1. Chapter 6 – Parallel Circuits
Introductory Circuit Analysis
Robert L. Boylestad 1

2. 6.1 - Introduction
There are two network configurations – series and parallel
In Chapter 5 we covered a series network, and in this chapter we will cover the parallel circuit and all the methods and laws associated with it 2

3. 6.2 - Parallel Elements
Two elements, branches, or networks are in parallel if they have two points in common as in the figure below
Insert Fig 6.2 3

4. 6.3 - Total Conductance and Resistance
For parallel elements, the total conductance is the sum of the individual conductances.
GT = + G1 + G2 + G3 +… + Gn
As the number of resistors in parallel increases, the input current level will increase for the same applied voltage
This is the opposite effect of increasing the number of resistors in a series circuit 4

5. Total Conductance and Resistance
Since G = 1/R the total resistance for a network can be determined by the equation below
Note that the equation is for 1 divided by the total resistance rather than the total resistance
Once the right side of the equation has been determined, it is necessary to divide the result into 1 to determine the total resistance 5

6. Total Conductance and Resistance
The total resistance of a parallel resistor is always less than the value of the smallest resistor
Additionally, the wider the spread in numerical value between two parallel resistors, the closer the total resistance will be to the smaller resistor
The equation becomes significantly easier to apply for equal resistors in parallel
Total resistance of N parallel resistors of equal value is the resistance of one resistor divided by the number (N) of parallel elements 6

7. Total Conductance and Resistance
The total resistance of two resistors is the product of the two divided by their sum
The equation was developed to reduce the effects of the inverse relationship when determining RT 7

8. Total Conductance and Resistance
Parallel elements can be interchanged without changing the total resistance or input current
For parallel resistors, the total resistance will always decrease as additional elements are added in parallel 8

9. 6.4 - Parallel Circuits
Total resistance is determined by RT = R1 R2 / (R1 + R2 ) and the source current by Is = E / RT .
The subscript s will be used to denote a property of the source
The voltage across parallel elements is the same
V1 = V2 = E
Voltage across resistor 1 is equal to the voltage across resistor 2 9

10. Is = I1 + I2 Parallel Circuits
For single-source parallel networks, the source current (Is) is equal to the sum of the individual branch currents
Is = I1 + I2 10

11. 6.5 - Kirchhoff’s Current Law
Kirchhoff’s voltage law provides an important relationship among voltage levels around any closed loop of a network
Now consider Kirchhoff’s current law (KCL)
Kirchhoff’s current law states that the algebraic sum of the currents entering and leaving an area, system, or junction is zero
The sum of the current entering an area, system or junction must equal the sum of the current leaving the area, system, or junction
Ientering = Ileaving 11

12. Kirchhoff’s Current Law
Most common application of the law will be at the junction of two or more paths of current flow
Determining whether a current is entering or leaving a junction is sometimes the most difficult task
One approach to understanding the flow is to picture yourself as standing on the junction point and treating the path currents as arrows
If the arrow appears to be heading toward you, the current is entering the junction
If you see the tail of the arrow as it travels down its path away from you, the current is leaving the junction 12

13. 6.6 - Current Divider Rule
The current divider rule (CDR) will determine how the current entering a set of parallel branches will split between the elements
For two parallel elements of equal value, the current will divide equally
For parallel elements with different values, the smaller the resistance, the greater the share of input current
For parallel elements of different values, the current will split with a ratio equal to the inverse of their resistor values 13

14. Current Divider Rule Current seeks the path of least resistance
The current entering any number of parallel resistors divides into these resistors as the inverse ratio of their ohmic value 14

15. 6.7 - Voltage Sources in Parallel
Voltage sources are placed in parallel only if they have the same voltage rating
The purpose for placing two or more batteries in parallel would be to increase the current rating
The formula to determine the total current is:
Is = I1 + I2 +… IN
at the same terminal voltage 15

16. Voltage Sources in Parallel
Two batteries of different terminal voltages placed in parallel
When two batteries of different terminal voltages are placed in parallel, the larger battery tries to drop rapidly to the lower supply
The result is the larger battery quickly discharges to the lower voltage battery, causing the damage to both batteries 16

17. 6.8 - Open and Short Circuits
An open circuit can have a potential difference (voltage) across its terminal, but the current is always zero amperes
Two isolated terminals not connected by any element: 17

18. Open and Short Circuits
A short circuit can carry a current of a level determined by the external circuit, but the potential difference (voltage) across its terminals is always zero volts
Insert Fig 6.44 18

19. 6.9 - Voltmeters: Loading Effect
Voltmeters are always placed across an element to measure the potential difference
The resistance of two parallel resistors will always be less than the resistance of the smallest resistor
A DMM has internal resistance which will alter, somewhat, the network being measured
The loading of a network by the insertion a meter is not to be taken lightly, especially if accuracy is a primary consideration 19

20. Voltmeters: Loading Effect
A good practice is to always check the meter resistance level against the resistive elements of the network before making a measurement
Most DMMs have internal resistance levels in excess of MW on all voltage scales
Internal resistance of VOMs is sensitive to the scale chosen
Internal resistance is determined by multiplying the maximum voltage of the scale setting by the ohm/volt ( / V) rating of the meter, normally found at the bottom of the face of the meter 20

21. 6.10 - Troubleshooting Techniques
Troubleshooting is a process by which acquired knowledge and experience are employed to localize a problem and offer or implement a solution
Experience and a clear understanding of the basic laws of electrical circuits is vital
First step should always be knowing what to expect 21

22. 6.11 - Applications Car system Parallel computer bus connections
The electrical system on a car is essentially a parallel system
Parallel computer bus connections
The bus connectors are connected in parallel with common connections to the power supply, address and data buses, control signals, and ground 22

23. Applications House wiring
Except in some very special circumstances the basic wiring of a house is done in a parallel configuration
Each parallel branch, however, can have a combination of parallel and series elements
Each branch receives a full 120 V or 208 V, with the current determined by the applied load 23