# Effect of Harmonics on Ferroresonance in Low Voltage Power Factor Correction System--A Case Study
## 2 Feroresonance  
Feroresonance is a complex nonlinear phenomenon that occurs rarely and is difficult to predict [12; 16; 21; 32]. Feroresonance can occur in electrical networks where there is an inductive element that can go into saturation ( reactor, transformer), a capacitive element (capacitor, cables, long lines), an AC source, and low damping (light load or no load) [12; 21; 33]. In feroresonance, energy oscillates back and forth between the capacitive element and the nonlinear inductive element, which alternately saturates. As a result, large overcurrents and overvoltages occur with abnormal waveforms and high harmonic content, which are dangerous and can damage equipment. Since the core of the inductive element is made of ferromagnetic material, the inductance is nonlinear and decreases sharply in saturation, which increases the magnetizing current. The magnetizing current can include both even and odd harmonics and harmonics that are not multiples of the fundamental. When the core is driven into saturation, the nonlinear circuit can have multiple values of the inductances, so that feroresonance can occur for a wide range of system parameters (large range of values C), which is shown in Figure 1[14]. The voltage of the inductive element is described by:  
\[U_{L}=\frac{1}{\omega C}I+U, \tag{1}\]  
where \(U_{L}\) is the inductor voltage, \(I\) is the current in the series circuit, \(U\) is the supply voltage, \(\omega=2\pi f\) is the frequency of the supply voltage, and \(C\) is the capacitance. The frequency of currents and voltages in feroresonance may be different from the frequency of the source and need not be expressed in terms of the resonant frequency [12]. In feroresonance, there are multiple stable states for the same configuration and set of system parameters, one of which is the allowed state (linear behavior), and other states involve feroresonance [12; 16; 21]. For feroresonance to occur, feroresonance conditions must exist, and an event must occur that triggers a jump from an allowed steady-state  
Figure 1: Graphical view of feroresonance circuit. to a ferroresonance state. Various network events such as transients, switching on and off transformers and loads, switching of compensation stages, faults, etc., can trigger the ferroresonance. The ferroresonance is very sensitive to initial conditions (switching time, residual flux in the core, harmonics in the network, the initial voltage at the capacitor, etc.) and system parameter values that determine in which of the possible steady states the system will end [12; 14]. In the steady states of ferroresonance, the energy oscillations between the capacitance and the nonlinear inductance can take four different types, and the resonant frequency need not be specified [21; 34]. Possible modes of oscillation of the ferroresonance, i.e., possible steady states of the ferroresonance, are [12]  
fundamental oscillation,  
subharmonic oscillation,  
quasi-periodic oscillation,  
chaotic oscillation. The characteristics of the fundamental oscillation are shown in Figure 2. Voltages and currents are periodic signals with the same period as the fundamental components. The spectrum is discontinuous and consists of the fundamental and its multiples. The characteristics of the subharmonic mode are shown in Figure 3. Voltages and currents are periodic signals with a period that is a multiple of the period of the fundamental component. The spectrum is discontinuous and consists of submultiples of the fundamental, with the fundamental also being part of the spectrum. The characteristics of the quasi-periodic mode are shown in Figure 4. Voltages and currents are not periodic signals. The spectrum is discontinuous with frequencies that are not multiples. Figure 4: Ferroresonance characteristics for quasi-periodic oscillations. Figure 3: Ferroresonance characteristics for subharmonic oscillations. Figure 2: Ferroresonance characteristics for fundamental oscillations.