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

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

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

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

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

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

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

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

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

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

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 • ÂIentering = ÂIleaving

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

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

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

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

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

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:

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

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

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 10 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 (W / V) rating of the meter, normally found at the bottom of the face of the meter

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

22. 6.11 - Applications • Car system • 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

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