Chapter 5 Series dc Circuits

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Chapter 5 Series dc Circuits

• Introductory Circuit Analysis
• Robert L. Boylestad

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5.1 - Introduction

• Two types of current are readily available,
  direct current (dc) and sinusoidal alternating
  current (ac)
• We will first consider direct current (dc)

Insert Fig 5.1
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Introduction

• If a wire is an ideal conductor, the potential
  difference (V) across the resistor will equal the
  applied voltage of the battery.
• V (volts) E (volts)
• Current is limited only by the resistor (R).
  The higher the resistance, the less the current.

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5.2 - Series Resistors

• The total resistance of a series configuration is
  the sum of the resistance levels.
• The more resistors we add in series, the greater
  the resistance (no matter what their value).
• Current through all resistors in a series circuit
  is the same.

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Series Resistors

• When series resistors have the same value,
• Where N the number of resistors in the string.
• The total series resistance is not affected by
  the order in which the components are connected.

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5.3 Series Circuits

• Total resistance (RT) is all the source sees.
  
• Once RT is known, the current drawn from the
  source can be determined using Ohms law
• Since E is fixed, the magnitude of the source
  current will be totally dependent on the
  magnitude of RT .

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Series Circuits

• The polarity of the voltage across a resistor is
  determined by the direction of the current.
• When measuring voltage, start with a scale that
  will ensure that the reading is lower than the
  maximum value of the scale. Then work your way
  down until a reading with the highest level of
  precision is made.

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5.4 Power Distribution in a Series Circuit

• The power applied by the dc supply must equal
  that dissipated by the resistive elements.

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Series connection of resistors.
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Resistance seen at the terminals of a series
circuit.
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Using an ohmmeter to measure the total resistance
of a series circuit.
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I E/RT (8.4 V)/(140 ?) 0.06 A 60 mA
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Using voltmeters to measure the voltages across
the resistors
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Protoboard with areas of conductivity defined
using two different approaches.
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Two setups for a network on a protoboard with
yellow leads added to each configuration to
measure voltage V3 with a voltmeter.
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5.5 - Voltage Sources in Series

• Voltage sources can be connected in series to
  increase or decrease the total voltage applied to
  the system.
• Net voltage is determined by summing the sources
  having the same polarity and subtracting the
  total of the sources having the opposite polarity.

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Reducing series dc voltage sources to a single
source.
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Series connection of dc supplies (a) four 1.5 V
batteries in series to establish a terminal
voltage of 6 V (b) incorrect connections for two
series dc supplies (c) correct connection of two
series supplies to establish 60 V at the output
terminals.
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5.6 - Kirchhoffs Voltage Law

• Kirchhoffs voltage law (KVL) states that the
  algebraic sum of the potential rises and drops
  around a closed loop (or path) is zero.

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Kirchhoffs Voltage Law

• The applied voltage of a series circuit equals
  the sum of the voltage drops across the series
  elements
• The sum of the rises around a closed loop must
  equal the sum of the drops.
• The application of Kirchhoffs voltage law need
  not follow a path that includes current-carrying
  elements.
• When applying Kirchhoffs voltage law, be sure
  to concentrate on the polarities of the voltage
  rise or drop rather than on the type of element.
• Do not treat a voltage drop across a resistive
  element differently from a voltage drop across a
  source.

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Applying Kirchhoffs voltage law to a series dc
circuit.
E V1 V2
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16 V1 - 4.2 - 9 0 ?V1 2.8 V
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32 12 Vx 0 ? Vx 20 V or,
Vx 6 14 0 ? Vx 20 V
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60 40 Vx 30 0 ? Vx 50 V
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5.7 Voltage Division in a Series Circuit

• The voltage across the resistive elements will
  divide as the magnitude of the resistance levels.
• The greater the value of a resistor in a series
  circuit, the more of the applied voltage it will
  capture.
• Voltage Divider Rule (VDR)
• The VDR permits determining the voltage levels of
  a circuit without first finding the current.

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Voltage Division in a Series Circuit

• The voltage across a resistor in a series
  circuit is equal to the value of the resistor
  times the total impressed voltage across the
  series elements divided by the total resistance
  of the series elements.
• The rule can be extended to voltage across two
  or more series elements if the resistance
  includes total resistance of the series elements
  that the voltage is to be found across.

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How the voltage will divide across series
resistive elements
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How the voltage will divide across series
resistive elements
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The largest of the series resistive elements will
capture the major share of the applied voltage.
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V (7?)/(15?)(37.5V) 17.5 V
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5.8 - Interchanging Series Elements

• Elements of a series circuit can be interchanged
  without affecting the total resistance, current,
  or power to each element
• In the Figures below, resistors 2 and 3 are
  interchanged without affecting the total
  resistance

Insert Fig 5.20
Insert Fig 5.19
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5.9 - Notation

• Voltage sources and grounds

Ground symbol
Voltage source symbol
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Notation

• Double-subscript notation
• Because voltage is an across variable and
  exists between two points, the double-subscript
  notation defines differences in potential.
• The double-subscript notation Vab specifies
  point a as the higher potential. If this is not
  the case, a negative sign must be associated with
  the magnitude of Vab .
• The voltage Vab is the voltage at point (a) with
  respect to point (b).

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Notation

• Single-subscript notation
• The single-subscript notation Va specifies the
  voltage at point a with respect to ground (zero
  volts). If the voltage is less than zero volts,
  a negative sign must be associated with the
  magnitude of Va .

Va 10 V Vb 4 V ?Vab 10 4 6 V
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Notation

• General Relationship
• If the voltage at points a and b are known with
  respect to ground, then the voltage Vab can be
  determined using the following equation
• Vab Va V b

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Va 10 V and Vab 4 V Vb 6 V and Vbc 20
V ?Vc -14 V
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5.10 Voltage Regulation and the Internal
Resistance of Voltage Sources

• The ideal voltage source has no internal
  resistance and an output voltage of E volts with
  no load or full load.
• Every practical voltage source (generator,
  battery, or laboratory supply) has some internal
  resistance.
• Voltage across the internal resistance lowers
  the source output voltage when a load is
  connected.
• For any chosen interval of voltage or current,
  the magnitude of the internal resistance is given
  by
• Rint ?VL / ?IL

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(a) Sources of dc voltage (b) equivalent
circuit.
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Demonstrating the effect of changing a load on
the terminal voltage of a supply.
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Rint ?VL/?IL (20.1 18.72)/(275.34 mA) 5?
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Voltage Regulation and the Internal Resistance of
Voltage Sources

• For any supply, ideal conditions dictate that
  for a range of load demand (IL), the terminal
  voltage remains fixed in magnitude.
• If a supply is set at 12 V, it is desirable that
  it maintain this terminal voltage, even though
  the current demand on the supply may vary.
• Voltage regulation (VR) characteristics are
  measures of how closely a supply will come to
  maintaining a supply voltage between the limits
  of full-load and no-load conditions.

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Voltage Regulation and the Internal Resistance of
Voltage Sources

• Ideal conditions VFL VNL and VR 0
• The lower the voltage regulation, the less the
  variation in terminal voltage with changes in
  load

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Defining the properties of importance for a power
supply.
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• VNL 20.1 V and VFL 18.72 V
• ?VR (20.1 18.72)/(18.72) 0.0737 7.37

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5.11 Loading Effects of Instruments

• For an up-scale (analog meter) or positive
  (digital meter) reading an ammeter must be
  connected with current entering the positive
  terminal and leaving the negative terminal
• Ammeters are placed in series with the branch in
  which the current is to be measured

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Loading Effects of Instruments

• Voltmeters are always hooked up across the
  element for which the voltage is to be determined
• For a double-script notation Always hook up the
  red lead to the first subscript and the black
  lead to the second.
• For a single-subscript notation Hook up the red
  lead to the point of interest and the black lead
  to the ground

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Including the effects of the internal resistance
of an ammeter (a) 2 mA scale (b) 2 A scale.
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Applying an ammeter, set on the 2 mA scale, to a
circuit with resistors in the kilohm range (a)
ideal (b) practical.
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5.13 Applications

• Holiday lights
• Holiday lights are connected in series if one
  wire enters and leaves the casing.
• If one of the filaments burns out or is broken,
  all of the lights go out unless a fuse link is
  used.
• A fuse link is a soft conducting metal with a
  coating on it that breaks down if the bulb burn
  out, causing the bulb to be by-passed, thus only
  one bulb goes out.

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Holiday lights
(a) a 50-unit set
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(b) Bulb construction
Holiday lights
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(a) Single-set wiring diagram (b) special wiring
arrangement (c) redrawn schematic
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(d) special plug and flasher unit.
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Applications

• Microwave oven
• A series circuit can be very useful in the
  design of safety equipment.
• In a microwave, it is very dangerous if the oven
  door is not closed or sealed properly. Microwaves
  use a series circuit with magnetic switches on
  the door to ensure that the door is properly
  closed.
• Magnetic switches are switches where the magnet
  draws a magnetic conducting bar between two
  conductors to complete the circuit.

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Series safety switches in a microwave oven.
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Applications

• A Series Alarm Circuit