## 2 Spectral Components of Brushed DC Motor Current
### Other Spectral Components  
Brushed dc motors are composed of two circuits: the armature circuit and the field circuit. The interaction of the magnetic field produced by each circuit develops torque on the rotor, causing rotation. The rotation is produced according to [PERSON]'s law and the Lorentz force law. The back-EMF is produced according to [PERSON]'s law and the torque is developed according to the Lorenz force law ([PERSON] et al., 2017). According to [PERSON]'s law, the back-EMF induced in coil \(i\) of a brushed dc motor is:  
\[V_{i}\ =\ -(d\phi_{i}/dt) \tag{3}\]  
where the magnetic flux in the coil \(i\), \(\phi_{i}\), is,  
\[\phi_{i}=\iint_{S_{i}}\mathbf{B}\cdot\mathbf{da} \tag{4}\]  
and \(\mathbf{B}\) is the magnetic flux density, \(\mathbf{da}\) is the differential area element, and \(S_{i}\) is the area of the coil \(i\). The brush voltage in the brushed dc motor, \(V\), is the sum of the rectified induced voltage of every coil:  
\[V=\sum_{i}\lvert V_{i}\rvert \tag{5}\]  
Because each coil and pole in a brushed dc motor is unique, the induced voltage associated with each coil, \(V_{i}\), has a slightly different shape compared to the voltage of other coils. Furthermore, the phases are also different because the position of every coil in the rotor is different. Because of this, and according to Equation (5), the back-EMF at the brushes, \(V\), has a characteristic shape that is repeated at every rotation of the rotor. The period of this repetition is the inverse of the number of revolutions per second of the motor, and its frequency is the motor speed in number of revolutions per second. The Fourier series can be used to decompose \(V\) as:  
\[V(t)=\sum_{l=0}^{\infty}A_{l}\cos(2\pi l(n/60)t+\theta_{l}) \tag{6}\]  
where \(l\) is the index of the Fourier series, \(A_{l}\) and \(\theta_{l}\) are the coefficients of the Fourier series, \(n/60\) is the revolutions per second of the motor, and \(n\) is the revolutions per minute ([PERSON] et al., 2017). The Fourier transform of Equation (6) results in:  
\[V(f)=\sum_{l=0}^{\infty}B_{l}\delta(f-l(n/60)) \tag{7}\]  
where \(f\) is the frequency, \(B_{l}\) is the complex coefficient that depends on the coefficients of the Fourier series \(A_{l}\) and \(\theta_{l}\), and \(\delta(\cdot)\) is the Dirac delta function. According to Equation (7), the spectrum of the voltage has different spectral components, with a component at each of the following frequencies:  
\[f_{l}=l(n/60)\]and the distance between components is equal to:  
\[df_{m}=n/60 \tag{9}\]  
As pointed out in Section 2, the current and voltage are related, so their spectra are also related. Therefore, every component in the voltage spectrum is also present in the current spectrum, and there is an equivalent expression from Equations (6) to (9) available for the current.  
The frequency of the ripple component is in a multiple of \(n/60\) according to Equation (2), thus, it coincides with a frequency produced by the method explained in this subsection according to Equation (8). The ripple frequency corresponds to the index \(l=\ (2p\cdot k)/\eta\) and it is the main component related to the motor speed.  
Because each \(V_{i}\) is only slightly different from the rest, the variations in the voltage and current are small, and the weights of the coefficients \(A_{l}\) are also small.  
The spectral components produced by this effect can be easily detected in the current when the amplitude of the ripple component is similar to the amplitude of these components. This is the case for brushed dc motors with a large number of coils, such as integral horsepower brushed dc motors.  
An example of the current amplitude spectrum of a real integral horsepower brushed dc motor is shown in Figure 2.