Matrix impedance

Torque-based impedance controller, x is the robot actual pose in the task space computed from the actual joint configuration q with the forward kinematics (FK) block J is the robot Jacobian is the desired pose in the task space x is the equilibrium pose of the environment is the net stiffness of the sensor and of the envirotunent f j and are the external enviroiunent forces expressed in the task space and in the joint space, respectively fj is the desired force vector is the desired torque vector computed from the force equilibrium r is the torque input vector of the inner torque control loop and is the commanded motor torque vector. The command force f is defined as f = Z(x - x), where Z is the impedance matrix. When the environmental forces are available (dotted lines), the measurements are used to decouple the dynamic of the system. 

Equations (10.206) and (10.207) are used to compute a square x n impedance matrix, where Zu are the diagonal terms and Z y are the symmetrical off-diagonal terms. Thus, for a four-wire overhead wye line section, it will result in a 4 x 4 matrix. 

The impedance matrix needs to be reduced to a 3 x 3 phase frame matrix consisting of the self- and mutual equivalent impedances for the three phases. A standard method of reduction is the Rron reduction where the assumption is made that the line has a multigrounded neutral. Each element of the reduced matrix is determined by the equation... 

Often, the analysis of a feeder uses the positive and zero sequence impedances for the line sections. The 3x3 sequence impedance matrix can be obtained by the following transformation ... 

The matrices (T]j, [L lj and [L2lj characterize the connection of the state vector Zj, the vector of the deflections of the foundation X and of the external forces Fj with the state vector Zj . These matrices can be Identified by seting up the equations of motion of the respective storey unit and rearranging these equations. Considering the Impedance matrix of the soil - based on the mechanical model shown above - and the boundary conditions for this system, the response of the structure can be computed e g. in the time domain by time step Integration. 

The characteristic impedance and admittance matrices are S5mimetrical matrices. For example, for the characteristic impedance matrix ... 

Let us define matrix P as a product of series impedance matrix Z and shunt admittance matrix 7 for a multiconductor system ... 

Thus, the sending-end voltage (Vj) is calculated by using the characteristic impedance matrix from the three-phase currents ... 

It is often observed in practice that the number of conductors changes at a boundary, as shown in Figure 1.53, where phases a and b are short-circuited at node 1. In the case of a cross-bonded cable, three-phase metallic sheaths are rotated at every cross-bonding point. In such a case, it is required to reduce the order of an impedance matrix and/or to rotate the matrix elements. 

Five towers are included in the model. The span distance of the transmission line between adjacent towers is 450 m, and that from tower No. 1 to the substation is 100 m. The end of the transmission line is terminated with the surge impedance matrix or, approximately, with matching resistances Rp = 350 Q for a PW and Rg = 560 Q for a GW. 

The surge impedance matrix [Z ] of the source circuit and [Z ] at the right of node r are... 

Each element of the impedance matrix is composed of the cable internal impedance and the cable outer media (earth-return) impedance, as explained in Chapter 1 of this volume. In the overhead line case, the conductor internal impedance is composed of only one impedance (i.e., the outer surface impedance of a conductor). The cable internal impedance consists of the following six components [1] ... 

The impedance matrix [Z"] and the admittance matrix [F ] are 4x4 matrices composed of three cores and a reduced single sheath. 

Because the propagation mode of the cable can be expressed by both -coaxial- and sheath-propagation mode, in a high-frequency region where skin depth is smaller than sheath thickness, the impedance matrix is composed of the following two submatrices ... 

Finally, the sheath impedance is the average of all the elements of the original sheath impedance matrix ... 

One cable system corresponds to six conductor systems composed of three cores and three metallic sheaths. As in the last section, the 6 x 6 impedance matrix of the cable system is represented by the following equation [1] ... 

By reducing the sheath conductors, the six-conductor system is reduced to a four-conductor system composed of three cores and one equivalent metallic sheath, as shown in Figure 3.7. The 4x4 reduced impedance matrix can be expressed as... 

The calculation process in the case of a cross-bonded cable using the proposed formulas is shown as follows (the 6x6 impedance matrix Z is obtained using cable constants [1,11,12]) ... 

The proposed formulas are known to have a satisfactory accuracy for planning and implementation studies. An acceptable level of error is introduced by the impedance matrix reduction discussed earlier. Owing to the matrix reduction, unbalanced sheath currents that flow into the earth at earthing joints are not considered in the proposed formulas. 

The capacitance matrix looks similar to the impedance matrix in Table 3.3a-2. The capacitance between the core and sheath of the homogeneous model is identical to that of the solidly bonded cable. The equivalent capacitance Q4 of the cross-bonded cable in Table 3.3b-2 is given as the sum of the elements as shown in Equations 3.44 and 3.45. 

The dynamic impedance model is based on earlier studies on machine foundation vibrations, in which, it is assumed that the response of rigid foundations excited by harmonic external forces can be characterized by the impedance or dynamic stiffness matrix for the foundation. The impedance matrix depends on the frequency of excitation, the geometry of foundation and the properties of the underlying soil deposit. 

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