Methods of Analysis

In analyzing a circuit using Kirchhoff's circuit laws, one can either do nodal analysis using Kirchhoff's current law (KCL) or mesh analysis using Kirchhoff's voltage law (KVL). Nodal analysis writes an equation at each electrical node, requiring that the branch currents incident at a node must sum to zero. The branch currents are written in terms of the circuit node voltages. As a consequence, each branch constitutive relation must give current as a function of voltage; an admittance representation. For instance, for a resistor, I_branch = V_branch * G, where G (=1/R) is the admittance (conductance) of the resistor.

First, we will have the Nodal Analysis

In this method, we set up and solve a system of equations in which the unknowns are the voltages at the principal nodes of the circuit. From these nodal voltages the currents in the various branches of the circuit are easily determined.

The steps in the nodal analysis method are:

• Count the number of principal nodes or junctions in the circuit. Call this number n. (A principal node or junction is a point where 3 or more branches join. We will indicate them in a circuit diagram with a red dot. Note that if a branch contains no voltage sources or loads then that entire branch can be considered to be one node.)
• Number the nodes N₁, N₂, . . . , N_n and draw them on the circuit diagram. Call the voltages at these nodes V₁, V₂, . . . , V_n, respectively.
• Choose one of the nodes to be the reference node or ground and assign it a voltage of zero.
• For each node except the reference node write down Kirchoff's Current Law in the form "the algebraic sum of the currents flowing out of a node equals zero". (By algebraic sum we mean that a current flowing into a node is to be considered a negative current flowing out of the node.)
  
  For example, for the node to the right KCL yields the equation:

 I_a + I_b + I_c = 0

Express the current in each branch in terms of the nodal voltages at each end of the branch using Ohm's Law (I = V / R). Here are some examples:



The current downward out of node 1 depends on the voltage difference V1 - V3 and the resistance in the branch.




In this case the voltage difference across the resistance is V1 - V2 minus the voltage across the voltage source. Thus the downward current is as shown.




In this case the voltage difference across the resistance must be 100 volts greater than the difference V1 - V2. Thus the downward current is as shown.


The result, after simplification, is a system of m linear equations in the m unknown nodal voltages (where m is one less than the number of nodes; m = n - 1). The equations are of this form:

where G₁₁, G₁₂, . . . , G_mm and I₁, I₂, . . . , I_m are constants.

Alternatively, the system of equations can be gotten (already in simplified form) by using the inspection method.

• Solve the system of equations for the m node voltages V₁, V₂, . . . , V_m

Series and Parallel Connections

What are "series" and "parallel" circuits?

Circuits consisting of just one battery and one load resistance are very simple to analyze, but they are not often found in practical applications. Usually, we find circuits where more than two components are connected together.

There are two basic ways in which to connect more than two circuit components: series and parallel. First, an example of a series circuit: 

Here, we have three resistors (labeled R₁, R₂, and R₃), connected in a long chain from one terminal of the battery to the other. (It should be noted that the subscript labeling -- those little numbers to the lower-right of the letter "R" -- are unrelated to the resistor values in ohms. They serve only to identify one resistor from another.) The defining characteristic of a series circuit is that there is only one path for electrons to flow. In this circuit the electrons flow in a counter-clockwise direction, from point 4 to point 3 to point 2 to point 1 and back around to 4.

The equivalent resistance of any number of resistors connected in series the sum of the individual resistances. 

For N resistors in series,

To determine the voltage across each resistor,

Now, let's look at the other type of circuit, a parallel configuration: 

Again, we have three resistors, but this time they form more than one continuous path for electrons to flow. There's one path from 8 to 7 to 2 to 1 and back to 8 again. There's another from 8 to 7 to 6 to 3 to 2 to 1 and back to 8 again. And then there's a third path from 8 to 7 to 6 to 5 to 4 to 3 to 2 to 1 and back to 8 again. Each individual path (through R₁, R₂, and R₃) is called a branch.

The defining characteristic of a parallel circuit is that all components are connected between the same set of electrically common points. Looking at the schematic diagram, we see that points 1, 2, 3, and 4 are all electrically common. So are points 8, 7, 6, and 5. Note that all resistors as well as the battery are connected between these two sets of points.

As for a parallel circuit, the equivalent resistance of two parallel resistors is equal to the product of their resistances divided by their sum,

Or in general,

To get the current in a parallel circuit, we may use,

 And, of course, the complexity doesn't stop at simple series and parallel either! We can have circuits that are a combination of series and parallel, too:

In this circuit, we have two loops for electrons to flow through: one from 6 to 5 to 2 to 1 and back to 6 again, and another from 6 to 5 to 4 to 3 to 2 to 1 and back to 6 again. Notice how both current paths go through R₁ (from point 2 to point 1). In this configuration, we'd say that R₂ and R₃ are in parallel with each other, while R₁ is in series with the parallel combination of R₂ and R₃.

The basic idea of a "series" connection is that components are connected end-to-end in a line to form a single path for electrons to flow: 

The basic idea of a "parallel" connection, on the other hand, is that all components are connected across each other's leads. In a purely parallel circuit, there are never more than two sets of electrically common points, no matter how many components are connected. There are many paths for electrons to flow, but only one voltage across all components: 

Series and parallel resistor configurations have very different electrical properties.