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

2. 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

3. 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

4. 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 to 1000 in decimal steps

5. 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 = 

6. 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

7. 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

8. 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

9. Nodal Analysis: Example #1R 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.

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

13. 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.

14. 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:

16. 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

17. 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.

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