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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 (R T ) is all the source “sees.” Once R T is known, the current drawn from the source can be determined using Ohm’s law: Since E is fixed, the magnitude of the source current will be totally dependent on the magnitude of R T.

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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/R T = (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 V 3 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 - Kirchhoff’s Voltage Law Kirchhoff’s 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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Kirchhoff’s 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 Kirchhoff’s voltage law need not follow a path that includes current-carrying elements. When applying Kirchhoff’s 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 Kirchhoff’s voltage law to a series dc circuit. E = V 1 + V 2

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+16 – V = 0 V 1 = 2.8 V

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+32 – 12 – V x = 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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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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Voltage sources and grounds Notation 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 V ab specifies point a as the higher potential. If this is not the case, a negative sign must be associated with the magnitude of V ab. The voltage V ab is the voltage at point ( a) with respect to point ( b).

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Notation Single-subscript notation The single-subscript notation V a 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 V a. V a = +10 V V b = + 4 V V ab = 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 V ab can be determined using the following equation: V ab = V a – V b

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V a = 10 V and V ab = 4 V V b = +6 V and V bc = 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 R int = V L / I L

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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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R int = ∆V L /∆I L = (20.1 – 18.72)/( 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 ( I L ), 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: V FL = V NL 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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V NL = 20.1 V and V FL = V V R = (20.1 – 18.72)/(18.72) = = 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

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