SINGLE LOOP CIRCUITS A single loop circuit is one which has only a single loop. The same current flows through each element of the circuit-the elements are in series. We will consider circuits consisting of voltage sources and resistors.

VOLTAGE DIVIDER Consider two resistors in series with a voltage v(t) across them: R1R1 R2R2 - v 1 (t) v 2 (t) + - v(t) IMPORTANT VOLTAGE DIVIDER EQUATIONS

FIRST GENERALIZATION: MULTIPLE SOURCES i(t) KVL Collect all sources on one side Voltage sources in series can be algebraically added to form an equivalent source. We select the reference direction to move along the path. Voltage drops are subtracted from rises

SECOND GENERALIZATION: MULTIPLE RESISTORS APPLY KVL TO THIS LOOP VOLTAGE DIVISION FOR MULTIPLE RESISTORS

Multiple Sources/Resistors _ _ _ _ + These two circuits are equivalent where and

I R1R1 R2R2 V + - I1I1 I2I2 Single Node Pair Circuit How do we find I 1 and I 2 ?

Apply KCL at the Top Node I 1 + I 2 = I R1R1 R2R2 V + - I1I1 I2I2 I

Solve for V

Equivalent Resistance If we wish to replace the two parallel resistors with a single resistor whose voltage-current relationship is the same, the equivalent resistor has a value of:

Now to find I 1 This is the current divider formula. It tells us how to divide the current through parallel resistors.

What is the formula for I 2 ?

Is2Is2 V R1R1 R2R2 + - I1I1 I2I2 More Than One Source How do we find I 1 or I 2 ? Is1Is1

Apply KCL at the Top Node I 1 + I 2 = I s1 - I s2

Multiple Current Sources We find an equivalent current source by algebraically summing current sources. We find an equivalent resistance. We find V as equivalent I times equivalent R. We then find any necessary currents using Ohm’s law.

SERIES AND PARALLEL RESISTOR COMBINATIONS For analysis, series resistors can be replaced by an equivalent resistor. Parallel resistors can be replaced by an equivalent resistor/ impedance. Complicated networks of resistors can be replaced by a single equivalent resistor.

Equivalent Resistance i(t) + - v(t) i(t) + - v(t) R eq R eq is equivalent to the resistor network on the left in the sense that they have the same i-v characteristics.

Equivalent Resistance The rest of the circuit cannot tell whether the resistor network or the equivalent resistor is connected to it. The equivalent resistance cannot be used to find voltages or currents internal to the resistor network.

Series Resistance R1R1 R3R3 R2R2 R eq R eq = R 1 + R 2 + R 3 Two elements are in series if the current that flows through one must also flow through the other.

Parallel Resistance R eq 1/R eq = 1/R 1 + 1/R 2 + 1/R 3 R3R3 R2R2 R1R1 Two elements are in parallel if they are connected between the same two nodes.

Circuits with Series and Parallel Combinations The combination of series and parallel resistances can be used to find voltages and currents in circuits Simplification –Resistances are combined to create a simple circuit (usually one source and one resistance), from which a voltage or current can be found. Start from the furthest branch from the source. Backtracking –Once the voltage or current is found, KCL and KVL, Ohm’s Law, Voltage and Current Dividers are used to work back through the network to find voltages and currents.

FIRST WE PRACTICE COMBINING RESISTORS 6k||3k (10K,2K)SERIES SERIES

EXAMPLES COMBINATION SERIES-PARALLEL AN EXAMPLE WITHOUT REDRAWING RESISTORS ARE IN SERIES IF THEY CARRY EXACTLY THE SAME CURRENT RESISTORS ARE IN PARALLEL IF THEY ARE CONNECTED EXACTLY BETWEEN THE SAME TWO NODES

AN “INVERSE SERIES PARALLEL COMBINATION” SIMPLE CASE NOT SO SIMPLE CASE

FIRST REDUCE IT TO A SINGLE LOOP CIRCUIT SECOND: “BACKTRACK” USING KVL, KCL OHM’S …OTHER OPTIONS...