We study the effect of a passive harmonic cavity, introduced to cause bunch lengthening, in an electron storage ring. We derive a formula for the induced voltage from such a cavity with high $Q$, excited by a sequence of bunches, allowing for arbitrary gaps in the sequence and arbitrary currents. Except for a minor term that can be determined iteratively, the voltage is given in terms of a single mode of the Fourier transforms of the bunch forms, namely the mode at the resonant frequency of the cavity. Supposing that the only wake field is from the harmonic cavity, we derive a system of coupled Ha"issinski equations which determine the bunch positions and profiles in the equilibrium state. The number of unknowns in the system is only twice the number of bunches, and it can be solved quickly by a Newton iteration, starting with a guess determined by path-following from a solution at weak current. We explore the effect of the fill pattern on the bunch lengthening, and also the dependence on the shunt impedance and detuning of the cavity away from the third harmonic of the main accelerating cavity. We consider two measures to reduce the effects of gaps: 1)~ distribution of the gaps around the ring to the greatest extent allowed, and 2)~" guard bunches" with higher charges adjacent to the gaps, compensating for the charge missing in gaps. Results for parameters of the forthcoming ALS-U light source are presented.

中文翻译：

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无源谐波腔中任意束链的平衡：通过耦合Haïssinski方程的解

我们研究了在电子存储环中引入的无源谐波腔的影响，该腔会引起束加长。我们推导了这样一个具有高谐振腔的感应电压的公式$问$，由一系列束激励，允许序列中有任意间隙和任意电流。除了可以迭代确定的次要术语外，电压是根据束形式的傅立叶变换的单一模式（即谐振腔共振频率下的模式）给出的。假设唯一的唤醒场来自谐波腔，我们导出了一个耦合的Ha“ issinski方程组，该方程组确定了平衡状态下的束位置和轮廓。系统中的未知数仅为束数的两倍，并且可以通过牛顿迭代快速解决问题，首先从在弱电流下的解决方案进行路径跟踪确定猜测，然后研究填充模式对束延长的影响，以及对分流阻抗的依赖以及腔远离主加速腔的三次谐波的失谐。我们考虑了两种措施来减小间隙的影响：1）〜在环周围最大程度地分配间隙，以及2）〜在间隙附近具有较高电荷的“保护束”，以补偿间隙中丢失的电荷。给出了即将到来的ALS-U光源的参数结果。