficient \(\lambda\) the ratio of the energy \(A_{\mathrm{rp}}\) to the energy stored in the accelerator at current \(I_0\):

\[ \lambda=\frac{A_{\mathrm{rp}}}{A_{\mathrm{po}}}. \tag{7} \]

Taking into account that

\[ I_{\mathrm{p}\sim}=I_{\mathrm{y}\sim}\cdot\frac{L_{\mathrm{y}}}{L_{\mathrm{p}}}, \tag{8} \]

from relation (7), with allowance for (4) and (6), we obtain

\[ \lambda=2\frac{I_{\mathrm{y}\sim}}{I_0} -\left(\frac{I_{\mathrm{y}\sim}}{I_0}\right)^2\cdot \frac{L_{\mathrm{y}}}{L_{\mathrm{p}}} \tag{9} \]

or

\[ \lambda=2\alpha-\alpha^2\frac{L_{\mathrm{y}}}{L_{\mathrm{p}}}, \tag{9,a} \]

where \(\alpha\) is the degree of magnetization of the accelerator by direct current.

It follows from (9) that the maximum transfer of energy from the reactor to the accelerator will occur for \(L_{\mathrm{p}}\to\infty\). In this case \(\lambda=2\alpha\), \(C_{\mathrm{p}}=0\), that is, under the condition of the maximum value of the energy transferred from the reactor, the need for the capacitor bank \(C_{\mathrm{p}}\) disappears.

The degree of magnetization \(\alpha\) in accelerators is usually close to 1. Fig. 2 gives the function of the variation of the ratio of the inductances of the reactor and the accelerator as a function of \(\lambda\), plotted according to the equation

\[ \frac{L_{\mathrm{p}}}{L_{\mathrm{y}}}=\frac{\alpha^2}{2\alpha-\lambda} \tag{10} \]

for the case \(\alpha=1\). It follows from Fig. 2 that \(\lambda\) can have not only positive, but also negative values. The sign of the energy-transfer coefficient changes at \(L_{\mathrm{p}}=0.5\,L_{\mathrm{y}}\). Physically this means that, at \(L_{\mathrm{p}}=0.5\,L_{\mathrm{y}}\), at the instant of the field with maximum energy in the accelerator, the current in the reactor has changed direction, and its instantaneous value is equal to the current \(I_0\).

Fig. 2. Curve of the dependence of the ratio of the inductances \(L_{\mathrm{p}}\) and \(L_{\mathrm{y}}\) on the transfer coefficient \(\lambda\).

For the current in the reactor corresponding to the instant when the current in the accelerator is maximal, one may write

\[ I_{\mathrm{p}}=I_0-I_{\mathrm{p}\sim}. \]

Taking into account relations (8) and (10), after transformations we have

\[ I_{\mathrm{p}} = I_{\mathrm{y}}\left(\frac{\lambda-a}{a^2}\right). \]

for \(a = 1 - I_{\mathrm{p}} = I_{\mathrm{y}\sim}(\lambda - 1)\).

Figure 3 gives the current curves of the reactor and the accelerator corresponding to various values of \(\lambda\). For \(\lambda < 0\), at the instant of the maximum current in the accelerator, the energy of the reactor is greater than it was at the current \(I_0\).

Fig. 3. Current curves in the reactor and in the electromagnet:
a) reactor currents for \(a = 1\). 1. \(L_{\mathrm{p}} = 0.4L_{\mathrm{y}}, \lambda = -0.5\);
2. \(L_{\mathrm{p}} = 0.5L_{\mathrm{y}}, \lambda = 0\); 3. \(L_{\mathrm{p}} = L_{\mathrm{y}}, \lambda = 1\); 4. \(L_{\mathrm{p}} = 2L_{\mathrm{y}}, \lambda = 1.5\).
b) accelerator current for \(a \ne 1\).

If the maximum energy in the field of the accelerator electromagnet at the current \(I_0 + I_{\mathrm{y}\sim}\) is taken as \(A\), then it can be shown that the energy is distributed among the reactor and accelerator electromagnets and the capacitor banks as follows:

\[ A_{\mathrm{y}0} = \frac{1}{(1+a)^2}A; \tag{11} \]

\[ A_{\mathrm{y}\mathrm{c}} = \frac{a^2}{(1+a)^2}A; \tag{12} \]

\[ A_{\mathrm{pc}} = \frac{2a-\lambda}{(1+a)^2}\cdot A; \tag{13} \]

\[ A_{\mathrm{p}\mathrm{n}} = \frac{\lambda}{(1+a)^2}A; \tag{14} \]

\[ A_{\mathrm{p}0} = \frac{a^2}{(1+a)^2(2a-\lambda)}A. \tag{15} \]

It follows from these relations that only the distribution of energy between the reactor and the capacitor bank shunting it depends on the value of \(\lambda\), or, equivalently, on the ratio \(L_p\) to \(L_y\). Table 1 gives the distribution of energy in the elements of the supply circuit for various transfer coefficients \(\lambda\), in the case \(\alpha = 1\).

Table 1

As is seen from the table, with increasing \(\lambda\) the required value of the capacitor bank \((A_{pc}+A_{yc})\) decreases, while the value of the reactor \((A_{p0})\) increases. The choice of the parameter \(\lambda\) is determined by the requirement of a minimum reactor size, a minimum value of the capacitor bank \(C_p\), or the minimum cost of the accelerator power-supply circuit as a whole. Of greatest interest is usually the choice of \(\lambda\) from the condition of minimum cost, since the cost of the excitation circuits of the controlling magnetic fields of accelerators is large and to a considerable extent determines the cost of accelerators.

Choice of the parameter \(\lambda\) from the condition of minimum cost

The cost of the reactor and of the capacitor bank \(C_p\) is

\[ B = B_1 + B_2 = k_1 A_{p0} + k_2 A_{pc} = \frac{\alpha^2 A \cdot k_1}{(1+\alpha)^2(2\alpha-\lambda)} + \frac{(2\alpha-\lambda)A \cdot k_2}{(1+\alpha)^2}, \tag{16} \]

where
\(B_1\) is the cost of the reactor, rubles;
\(B_2\) is the cost of the bank \(C_p\), rubles;
\(k_1\) is the cost of storing 1 joule in the reactor, taking into account the cost of the dc-voltage source, rubles/J;
\(k_2\) is the cost of storing 1 joule in the capacitors, rubles/J.

Equating the derivative of expression (16) with respect to \(\lambda\) to zero, after transformations we obtain the value \(\lambda_1\) that ensures the minimum cost of the supply circuit:

\[ \lambda_1 = \alpha \left( 2 - \sqrt{\frac{k_1}{k_2}} \right). \tag{17} \]

The inductance of the reactor corresponding to the minimum cost of the supply circuit is obtained by substituting (17) into (10):

\[ L_{p1} = L_y \cdot \alpha \sqrt{\frac{k_2}{k_1}} . \tag{18} \]

The cost of the reactor and of the capacitance \(C_p\) corresponding to \(\lambda_1\) is determined by substituting (17) into (16):

\[ B_{\min} = \frac{2A\alpha}{(1+\alpha)^2}\sqrt{k_1k_2} = 2A_{y0}\alpha \sqrt{k_1k_2}. \tag{19} \]

The cost of the reactor corresponding to \(\lambda_1\) is obtained by substituting (17) into the expression for \(B_1\) of equality (16):

\[ B_1=\frac{A\cdot\alpha}{(1+\alpha)^2}\sqrt{\kappa_1\kappa_2} =A_{y0}\alpha\sqrt{\kappa_1\kappa_2}, \tag{20} \]

that is, the cost of the reactor is equal to one half of the total minimum cost of the reactor and the battery \(C_p\).

If the costs of the reactor and the battery \(C_p\) are expressed not as a function of \(\lambda\), but as a function of \(L_p\), we obtain

\[ B_1=\frac{L_p I_0^2}{2}\,\kappa_1, \tag{21} \]

\[ B_2=\frac{C_p U^2}{2}\,\kappa_2 =\frac{\kappa_2 \alpha^2 I_0^2 L_y}{2L_p}. \tag{22} \]

It follows from these expressions that the cost of the reactor varies as a linear function of \(L_p\), while the cost of the capacitor bank varies hyperbolically. From Fig. 4 it is seen that the minimum

Fig. 4. Dependence of the cost of the reactor \((B_1)\), the cost of the battery \(C_p\) \((B_2)\), and their total cost \((B)\) on the inductance of the reactor.

of the curve \(B\) with respect to \(L_p\) is not sharply expressed, and when \(L_p\) is varied by \(10\div15\%\), the cost of the reactor and the battery \(C_p\) changes insignificantly.

The maximum value of the current in the reactor can also be expressed through the bias current and the coefficients \(\kappa_1\) and \(\kappa_2\)

\[ I_{pm}=I_0+I_{p\sim} \]

and, after transformations taking formulas (8) and (18) into account, we obtain

\[ I_{pm}=I_0\left(1+\sqrt{\frac{\kappa_1}{\kappa_2}}\right). \tag{23} \]

The values of the optimal inductance and of the maximum current in the reactor winding determine the volume of the reactor air gap, the maximum energy stored in the reactor, and its dimensions.

The reactor winding can be calculated from the effective value of the current, which, like \(I_{pm}\), is expressed only through the bias current \(I_0\) and the coefficients \(k_1\) and \(k_2\)

\[ I_{p\,\mathrm{eff}}=\sqrt{I_0^2+\frac{I_p^2}{2}} =I_0\sqrt{1+\frac{k_1}{2k_2}} . \tag{24} \]

Conclusions

1. During operation of the power-supply circuit of a betatron or synchrotron with DC biasing, an exchange of energies takes place between the magnetic fields of the reactor and of the accelerator.

2. The minimum cost of the power-supply circuit is determined by the ratio between the coefficients expressing the cost of accumulating 1 joule of energy in the inductive energy store—the reactor \((k_1)\)—and 1 joule of energy in the capacitive store—the capacitor \((k_2)\). There exists an optimal reactor inductance \((L_{p1})\) and an optimal coefficient of energy transfer from the reactor to the accelerator \((\lambda_1)\), corresponding to the minimum cost of the power-supply circuit as a whole.

3. The maximum reactor current, which together with the inductance \(L_{p1}\) determines the volume of its air gap, and the effective reactor current are likewise expressed only through the coefficients \(k_1\), \(k_2\) and the bias current \(I_0\). These quantities determine all the basic parameters of the power-supply circuit, the criterion for which is the condition of its minimum cost.

References

1. M. G. White. CERN Symposium, 1956.
2. S. Waaben. Nuclear Instrument and Methods, 9, 1960.
3. L. M. Anan’ev et al. Induction Electron Accelerator—Betatron. Gosatomizdat, Moscow, 1961.