3.1 Resistors in Series When two elements connected at a single node, they are said to be in series Series-connected circuit elements carry the same current The resistors in the circuit shown are connected on series The resistors carry the same current as can be shown KCL @ node a KCL @ node b We can apply KCL t o the other nodes b,c,e,f,g,h to conclude that the current is the same namely is

Applying KVL around the loop we obtain The significant of the equation is that the seven resistors can be replaced by a single Resistor whose value is the sum of the individual resistor

In general, if k resistors are connected in series, the equivalent single resistor has a resistance equal to the sum of the k resistances

3.2 Resistors in Parallel When two elements connected at a single node pair, they are said to be in parallel Parallel-connected circuit elements have the same voltage across their terminals The resistors in the circuit shown are connected on parallel Do not make the mistake of assuming that two elements are parallel connected merely Because they are lined up in parallel We can show that the parallel connection have the same voltage by applying KVL to the each loop in the circuit

Applying KCL at node a From the parallel connection the elements have the same voltage and polarity namely vs From which

The four resistors on the circuit can be replaced by a single equivalent resistor In general, if k resistors are connected in parallel, the equivalent single resistor has a resistance equal to Using conductance when dealing with resistor in parallel is more convenient

3.3 The voltage-Divider and Current-Divider Circuits In some applications, we need to develop more than one voltage level from a single voltage supply One way of doing this is by using a voltage-divider-circuit We analyze this circuit as follows Using Ohm’s law, we have

We can see that v1 and v2 are each fraction of vs . Each fraction is the ratio of the resistance across which the divided voltage is defined to the Sum of the two resistances. Because this ratio is always less than 1.0 , the divided voltages v1 and v2 are always less than the source voltage vs If vs is specified , say 15 V and a particular value of v2 is desire say 5 V There are an infinite combinations of R1 and R2 that will satisfy this ratio

Consider connecting a resistor RL in parallel with R2 as shown The resistor RL acts as a load on the voltage-divider circuit The output voltage vo becomes Note as RL → ∞ , vo reduce to as it should be

The Current-Divider The current divider consist of two resistors connected in parallel as shown We will find the relationship between is and i1 and i2 as follows The equation shows that the current is divided between two resistors in parallel such that The current in one resistor is controlled by the other resistor This is similar to a main water pipe that split into two pipes with different dirt and sand and obstacle ( معيقات). The pipe with the less obstacle will have the current flow in it more than the Other pipe with more obstacle

3.4 The Voltage and Current Division We can generalize the voltage and current division as follows Voltage division Current division

Example 3.4 Use current division to find the current io and use voltage division to find the voltage vo

3.7 Delta to Wye Equivalent Circuit Consider the following circuit

a a b c

a a b c Solving for we have,

Solving for we have,

Example 3.7 : Find the current and power supplied by the 40 V source ?

i