1. 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

2. 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

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

4. 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

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

6. 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

7. 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

8. 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

9. 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

10. 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

11. 3.4 The Voltage and Current Division
We can generalize the voltage and current division as follows
Voltage division
Current division

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

13. 3.7 Delta to Wye Equivalent Circuit
Consider the following circuit

15. a a b c

16. a a b c
Solving for
we have,

17. Solving for
we have,

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

20. i