Circuit w/ Dependent Source Example Find i2, i1 and io

Using Equivalent Resistances Simplify a circuit before applying KCL and/or KVL: Example: Find I I R1 R1 = R2 = 3 kW R3 = 6 kW R3 R2  + 7 V R4 R4 = R5 = 5 kW R6 = 10 kW R6 R5

The Wheatstone Bridge Circuit used to precisely measure resistances in the range from 1 W to 1 MW, with ±0.1% accuracy R1 and R2 are resistors with known values R3 is a variable resistor (typically 1 to 11,000W) Rx is the resistor whose value is to be measured battery R1 R2 + V – current detector R3 Rx variable resistor

Finding the value of Rx Adjust R3 until there is no current in the detector Then, R2 R1 Rx = R3 Derivation: i1 = i3 and i2 = ix i3R3 = ixRx and i1R1 = i2R2 i1R3 = i2Rx KCL => KVL => R1 R2 i1 i2 + V – i3 ix R3 Rx R3 R1 Rx R2 = Typically, R2 / R1 can be varied from 0.001 to 1000 in decimal steps

Identifying Series and Parallel Combinations Some circuits must be analyzed (not amenable to simple inspection) R1  + R4 R5 R2 V R3 Special cases: R3 = 0 OR R3 = 

Resistive Circuits: Summary Equivalent resistance of k resistors in series: Req = S Ri = R1 + R2 + ••• + Rk Equivalent resistance of k resistors in parallel: Voltage divided between 2 series resistors: Current divided between 2 parallel resistors: k i=1 1 Req k 1 Ri 1 R1 1 R2 1 Rk = S = + + ••• + i=1 R1 R1 + R2 v1 = vs R2 R1 + R2 i1 = is

Circuit Analysis Methods … Given: a circuit with voltage and/of current sources, and components (Rs, Cs, Ls, diodes, transistors, etc.) connected by wires Task: find all the voltages and currents of interest Approaches: Use Ohm’s Law, KCL and KVL piecemeal Nodal analysis – node voltages are the unknowns Mesh (loop) analysis – mesh currents are the unknowns Superposition (linear circuits) – work with only one voltage or current source at a time) and use #1, #2 or #3 Computer simulation of circuit behavior

Node-Voltage Circuit Analysis Method Choose a reference node (“ground”) Look for the one with the most connections! Define unknown node voltages those which are not fixed by voltage sources Write KCL at each unknown node, expressing current in terms of the node voltages (using the I-V relationships of branch elements) Special cases: floating voltage sources Solve the set of independent equations N equations for N unknown node voltages

Nodal Analysis: Example #1 R 4 V1 2 + - IS 3 R1 Choose a reference node. Define the node voltages (except reference node and the one set by the voltage source). Apply KCL at the nodes with unknown voltage. Solve for unknown node voltages.

Nodal Analysis: Example #2 V 2 1 R 4 5 3 I Va

Nodal Analysis w/ “Floating Voltage Source” A “floating” voltage source is one for which neither side is connected to the reference node, e.g. VLL in the circuit below: R 4 2 I V a b + - LL 1 Problem: We cannot write KCL at nodes a or b because there is no way to express the current through the voltage source in terms of Va-Vb. Solution: Define a “supernode” – that chunk of the circuit containing nodes a and b. Express KCL for this supernode. Incorporate voltage source constraint into KCL equation.

Nodal Analysis: Example #3 supernode V V a LL V b - + I R R I 1 2 4 2 Eq’n 1: KCL at supernode Substitute property of voltage source:

Node-Voltage Method and Dependent Sources If a circuit contains dependent sources, what to do? Example: iD 20 W 10 W 200 W + – 2.4 A 80 V – + 5iD

Node-Voltage Method and Dependent Sources Dependent current source: treat as independent current source in organizing and writing node eqns, but include (substitute) constraining dependency in terms of defined node voltages. Dependent voltage source: treat as independent voltage source in organizing and writing node eqns, but include (substitute) constraining dependency in terms of defined node voltages.

Example: – + 80 V 5iD 20 W 10 W 200 W 2.4 A iD