An Example of Solving a Circuit 2.7 2000/07/06 2002/05/15 Don Johnson dhj@rice.edu Don Johnson dhj@rice.edu Roy Ha rha@rice.edu circuit KCL KVL An example of solving a circuit

The circuit shown is perhaps the simplest circuit that performs a signal processing function. The input is provided by the voltage source v in and the output is the voltage v out across the resistor labelled R 2 .

For the example circuit above, we have three v-i relations, two KCL equations, and one KVL equation for solving for the circuit's six voltages and currents. v v in v-i: v 1 R 1 i 1 v out R 2 i out KCL: i i 1 0 i 1 i out 0 KVL: v v 1 v out 0 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 v out . The KVL equation can be rewritten as v in v 1 v out . Substituting into it the resistor's v-i relations, we have v in R 1 i 1 R 2 i out . 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 i 1 i out , which means that v in R 1 i out R 2 i out R 1 R 2 i out . Solving for the current in the output resistor, we have i out v in R 1 R 2 . We have now solved the circuit: We have expressed one voltage or current in terms of sources 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. V out R 2 R 1 R 2 v in Referring back to , 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)? The circuit serves as an amplifier having a gain of R 2 R 1 R 2 .