Direct Determination of Z_{bus} Matrix
We shall now use the methods given in Sections 3.3.1 to 3.3.4 for the direct determination of the Z_{bus} matrix without forming the Y_{bus} matrix first. To accomplish this we shall consider the system of Fig. 3.2 and shall use the system data given in Example 3.1. Note that for the construction of the Z bus matrix we first eliminate all the voltage sources from the system.
Step1 : Start with bus1. Assume that no other buses or lines exist in the system. We add this bus to the reference bus with the impedance of j 0.25 per unit. Then the Z_{bus} matrix is

(3.43) 
Step2 : We now add bus2 to the reference bus using (3.31). The system impedance diagram is shown in Fig. 3.10. We then can modify (3.43) as

(3.44) 
Fig. 3.10 Network of step2.
Step3 : We now add an impedance of j 0.2 per unit between buses 1 and 2 as shown in Fig. 3.11. The interim Z bus matrix is then obtained by applying (3.41) on (3.44) as
Eliminating the last row and last column using the Kron's reduction of (3.31) we get

(3.45) 
Step4 : We now add bus3 to bus1 through an impedance of j 0.25 per unit as shown in Fig. 3.12. The application of (3.35) on (3.45) will then result in the following matrix
Fig. 3.11 Network of step3.

(3.46) 
Fig. 3.12 Network of step4.
Step5 : Connect buses 2 and 3 through an impedance of j 0.4 per unit as shown in Fig. 3.13. The interim Z_{bus} matrix is then formed from (3.41) and (3.46) as
Fig. 3.13 Network of step5.
Using the Kron's reduction we get the following matrix

(3.47) 
Step6 : We now add a new bus4 to bus2 through an impedance of j 0.5 as shown in Fig. 3.14. Then the application of (3.35) on (3.47) results in the following matrix

(3.48) 
Fig. 3.14 Network of step6.
Step7 : Finally we add buses 3 and 4 through an impedance of j 0.4 to obtain the network of Fig. 3.3 minus the voltage sources. The application of (3.41) on (3.48) results in the interim Z_{bus} matrix of
Eliminating the 5^{th} row and column through Kron's reduction we get the final Z_{bus} as

(3.49) 
The Z_{bus} matrix given in (3.49) is the as that given in Example 3.1 which is obtained by inverting the Y_{bus} matrix.
