Chapter 6

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Chapter 6 Parallel dc Circuits

• Introductory Circuit Analysis
• Robert L. Boylestad

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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.

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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
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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

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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.

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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.

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Parallel Resistors

• For equal resistors in parallel
• Where N the number of parallel resistors.

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(No Transcript)
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1/RT 1/1 ¼ 1/5 1 0.25 0.2 1.45 ?RT
1/1.45 0.69?
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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

RT PRODUCT/SUM
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RT (3 x 6)/(3 6) 18/9 2?
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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.

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Using a protoboard to set up the circuit
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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.

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Measuring the voltages of a parallel dc
network.
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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.

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Parallel Circuits

• For parallel circuits, the greatest current will
  exist in the branch with the lowest resistance.

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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.

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Measuring the source current of a parallel
network.
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Measuring the current through resistor R1.
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6.5 - Kirchhoffs Current Law

• Kirchhoffs voltage law provides an important
  relationship among voltage levels around any
  closed loop of a network.
• Kirchhoffs 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.

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Kirchhoffs 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.

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Kirchhoffs current law.
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(a) Demonstrating Kirchhoffs current law (b)
the water analogy for the junction in (a).
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I3 5A and I4 4A
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I1 1A I3 I1 1A I4 I2 4A I5 I3 I4
5A
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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.

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Current Divider Rule
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Using the current divider rule to calculate
current I1
1/RT 1/1k 1/10k 1/22k ?RT 873? I1
(RT/R1)IT (873/1000)(12 mA) 10.5 mA
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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.

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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

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Examining the impact of placing two lead-acid
batteries of different terminal voltages in
parallel.
I (12 6)/(0.03 0.02) 120A
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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.

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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
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I (6V)/(12?) 0.5A and V (0.5A)(10?) 5V
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I (6V)/(2?) 3A and V 0
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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.

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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) rating of the
  meter, normally found at the bottom of the face
  of the meter.

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Vab 20V
Vab (11M?)/(12M?)(20V)
18.33V
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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

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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.

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Expanded view of an automobiles electrical
system.
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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.

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Single phase of house wiring (a) physical
details (b) schematic representation.
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Continuous ground connection in a duplex outlet.