# Precision Analysis and Design of Rotating Coil Magnetic Measurements System
## 2 Mathematical Modeling  
Measuring the magnetic field using a rotating coil technique is based on [PERSON]'s law of induction. It makes a rectangular magnetic field measuring instruments with width (H) and length (L) known as a \"search coil\" or \"harmonic coil\" as shown in Figure 1.  
From Figure 1, the electromagnetic force can be calculated as Equation (1) below:  
\[V\ =-\frac{d\Phi}{dt}\Rightarrow\Phi=-\int V\cdot dt, \tag{1}\]  
where \(V\) is the electromotive force of the search coil (m), and \(\Phi\) is the magnetic flux (weber).  
Figure 1: Rotating coil: (**a**) search coil; (**b**) orbit of a dipole magnetic field.  
The magnetic flux can be calculated from Equation (2):  
\[\Phi(\theta)=N\int\limits_{\sum_{col}}B\cdot da=N\cdot L\int\limits_{R1}^{R2}B_{ \theta}\cdot dr, \tag{2}\]  
where \(B\) is the magnetic strength (tesla), \(N\) is the number of turns, \(L\) is the length of the search coil (m), \(R_{1}\) is radial of the rotation of the inner coil (m), and \(R_{2}\) is the radial of the rotation of the outer coil (m).  
The search coil's surface can be calculated from Equation (3):  
\[\Sigma_{coil}=N\cdot L\cdot\int\limits_{R_{1}}^{R_{2}}dr=N\cdot L\cdot(R_{2}-R _{1}), \tag{3}\]  
The flux component obtained by measuring the magnetic field can be represented by Equation (4):  
\[\Phi(\theta)=\sum\limits_{n=1}^{\infty}N\cdot L\cdot\frac{R_{2}^{n}-R_{1}^{n}} {n\cdot R_{0}^{n-1}}\cdot[B_{n}(R_{0})\cos n\theta-A_{n}(R_{0})\sin n\theta], \tag{4}\]  
where \(B_{n}\) is the normal dipole coefficients of the field, and \(A_{n}\) is skew dipole coefficients of the field.  
From the abovementioned principle, measuring the intensity of the magnetic field using a rotating coil technique, one can measure the signal as a sinusoidal waveform as shown in Figure 2 below. It is a simulation of the flux obtained by measuring the magnetic field in the vertical axis (normal dipole) and horizontal axis (skew dipole).  
Figure 2: (**a**) Normal dipole signal; (**b**) skew dipole signal.  
From Equation (1) and Figure 2, the electromotive force obtained from the search coil can be measured as in Equation (5):  
\[V(t)=-\frac{d}{dt}\Biggl{\{}\sum_{n=1}^{\infty}N\cdot L\cdot\frac{R_{2}^{n}-R_{1} ^{n}}{n\cdot R_{0}^{n-1}}\cdot[B_{n}(R_{0})\cos n\theta-A_{n}(R_{0})\sin n\theta ]\Biggr{\}}, \tag{5}\]  
\[V(t)=\sum_{n=1}^{\infty}N\cdot L\cdot\omega\cdot\frac{R_{2}^{n}-R_{1}^{n}}{R_{ 0}^{n-1}}\cdot[B_{n}(R_{0})\sin\theta-A_{n}(R_{0})\cos n\theta], \tag{6}\]  
where \(\theta=\omega\cdot t+\delta\).  
From Equation (6), it can be noticed that the electromotive force obtained from the search coil directly varies with the rotating speed of the coil. Magnetic field measurements with this technique can result in an approximately 0.1% error in rotation. Therefore, it is necessary to have a constant speed control system for the search coil, because the electromotive force varies directly with the speed used in the rotation. This study used a Direct Current (DC) motor to drive the packed coil into the cylinder rod and used a closed-loop control system with the PI controller. A block diagram of the speed control system of the rotating coil is shown in Figure 3, and the PI controller is shown in Equation (7):  
\[\textit{Controller}=K_{p}+\frac{K_{i}}{s}, \tag{7}\]  
where \(K_{p}\) is the proportional control, and \(K_{i}\) is the integral control.