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Electric Circuits and Interconnection Laws

Module by: Don Johnson. E-mail the author

Summary: A circuit is the electrical/physical analogue of a system. Kirchoff's laws provde connection rules about "nodes" in a circuit to determine relations between various circuit elements.

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A circuit connects circuit elements together in a specific configuration designed to transform the source signal (originating from a voltage or current source) into another signal—the output—that corresponds to the current or voltage defined for a particular circuit element. A simple resistive circuit is shown in Figure 1. This circuit is the electrical embodiment of a system having its input provided by a source system producing v in t v in t .

Figure 1: The circuit shown in the top two figures is perhaps the simplest circuit that performs a signal processing function. On the bottom is the block diagram that corresponds to the circuit. The input is provided by the voltage source v in v in and the output is the voltage v out v out across the resistor label R 2 R 2 . As shown in the middle, we analyze the circuit—understand what it accomplishes—by defining currents and voltages for all circuit elements, and then solving the circuit and element equations.
Figure 1
(a)
Figure 1(a) (circuit4.png)
(b)
Figure 1(b) (circuit4a.png)
(c)
Figure 1(c) (system4.png)

To understand what this circuit accomplishes, we want to determine the voltage across the resistor labeled by its value R 2 R 2 . Recasting this problem mathematically, we need to solve some set of equations so that we relate the output voltage v out v out to the source voltage. It would be simple—a little too simple at this point—if we could instantly write down the one equation that relates these two voltages. Until we have more knowledge about how circuits work, we must write a set of equations that allow us to find all the voltages and currents that can be defined for every circuit element. Because we have a three-element circuit, we have a total of six voltages and currents that must be either specified or determined. You can define the directions for current flow and positive voltage drop any way you like. When two people solve a circuit their own ways, the values of their variables may not agree, but current and voltage values for each element will agree. Do recall in defining your voltage and current variables that the v-i relations for the elements presume that positive current flow is in the same direction as positive voltage drop. Once you define voltages and currents, we need six nonredundant equations to solve for the six unknown voltages and currents. By specifying the source, we have one; this amounts to providing the source's v-i relation. The v-i relations for the resistors give us two more. We are only halfway there.

What we need to solve every circuit problem is mathematical statements that express what the interconnection of elements is. Said another way, we need the laws that govern the electrical connection of circuit elements. First of all, the places where circuit elements attach to each other are called nodes. Two nodes are explicitly indicated in Figure 1; a third is at the bottom where the voltage source and resistor R 2 R 2 are connected. Electrical engineers tend to draw circuit diagrams—schematics— in a rectilinear fashion. Thus the long line connecting the bottom of the voltage source with the bottom of the resistor is intended to make the diagram look pretty. This line simply means that the two elements are connected together. Kirchoff's Laws, one for voltage and one for current, determine what a connection among circuit elements means. These laws can help us analyze this circuit.

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