R 十 v1R15= FIGURE 7. 1 An active network used to illustrate the procedure for writing the state equations in normal for Example 1 e write the state equations for the network N of Fig. 7. 1 by following the eight steps outlined above. Step I The voltage and current references of the branches of the active network n are as indicated in Fig.7.1 Step Select a normal tree T consisting of the branches Rp, C3, and ve. The subnetwork Cisv is another example of a normal tree Step 3 The tree branches R, C3, and v, are assigned the voltage symbols vp, v, and ve; and the cotree -links R2, Lep is and i are assigned the current symbols i, i4, i3,, and i, respectively. The controlled current source isis given the current symbol i because its current is controlled by the current of the branch 3, which is i3 Step 4 Applying Kirchhoff's current law, the branch currents in, i,, and i, can each be expressed as the sums of cotree (7.5a) (7.5b) (7.5c) Applying Kirchhoff's voltage law, the cotree-link voltages v2, v4, Vs, and v can each be expressed as the sums of tree-branch voltages V2=18-V3 (7.6a) (7.6b) e 2000 by CRC Press LLC© 2000 by CRC Press LLC Example 1 We write the state equations for the network N of Fig. 7.1 by following the eight steps outlined above. Step l The voltage and current references of the branches of the active network N are as indicated in Fig. 7.1. Step 2 Select a normal tree T consisting of the branches R1, C3 , and vg . The subnetwork C3i5vg is another example of a normal tree. Step 3 The tree branches R1, C3 , and vg are assigned the voltage symbols v1, v3, and vg ; and the cotree-links R2 , L4, i5 , and ig are assigned the current symbols i2 , i4 , i3 , and i g, respectively. The controlled current source i5 is given the current symbol i3 because its current is controlled by the current of the branch C3 , which is i3. Step 4 Applying Kirchhoff’s current law, the branch currents i1, i3, and i7 can each be expressed as the sums of cotree￾link currents: i1 = i4 + ig – i3 (7.5a) i3 = i2 – i4 (7.5b) i 7 = –i2 (7.5c) Step 5 Applying Kirchhoff’s voltage law, the cotree-link voltages v2 , v4 , v5, and v6 can each be expressed as the sums of tree-branch voltages: v2 = vg – v3 (7.6a) v4 = v3 – v1 (7.6b) FIGURE 7.1 An active network used to illustrate the procedure for writing the state equations in normal form