**Chapter 5 – Series Circuits** Introductory Circuit Analysis Robert L. Boylestad

**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

**Introduction** • If a wire were an conductor (no opposition to flow), the potential difference V across the resistor will equal the applied voltage of the battery • V (volts) = E (volts) • Current then is limited only by the resistor (R) The higher the resistance, the less the current

**5.2 - Series Circuits** • Two elements are in series if • They have only one terminal in common • The common point between the two elements is not connected to another current-carrying element • If all elements in the circuit are in series, then the network is called a series circuit • Examples of a series circuit are the tying of small pieces of rope together to form a longer rope and the connecting of pipes to get water from one point to another

**Series Circuits** • Current is the same through series elements • Used to determine if two elements are in series • A branch of a circuit is any portion of the circuit that has one or more elements in series • The total resistance of a series circuit is the sum of the resistance levels RT = R1 + R2 + R3 + R4 ….+ RN

**E** Is= RT Series Circuits • Total resistance (RT) is all the source “sees” • Once RT 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 RT Insert Fig 5.5

**Series Circuits** • The fact that current is the same through each element of a series circuit permits a direct calculation of the voltage across each resistor using Ohm’s law • V1 = IR1, V2 = IR2, V3 = IR3, … VN = IRN • The total power delivered to a resistive circuit is equal to the total power dissipated by the resistive elements Pdel = P1 + P2 + P3 + …+ PN

**5.3 - Voltage Sources in Series** • Voltage source can be connected in series to increase or decrease the total voltage applied to the system • Net voltage is determined by summing the sources with the same polarity and subtracting the total of the sources with the opposite “pressure” • ET = E2 + E3 - E1 (assuming that E1 has a different polarity than E2 and E3 )

**5.4 - 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 Insert Fig. 5.12

**Kirchhoff’s Voltage Law** • The applied voltage of a series circuit equals the sum of the voltage drops across the series elements Vrises = Vdrops • (the sum of the rise around a closed loop must equal the sum of the drop) • 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

**5.5 - 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 5.19 Insert Fig 5.20

**RxE** Vx = RT 5.6 - Voltage Divider Rule • The voltage across the resistive elements will divide as the magnitude of the resistance levels • It is the ratio of resistor value that counts when it comes to voltage division and not the relative magnitude of all the resistors • Voltage Divider Rule (VDR) • Permits determining the voltage levels of a circuit without first finding the current

**Voltage Divider Rule** • 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

**5.7 - Notation** • Voltage sources and grounds • Ground symbol with its defined potential • Symbol for voltage source

**Notation** • Double-subscript notation • Because voltage is an “across” variable and exists between two points, the double-subscript notation define 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 (w.r.t.) point b

**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

**Notation** • General comments • 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 - Vb

**5.8 - Internal Resistance of Voltage Sources** • Every source of voltage (generator, battery, or laboratory supply) has some internal resistance • The ideal voltage source has no internal resistance and an output voltage of E volts with no load or full load • Internal voltage across the internal resistance is computed using the formula: Vint = IFLRint • For any chosen interval of voltage or current, the magnitude of the internal resistance is given by Rint = VE / IL

**5.9 - Voltage Regulation** • 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 characteristics (VR) are measures of how closely a supply will come to maintaining a supply voltage between the limits of full-load and no-load conditions

**Voltage Regulation** • Ideal conditions, VFL = VNL and VR% = 0 • The smaller the voltage regulation, the less the variation in terminal voltage with change in load VR% = (Rint / RL) X 100%

**5.10 - Measurement Techniques** • 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

**Measurement Techniques** • 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

**5.11 - 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.

**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 insure 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.

**Applications** • Series alarm circuits