The voltage law says that the sum of voltages around every closed loop in the circuit must equal zero. A closed loop has the obvious defnition: Starting at a node, trace a path through the circuit that returns you to the origin node. KVL expresses the fact that electric fields are conservative: The total work performed in moving a test charge around a closed path is zero. The KVL equation for our circuit is
In writing KVL equations, we follow the convention that an element's voltage enters with a plus sign when traversing the closed path, we go from the positive to the negative of the voltage's defnition.
For the example circuit (Simple circult), we have three v-i relations, two KCL equations, and one KVL equation for solving for the circuit's six voltages and currents.
We have exactly the right number of equations! Eventually, we will discover shortcuts for solving circuit problems; for now, we want to eliminate all the variables but vout and determine how it depends on vin and on resistor values. The KVL equation can be rewritten as vin = v1 + vout . Substituting into it the resistor's v-i relation, we have vin = R1i1 + R2iout . Yes, we temporarily eliminate the quantity we seek. Though not obvious, it is the simplest way to solve the equations. One of the KCL equations says i1 = iout, which means that vin = R1iout + R2iout =(R1 + R2) iout. Solving for the current in the output resistor, we have . We have now solved the circuit: We have expressed one voltage or current in terms of R1+R2sources and circuit-element values. To find any other circuit quantities, we can back substitute this answer into our original equations or ones we developed along the way. Using the v-i relation for the output resistor, we obtain the quantity we seek.
Referring back to Figure 3.6, a circuit should serve some useful purpose. What kind of system does our circuit realize and, in terms of element values, what are the system's parameter(s)?