ELECTRICAL CIRCUITS REDUCTION ON THE BASIS OF Y-∆ TRANSFORMATION (RLC REDUCTION PROCESS)

The essence of methods based on Y-∆ transformation is the following. Let i-th node and its k neighbors be situated as it is shown on Fig. 1. Then, the component equation of i-th line has the following form:

, (1)

where .

For excluding from (1) which is equal to excluding i-th node, can be defined as:

. (2)

Lets replace by (2) in k equations where it is present. Let us consider the first node which will be neighboring to the i-th. Its equation is:

, (3)

where is a sum of all capacitances of node 1 excluding the i-th node, k1 is a number of nodes connected to node 1.
Equation (3) can be simplified:

.

It should be mentioned that it is equal to adding k-1 new elements between the 1st node and k-1 former neighbors of i-th node on Fig. 1.

For any two nodes neighbouring to i-th one, e.g. aand b, excluding of i-th node results in adding a new element between these nodes. The capacitance of the new element will be:

. (4)

Thus, by repeating this process for all k neighbours of i-th node we will remove it by this. For every node in the circuit, two time constants are defined:

and

where is a sum of all capacitances is, is a sum of all reactive conductances (the value reverse to inductance), is a sum of all conductance connected to node i.
Node constant of i node time is defined as:

.

i-th node is considered to be fast if

where is a time constant defined by user which depends on maximal circuit frequency - . It should be noted that is proportional to .

On practice, low eigenfrequencies which are working frequencies for mechanical systems usually are of some interest. That is why by variation a compromise between accuracy and dimension of the models received can be reached.

The process of RLC reduction requires:

• Circuit description including only R, L and C components;
• The list of non-reducing nodes numbers;
• The value of MAX_TAU.

Lets consider a fixed uniform beam with one degree of freedom as an example. Beam equivalent circuit is shown on Fig. 2. For the model given, the following element values were received:

• C1..C99 = -0.11667 F,
• C100..C199 = 0.7 F,
• L1..L99 = 5E-12 H,
• L100 = 5E-12 H.

Reduction results are shown in Table 1, frequency analysis for range of 100 Hz - 10 kHz on Figs. 3-6. Reduced circuit descriptions in NetALLTED input language at different values are shown in Table 2. In schematic design, reduced beam with one degree of freedom model can be represented in the same way as the equivalent circuit for this beam (Fig. 2), but with the smaller amount of sections and with the other element values.

Table 1. Reduction results

Configuration file looks like:
MAX_TAU 3e-5
KEEP 0
KEEP 100

Simulation file for =3*10^-5 looks like:
Task
Option 47, 48;
&&
Object
Circuit Beam;
J1(100,0) = -100;
C_1(82,100) = -0.116667;
L_10(100,0) = -34161.8;
L_11(23,50) = 1.35e-10;
L_12(23,0) = 1.15e-10;
L_13(50,0) = -16435.9;
L_14(50,82) = 1.6e-10;
C_2(100,0) = 6.3;
C_3(23,50) = -0.116667;
C_4(23,0) = 17.3833;
C_5(50,82) = -0.116667;
C_6(0,50) = 20.65;
C_7(0,82) = 17.5;
L_8(82,100) = 9e-11;
L_9(82,0) = -580750;
&&
task
dc;
ac;
tf K1 = UJ1/IJ1;
lplot db.K1, ph.K1(1000);
const lfreq = 100, ufreq = 10000;
&&
end

Fig. 4. AFC, PFC of equivalent circuit for fixed uniform beam with one degree of freedom (reduced variant, = 3*10^-5)

Fig. 5. AFC, PFC of equivalent circuit for fixed uniform beam with one degree of freedom (reduced variant, = 10^-5)

Fig. 6. AFC, PFC of equivalent circuit for fixed uniform beam with one degree of freedom (reduced variant, = 5*10^-6)

Table 2. Reduced circuit descriptions in NetALLTED input language

C_1(82,100) = -0.116667;
L_10(100,0) = -34161.8;
L_11(23,50) = 1.35e-10;
L_12(23,0) = 1.15e-10;
L_13(50,0) = -16435.9;
L_14(50,82) = 1.6e-10;
C_2(100,0) = 6.3;
C_3(23,50) = -0.116667;
C_4(23,0) = 17.3833;
C_5(50,82) = -0.116667;
C_6(0,50) = 20.65;
C_7(0,82) = 17.5;
L_8(82,100) = 9e-11;
L_9(82,0) = -580750;

C_1(23,31) = -0.116667;
C_10(0,74) = 5.6;
C_11(82,90) = -0.116667;
C_12(0,82) = 5.6;
C_13(90,100) = -0.116667;
C_14(90,0) = 6.3;
C_15(100,0) = 3.5;
C_16(50,39) = -0.116667;
C_17(50,0) = 6.65;
C_18(39,0) = 6.65;
C_19(8,23) = -0.116667;
C_2(31,39) = -0.116667;
C_20(8,0) = 7.93333;
C_21(23,0) = 8.05;
L_22(23,31) = 4e-11;
L_23(31,0) = -1.78957e+08;
L_24(31,39) = 4e-11;
L_25(50,58) = 4e-11;
L_26(58,66) = 4e-11;
L_27(66,74) = 4e-11;
L_28(82,74) = 4e-11;
L_29(82,90) = 4e-11;
C_3(0,31) = 5.6;
L_30(90,100) = 5e-11;
L_31(90,0) = -322639;
L_32(100,0) = -35848.8;
L_33(50,39) = 5.5e-11;
L_34(50,0) = -16900.1;
L_35(39,0) = -354903;
L_36(8,23) = 7.5e-11;
L_37(8,0) = 4e-11;
L_38(23,0) = -90611.1;
C_4(50,58) = -0.116667;
C_5(58,66) = -0.116667;
C_6(0,58) = 5.6;
C_7(66,74) = -0.116667;
C_8(0,66) = 5.6;
C_9(82,74) = -0.116667;
      
      

C_1(15,19) = -0.116667;
C_10(35,39) = -0.116667;
C_11(0,35) = 2.8;
C_12(43,39) = -0.116667;
C_13(0,39) = 2.8;
C_14(50,54) = -0.116667;
C_15(54,58) = -0.116667;
C_16(0,54) = 2.8;
C_17(58,62) = -0.116667;
C_18(0,58) = 2.8;
C_19(62,66) = -0.116667;
C_2(23,19) = -0.116667;
C_20(0,62) = 2.8;
C_21(66,70) = -0.116667;
C_22(0,66) = 2.8;
C_23(4,8) = -0.116667;
C_24(4,0) = 2.68333;
C_25(70,74) = -0.116667;
C_26(0,70) = 2.8;
C_27(74,78) = -0.116667;
C_28(0,74) = 2.8;
C_29(82,78) = -0.116667;
C_3(0,19) = 2.8;
C_30(0,78) = 2.8;
C_31(82,86) = -0.116667;
C_32(0,82) = 2.8;
C_33(86,90) = -0.116667;
C_34(0,86) = 2.8;
C_35(90,94) = -0.116667;
C_36(0,90) = 2.8;
C_37(94,100) = -0.116667;
C_38(94,0) = 3.5;
C_39(100,0) = 2.1;
C_4(23,27) = -0.116667;
C_40(15,8) = -0.116667;
C_41(15,0) = 3.85;
C_42(8,0) = 3.85;
C_43(43,50) = -0.116667;
C_44(43,0) = 3.85;
C_45(50,0) = 3.85;
L_46(4,0) = 2e-11;
L_47(70,74) = 2e-11;
L_48(74,78) = 2e-11;
L_49(82,78) = 2e-11;
C_5(23,0) = 2.8;
L_50(82,86) = 2e-11;
L_51(86,90) = 2e-11;
L_52(90,94) = 2e-11;
L_53(94,100) = 3e-11;
L_54(94,0) = -193583;
L_55(100,0) = -38716.7;
L_56(15,8) = 3.5e-11;
L_57(15,0) = -42378.2;
L_58(8,0) = -85171.3;
L_59(43,50) = 3.5e-11;
C_6(27,31) = -0.116667;
L_60(43,0) = -225847;
L_61(50,0) = -17372.9;
L_62(15,19) = 2e-11;
L_63(23,19) = 2e-11;
L_64(23,27) = 2e-11;
L_65(23,0) = -5.36871e+07;
L_66(27,0) = -8.94785e+07;
L_67(27,31) = 2e-11;
L_68(31,35) = 2e-11;
L_69(35,39) = 2e-11;
C_7(0,27) = 2.8;
L_70(43,39) = 2e-11;
L_71(50,54) = 2e-11;
L_72(54,58) = 2e-11;
L_73(58,62) = 2e-11;
L_74(62,66) = 2e-11;
L_75(66,70) = 2e-11;
L_76(4,8) = 2e-11;
C_8(31,35) = -0.116667;
C_9(0,31) = 2.8;