**Lesson 4: Series Circuits and Kirchhoff’s Voltage Law**

**Learning Objectives** • Identify elements that are connected in series. • State and apply KVL in analysis of a series circuit. • Determine the net effect of series-aiding and series-opposing voltage sources. • Compute the power dissipated by each element and the total power in a series circuit. • Describe the basic function of a fuse or a switch. • Draw a schematic of a typical electrical circuit, and explain the purpose of each component and indicate the polarity and current direction.

**INTRODUCTION** • Two types of current are readily available to the consumer today. • One is direct current (dc), in which ideally the flow of charge (current) does not change in magnitude (or direction) with time. • The other is sinusoidal alternating current (ac), in which the flow of charge is continually changing in magnitude (and direction) with time.

**FIG. 5.4 Series connection of resistors.** SERIES RESISTORS • Before the series connection is described, first recognize that every fixed resistor has only two terminals to connect in a configuration—it is therefore referred to as a two-terminal device.

**FIG. 5.6 Series connection of resistors.** FIG. 5.7 Series connection of four resistors of the same value SERIES RESISTORS

**Series Circuits** Two elements in a series Connected at a single point (node) No other current-carrying connections at this node A series circuitis constructed by connecting various elements in series

**Series Circuits** Normally Current will leave the positive terminal of a voltage source and move through the resistors Return to negative terminal of the source Current is the same everywhere in a series circuit

**Series Circuits** Current is similar to water flowing through a pipe Current leaving the element must be the same as the current entering the element Current = water flow rate Pressure = potential difference = voltage Same current passes through every element of a series circuit

**Kirchhoff’s voltage law (1)** Kirchhoff’s voltage law (KVL) states that the algebraic sum of all voltages around a closed path (or loop) is zero. Mathematically, KVL implies ET - V1 - V2 - V3 - ∙∙∙ - Vn = 0

**Kirchhoff’s voltage law (2)** Another way of stating KVL is: Summation of voltage rises is equal to the summation of voltage drops around a closed loop V1 + V2 + V3 + ∙∙∙ + Vn = ET being ET= E1+E2+E3+…+En

**Kirchhoff’s Voltage Law (KVL) (3)** A closed loop is any path that: Originates at a point Travels around a circuit Returns to the original point without retracing any segments

**Kirchhoff’s Voltage Law (4)** Summation of voltage rises is equal to the summation of voltage drops around a closed loop E1 + E2 = v1 + v2 + v3

**Example Problem 1** Determine the unknown voltages in the network below:

**Example Problem 2** Use Kirchhoff’s Voltage Law to determine the magnitude and polarity of the unknown voltage ES in the circuit below:

**Resistors in Series** Most complicated circuits can be simplified For a series circuit V1 + V2 + V3 = E IR1 + IR2 + IR3 = E I(R1 + R2 + R3 )= E I(R1 + R2 + R3 )= IRtotal (Note: I’s cancel)

**Two resistors in series** Two resistors in series can be replaced by an equivalent resistance Req.

**Nresistors in series** The equivalent resistance Req of any number of resistors in series is the sum of the individual resistances.

**Resistors in Series**

**Example Problem 3** For the circuit below (with Rtot =800Ω), determine: • Direction and magnitude of current • Voltage drop across each resistor • Value of the unknown resistance

**Power in a Series Circuit** • Power dissipated by each resistor is determined by the power formulas: P = VI = V2/R = I2R

**Power in a Series Circuit** • Since energy must be conserved, power delivered by voltage source is equal to total power dissipated by resistors PT = P1 + P2 + P3 + ∙∙∙ + Pn

**Example Problem 4** For the circuit below, determine: • Power dissipated by each resistor and total power dissipated by the circuit. • Verify that the summation of the power dissipated by each resistor equals the total power delivered by the voltage source.

**Voltage Sources in Series** • In a circuit with more than one source in series • Sources can be replaced by a single source having a value that is the sum or difference of the individual sources • Polarities must be taken into account

**Voltage Sources in Series** • Resultant source • Sum of the rises in one direction minus the sum of the voltages in the opposite direction

**Simplifying Sources**

**Interchanging Series Components** • Order of series components • May be changed without affecting operation of circuit • Sources may be interchanged, but their polarities can not be reversed • After circuits have been redrawn, it may become easier to visualize circuit operation

**Interchanging Series Components**

**Example Problem 5** Redraw the circuit below, showing a single voltage source and single resistor. Solve for the current in the circuit.

**Switches** The most basic circuit components is a switch. The switch below is known as a single-pole, single-throw (SPST) switch.

**Fuses** A fuse is a device that prevents excessive current to protect against overloads or possible fires. A fuse literally “blown” can not be reset.

**Circuit breakers** A circuit breaker also prevents excessive current in circuits however is uses an electro-mechanical mechanism that opens a switch. A “popped” circuit break can be reset.

**Consolidated Schematic** Circuit Breaker Ammeter Lamp Battery Fuse Voltmeter