# Review: A Survey on Configurations and Performance of Flow-Mode MR Valves
## 3 Fundamentals  
Here, we discuss the MR valve's operating principle, highlight basic fundamental relationships, and then propose a simple shear stress metric for the valve's performance characterization and comparison. Let us first consider the simple MR valve (electromagnet--EM) in Figure 5. The electromagnet produces a magnetic field of required strength. It consists of a core of specific dimensions, air gap(s), coil winding(s), and a ring. Contrary to other electromagnets the ones used in flow-mode actuators contain no moving parts. The EM's fixed-height air gap(s) form the flow channels. The air gap length is \(L_{g}\), and its size \(h=(D_{2}-D_{c})/2\). The coil window dimensions are \(W_{c}\times H_{c}\) (\(W_{c}\)--width, \(H_{c}\)--height), and it fits \(N_{c}\) number of wire turns. The outer diameter of the electromagnet is \(D_{p}\). The current in the coil results in \(N_{c}l\) ampere turns. Considering the EM's circuit, the following simple relationship can be obtained. Figure 4: Layout: valve decomposition. Figure 5: MR electromagnet (valve): 1—core, 2—coil, 3—annulus, 4—ring. Figure 3: Layout: actuator decomposition. \[\sum_{i}H_{i}l_{i}=N_{c}I \tag{1}\]  
As the annulus is the element of the largest reluctance, the above equation can be reduced as follows  
\[2H_{g}h\approx N_{c}I \tag{2}\]  
and \(B_{g}=\mu_{r}(B_{g})\mu_{0}H_{g}\). Then,  
\[2h\frac{B_{g}}{\mu_{r}(B_{g})\mu_{0}}-N_{c}I=0 \tag{3}\]  
where \(\mu_{r}\)--relative permeability of the fluid, \(\mu_{0}\)--absolute permeability (vacuum), \(l_{i}\)--length of the \(i\)th magnetic circuit section with a constant cross-section area. By solving the equation, the average gap flux density \(B_{g}\) can be computed, \(H_{g}\)--gap magnetic field strength. Using Gauss law, the magnetic flux \(\phi\) can be obtained  
\[\phi=B_{g}A_{m}=B_{g}\pi D_{m}L_{a}=B_{g}\pi(D_{c}+h)L_{a} \tag{4}\]  
where \(A_{m}\)--active surface (of one pole), \(D_{m}=D_{c}+h\). The obtained value of \(B_{g}\) is yet to be verified for magnetic saturation; the valve should not saturate under any circumstances or increasing the current in the coil will result in little or no effect. Given the core and the coil dimensions, the circuital resistance is  
\[R_{c}=\frac{\rho_{c}l_{w}}{A_{w}}=\frac{\rho_{c}4N_{c}(D_{c}-H_{c})}{d_{w}^{2}} \tag{5}\]  
and it is subject to the constraint \(U_{c}=1R_{c}<U_{bat}\), \(U_{c}\)--voltage drop, \(U_{bat}\)--supply voltage, \(I\)--circuital (coil) current, \(\rho_{c}\)--copper resistivity, \(A_{w}\)--single wire cross-section area, \(l_{w}\)--wire length. Let us then consider the simplest approximation of the Bingham plastic model usually used for modeling the output of MR (and ER) values [71; 72]  
\[\Delta P=\Delta P(B_{g},v)=\left\{ \begin{array}{ll}12\frac{\mu L_{g}}{b!}Q&B= 0\\ 4\frac{\tau_{0}(B_{g})L_{a}}{h}\operatorname{sgn}(Q)+12\frac{\mu L_{g}}{bh^{3 }}Q&|B|\
eq 0\end{array} \right. \tag{6}\]  
and the force \(F_{d}=A_{p}\Delta P\), \(\Delta P\)--pressure difference across the valve, \(Q=A_{p}v\)--flow rate, \(A_{p}\)--cross-section area (valve), \(b\)--gap width, \(v\)--velocity, \(\tau_{0}\)--yield stress. Thus, the (absolute) dynamic range at the reference velocity \(v_{ref}\) is given as follows  
\[\Delta F=F_{d}(B_{max},v_{ref})-F_{d}(0,v_{ref}) \tag{7}\]  
or the turn-up ratio \(K_{f}\) is  
\[K_{F}=\frac{F_{d}(B_{max},v_{ref})}{F_{d}(0,v_{ref})} \tag{8}\]  
Using the above simple or far more advanced formulas various optimization studies can then be performed in order to meet specific application criteria (range, force envelope, etc.) [73; 74]. While studying the simple model of the electromagnet in the context of the other configurations such as multiple gap valves, it should be appreciated that certain designs of the MR valves may present a serious challenge if one is to achieve a particular force envelope. For example, in a valve having \(N\) parallel annular flow paths, the flux is required to travel across \(2N\) air gaps, thus requiring an amplified number of ampere turns to meet the specified performance objectives.