The complex basis in electrical networks and its application to network-element-value calculation

A fundamental and novel concept is introduced regarding the independence of variables (such as branch voltages and currents) in an electrical network sinusoidally excited by multiple frequencies. The new concept is characterized by a set of variables called the complex basis. A complex basis of a set X of variables excluding exciting voltages and current is defined to be a minimal subset X_B of X such that any variable in X can be expressed as a linear combination of variables in X_B with constant coefficients. Graph-theoretical theorems and a derivation procedure for a complex basis in an active CR network excited by current sources are presented. Then the concept is applied to network-element-value calculation, an essential and important technique in network diagnosis and tuning. Matrices consisting of vectors of variables obtained by varying the network excitation, that is, by varying both the phasor values (the effective values and phases) and the frequency of the exciting currents, are investigated