# A Method for Broken Rotor Bars Diagnosis Based on Sum-Of-Squares of Current Signals
## 2 Modulation Effect of Broken Rotor Bars on the Current Signal  
In this section, a mathematical model is established to explain the modulation mechanism of current signals, which is caused by BRBs, by improving the signal model proposed in [29; 30]. When the motor is operated under healthy conditions, its current signals of phase A can be ideally expressed as  
\[i_{A}=\sqrt{2}l_{0}\cos(2\pi f_{e}t+\varphi_{i}) \tag{1}\]  
where \(l_{0}\) is the root mean squared amplitude of the current signal, \(f_{e}\) is the fundamental frequency, and \(\varphi_{i}\) is the initial phase of the current signal. Accordingly, the magnetic fluxes can be expressed as  
\[\phi_{A}=\sqrt{2}\phi\cos(2\pi f_{e}t+\varphi_{\phi}) \tag{2}\]  
where \(\phi\) is the root mean squared amplitude and \(\varphi_{\phi}\) is the initial phase of the magnetic flux. When the motor rotors are broken, the stator current causes a small additional periodic current component with a frequency \(f_{brb}=2g_{e}\)[31; 32]. The additional current can be expressed as  
\[i_{brb}=\sqrt{2}l_{brb}\cos(2\pi f_{brb}t-\varphi_{brb}) \tag{3}\]  
where \(l_{brb}\) is the root mean squared amplitude and the \(\varphi_{brb}\) is the initial phase of the additional current. The additional current causes the motor to produce torque ripple and it can be expressed as  
\[\Delta T=3p\phi l_{brb}\sin\Bigl{(}2\pi f_{brb}t-\varphi_{\phi}+\varphi_{brb} \Bigr{)} \tag{4}\]  
where \(p\) is the number of pole pairs. According to the dynamic formula \(J\frac{d\Delta\omega(t)}{dt}=\Delta T\)  
\[\Delta\theta(t)=\int\Delta\omega(t)\ dt=-\frac{3p\phi l_{brb}}{4\pi^{2}f_{brb}^ {2}J}\sin\Bigl{(}2\pi f_{brb}t-\varphi_{\phi}+\varphi_{brb}\Bigr{)} \tag{5}\]  
where \(J\) is the inertia of the motor rotor and the rotor system. Thus, the linkage flux can be derived as  
\[ \begin{array}{ll}\phi_{Abrb}(t)&=\sqrt{2}\phi\cos[2\pi f_{e}t-\Delta\theta( t)]\\ &=\sqrt{2}\phi\cos\Bigl{[}2\pi f_{e}t-\frac{3p\phi l_{brb}}{4\pi^{2}f_{brb}^{2} J}\sin\Bigl{(}2\pi f_{brb}t-\varphi_{\phi}+\varphi_{brb}\Bigr{)}\Bigr{]}\end{array}  \tag{6}\]  
Calculating the derivative of \(\phi_{Abrb}(t)\) with respect to \(t\), the fundamental electromotive force can be obtained as  
\[ \begin{array}{ll}E_{Abrb}(t)=&-\sqrt{2}\phi\Bigl{[}2\pi f_{e}-\frac{3p\phi l _{brb}}{2\pi f_{brb}J}\cos\Bigl{(}2\pi f_{brb}t-\varphi_{\phi}+\varphi_{brb} \Bigr{)}\Bigr{]}\\ &\cdot\sin\Bigl{[}2\pi f_{e}t-\frac{3p\phi l_{brb}}{4\pi^{2}f_{brb}^{2}J}\sin \Bigl{(}2\pi f_{brb}t-\varphi_{\phi}+\varphi_{brb}\Bigr{)}\Bigr{]}\end{array}  \tag{7}\]  
Assuming that the equivalent impedance of the motor winding is \(z=Ze^{\varphi_{z}}\), the fundamental frequency of stator current of the motor can be deduced as follows:  
\[ \begin{array}{ll}i_{Abrb}(t)=&-\frac{\sqrt{2}\phi_{0}}{Z}\cdot\Bigl{[}2\pi f _{e}-\frac{3p\phi l_{brb}}{2\pi f_{brb}J}\cos\Bigl{(}2\pi f_{brb}t-\varphi_{ \phi}+\varphi_{brb}\Bigr{)}\Bigr{]}\\ &\cdot\sin\Bigl{[}2\pi f_{e}t-\frac{3p\phi l_{brb}}{4\pi^{2}f_{brb}^{2}J}\sin \Bigl{(}2\pi f_{brb}t-\varphi_{\phi}+\varphi_{brb}\Bigr{)}\Bigr{]}\end{array}  \tag{8}\]  
The torque ripple, which is caused by BRBs, modulates the amplitude and phase of the fundamental frequency component of the three-phase current signals. The extra current \(i_{brb}\) generated by BRBs is a tiny component compared with the current fundamental frequency, and it is reasonable to ignore. Ignoring the initial phase and extra current, the stator current in the fault state can be expressed as  
\[i_{Abrb}(t)=I(t)\cdot\sin[2\pi f_{e}t+\beta\sin(2\pi f_{brb}t)] \tag{9}\]  
where \(I(t)=-\frac{\sqrt{2}\phi_{0}}{Z}\cdot\Bigl{[}2\pi f_{e}-\frac{3p\phi l_{brb} }{2\pi f_{brb}J}\cos(2\pi f_{brb}t)\Bigr{]}\) and \(\beta=-\frac{3p\phi l_{brb}}{4\pi^{2}f_{brb}^{2}J}\). It can be seen that \(I(t)\) contains the signature frequency components. Similarly, \(i_{Bbrb}(t)\) and \(i_{Cbrb}(t)\) can be derived. Therefore, the current signals of the faulty motor with BRB can be expressed as  
\[\left\{