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Chapter 6 – Parallel dc Circuits. Introductory Circuit Analysis Robert L. Boylestad. 6.1 - Introduction. There are two network configurations – series and parallel.

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chapter 6 parallel dc circuits

Chapter 6 – Parallel dc Circuits

Introductory Circuit Analysis

Robert L. Boylestad

6 1 introduction
6.1 - Introduction
  • There are two network configurations – series and parallel.
  • In Chapter 5 we covered a series network. In this chapter we will cover the parallel circuit and all the methods and laws associated with it.
6 2 parallel resistors
6.2 – Parallel Resistors
  • Two elements, branches, or circuits are in parallel if they have two points in common as in the figure below

Insert Fig 6.2

parallel resistors
Parallel Resistors
  • For resistors in parallel, the total resistance is determined from
  • Note that the equation is for the reciprocal of RT rather than for RT.
    • Once the right side of the equation has been determined, it is necessary to divide the result into 1 to determine the total resistance
parallel resistors1
Parallel Resistors
  • For parallel elements, the total conductance is the sum of the individual conductance values.
    • 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.
parallel resistors2
Parallel Resistors
  • The total resistance of any number of parallel resistors can be determined using
  • The total resistance of parallel resistors is always less than the value of the smallest resistor.
parallel resistors3
Parallel Resistors
  • For equal resistors in parallel:

Where N = the number of parallel resistors.

parallel resistors4
Parallel Resistors
  • A special case: 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
parallel resistors5
Parallel Resistors
  • Parallel resistors can be interchanged without changing the total resistance or input current.
  • For parallel resistors, the total resistance will always decrease as additional parallel elements are added.
6 3 parallel circuits
6.3 – Parallel Circuits
  • Voltage is always the same across parallel elements.

V1 = V2 = E

The voltage across resistor 1 equals the voltage across resistor 2, and both equal the voltage supplies by the source.

parallel circuits
Parallel Circuits
  • For single-source parallel networks, the source current (Is) is equal to the sum of the individual branch currents.
  • For a parallel circuit, source current equals the sum of the branch currents. For a series circuit, the applied voltage equals the sum of the voltage drops.
parallel circuits1
Parallel Circuits
  • For parallel circuits, the greatest current will exist in the branch with the lowest resistance.
6 4 power distribution in a parallel circuit
6.4 – Power Distribution in a Parallel Circuit
  • For any resistive circuit, the power applied by the battery will equal that dissipated by the resistive elements.
  • The power relationship for parallel resistive circuits is identical to that for series resistive circuits.
6 5 kirchhoff s current law
6.5 - Kirchhoff’s Current Law
  • Kirchhoff’s voltage law provides an important relationship among voltage levels around any closed loop of a network.
  • Kirchhoff’s current law (KCL) 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.
kirchhoff s current law
Kirchhoff’s Current Law
  • Most common application of the law will be at the junction of two or more paths of current.
  • Determining whether a current is entering or leaving a junction is sometimes the most difficult task.
    • If the current arrow points toward the junction, the current is entering the junction.
    • If the current arrow points away from the junction, the current is leaving the junction.
6 6 current divider rule
6.6 – Current Divider Rule
  • The current divider rule (CDR) is used to find the current through a resistor in a parallel circuit.
  • General points:
    • 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.
6 7 voltage sources in parallel
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 is to increase the current rating.
    • The formula to determine the total current is:
    • at the same terminal voltage.
voltage sources in parallel
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
6 8 open and short circuits
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.
open and short circuits
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

6 9 voltmeter loading effects
6.9 – Voltmeter Loading Effects
  • Voltmeters are always placed across an element to measure the potential difference.
    • The resistance of parallel resistors will always be less than the resistance of the smallest resistor.
    • A DMM has internal resistance which may alter the resistance of the network under test.
    • The loading of a network by the insertion of a meter is not to be taken lightly, especially if accuracy is a primary consideration.
voltmeter loading effects
Voltmeter Loading Effects
  • A good practice is to always check the meter resistance 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.
  • The internal resistance of a VOM depends on the scale chosen.
    • Internal resistance is determined by multiplying the maximum voltage of the scale setting by the ohm/volt ( / V) ratingof the meter, normally found at the bottom of the face of the meter.
6 11 troubleshooting techniques
6.11 – 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
6 13 applications
6.13 – 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.
  • 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.