# Analysis of Underlapped Symmetrically Ported Valve-Controlled Asymmetric Cylinder Drive
## 2 Theory
### Valve Model  
The control valve (electrohydraulic valve) is a Moog 76 model 102 flow control four-way servovalve. If current is applied to the armature coil in the torque motor, the T-shaped  
Figure 1: Schematic circuit of the Moog valve-controlled asymmetric cylinder system [8].  
Figure 2: Power transfer route.  
flapper rotates slightly. Assuming the flapper rotates clockwise, the offset between the left nozzle and flapper decreases, which leads to the decrease of flow passing through the left nozzle and causes the pressure to increase in the left control line. A similar process occurs in the right control line, leading to a decrease of pressure.  
The pressure difference in the control line moves the spool, therefore bends the feedback spring to balance the torque generated by the torque motor, so that the spool displacement is dependent on the rotation of the torque motor; furthermore, the spool displacement is dependent on the current.  
Appropriate transfer functions, Equations (1) and (2), are given by [PERSON] [12]. The expressions are linear, empirical relationships that approximate the responses of the actual servovalve. For the frequency range below 50 Hz, the valve dynamic can be approximated by a first-order transfer function given below [12]:  
\[\frac{Q(s)}{I(s)}=K\frac{1}{1+\tau_{s}s} \tag{1}\]  
where \(K\) is the flow gain \(\left[\frac{\text{m}^{3}}{s\cdot\text{amp}}\right]\) and \(\tau_{s}\) is the servovalve time constant \([\text{sec}]\).  
A second-order transfer function is necessary to represent a wider frequency range response as given below [12]:  
\[\frac{Q(s)}{I(s)}=K\frac{1}{1+\left(\frac{2\zeta}{\omega_{n}}\right)s+\left( \frac{s}{\omega_{n}}\right)^{2}} \tag{2}\]  
where \(\omega_{n}\) is the natural frequency [rad/s], and \(\zeta\) is the damping ratio. In a limit cycle oscillation study of a low-frequency phenomenon, a higher-order valve model does not show significant difference compared to a lower-order model [13]. [PERSON] [14] stated that a reduced-order transfer function can be used to approximate a higher-order transfer function, and a first-order model is used as a reference model by [PERSON] et al. [15].  
As all of the valve operational frequencies in the industrial sector are much lower than 50 Hz, the first-order transfer function Equation (1) is chosen for the valve model. As the displacement of the spool, \(X(s)\), controls the flow rate of the valve, Equation (1) can be modified as follows:  
\[\frac{X_{s}(s)}{I(s)}=K_{s}\frac{1}{1+\tau_{s}s} \tag{3}\]  
where \(K_{s}\) is the gain of the system [\(\frac{\text{m}}{\text{amp}}\)].  
The current is generated by an amplifier. Assuming the amplifier is ideal, then current is proportional to the input voltage as given by Equation (4):  
\[\frac{I(s)}{V_{v}(s)}=K_{A} \tag{4}\]  
where \(K_{A}\) is the gain of voltage to current [\(\frac{\text{amp}}{\text{volt}}\)]. Hence the valve dynamics are represented by:  
\[\frac{X_{s}(s)}{V_{v}(s)}=\frac{K_{A}K_{s}}{1+\tau_{s}s} \tag{5}\]  
A block diagram for this valve model is depicted in Figure 3. Valve model block diagram.  
Figure 3: Valve model block diagram.