Chapter 1 Mesh and Nodal Equations for an Active Circuit

1. Mesh Equations for an Active Circuit

To generalize equation (102) to fit a circuit containing vacuum tubes, we may suppose that only one of the E's on the right-hand side of (1-2) is an actual driving voltage and that the remaining E's are apparent plate generators representing the amplifications of the tubes. For example, in one particular tube, let us suppose that the jth mesh current flows from grid to cathode and the kth mesh current from cathode to plate as shown by Fig. 1.3. Following the usual assumptions, the amplification of the tube can then be represented by inserting an equivalent generator -e in series with plate impedance R0, where e is the grid voltage, as shown by Fig. 1.4. The passive impedances of the tube can be incorporated as part of the passive circuit and play no part in this analysis.

Since e = Zg Ij in Fig. 1.4, the equivalent plate generator voltage can also be written as - Zg Ij . The kth of equations (1-2) can therefore be written as

Zk1 I1 + + Zkj Ij + + Zkn In = - Zg Ij


Zk1 I1 + + (Zkj + - Zg) Ij + + Zkn In = 0 (1-4)

where Zkj is the passive coupling between the two meshes. It is obvious that the equation is still in the same form as the original kth equation of (1-2) provided we redefine Zkj to include the added quantity Zg. This is the familiar result that the amplifications of the tubes can be represented by modifications in the various coupling terms in the mesh equations. So far as the general form of the equations goes, the only distinction between active and passive structures is the fact that we can no longer assume in general that the principle of reciprocity holds. In other words, we can no longer assume that Zij = Zji. The quantity Zg will be called the mutual impedance or transimpedance of the tube, after the analogy with transconductance in the following discussion.

In order to prevent future confusion with signs, it is important to notice here the convention adopted in Fig. 1.3 for the positive direction of grid and plate currents. It has been so chosen that the transimpedances in the left sides of the mesh equations will be positive when the 's are positive, as they are in normal tubes, and also so that a uniform convention of sign can be adopted for a number of tubes in tandem coupled by ordinary interstage networks. With this choice, however, the equivalent plate generator voltage is negative, so that successive tubes in an amplifying circuit give successive phase reversals, in addition to any phase shifts which may be ascribed to the purely passive elements of the circuit. Similar remarks apply to the nodal analysis given later.