Kirchhoff’s rule

Kirchhoff and Bunsen s work created a new analytical tool for chemists, and, in part because of the introduction of the spectroscope, thallium was identified as an element, in 1861, and indium, in 1863. The tool was also used by astronomers, who used Kirchhoff s rules to determine the chemical composition of stars. In the long run, it was shown that all the heavy elements were by-products of stellar fusion. Even the atoms of our bodies were made from the stuff of stars. 

The equivalent impedance of the resistor/capacitor parallel circuit in Fig. 6.14(a), Z, must be determined by application of Kirchhoff s rule (i.e., the algebraic sum of the voltages of the voltage sources in any circuit loop must equal the algebraic sum of the voltage drops in the same loop). Thus ... 

Application of Kirchhoff s rule to the equivalent circuit yields ... 

Applying Kirchhoff s rule one can consider (jo-functional networks by the same analogy. Calculating the resistance of one chain (= wire) in the network and transforming it back to the mean square distances one finds... 

Be warned that a Web search for Kirchhoff will yield dozens of pages on Kirch-hoff s rules, which relate to electronic circuits. 

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Kirchhoff s laws - Basic rules for electric circuits, which state (a) the algebraic sum of the currents at a network node is zero and (b) the algebraic sum of the voltage drops around a closed path is zero. 

Algebra The rules for relating circuit variables are called Kirchhoff s laws. Table 1.1 describes the two sets of five algebraic equations used for modeling these circuits. 

The circuit model includes, among others, a resistor, capacitor and inductor. Such component models are available from standard libraries. The model equations are v = R i, i = C der(v) and V =L-der(i), respectively, where der() denotes the time derivative. Each component has two electric pin interfaces (filled and non-filled blue squares in Figure la) that include the voltage v as a potential and current i as a flow variable. Model equations are introduced for connected component (object) interfaces as follows The potential is the same, whereas the sum of flow equals zero according to Kirchhoff s node rule. For example, in case of the circuit model shown by Figure la) v =v = V7 and... 

The network equations constitute a set of A a 1 valence sum rule equations (eqn (3.3)) and A b Xa+1 loop equations (eqn (3.4)) where the network contains atoms and A b bonds. Alternatively one can use the equivalent Kirchhoff equations (2.7) and (2.11). One can readily write down equations of type 3.3 but one of these is redundant since the sum of all atomic valences in the crystal must be zero. There are many more than Ab — Aa + 1 possible loops in most bond graphs, but only Ab —Aa+ 1 are independent. Equations (3.3) and (3.4) thus constitute a set of Ab equations which is exactly the number needed to solve for the Ab unknown bond valences,. s. 

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