University of Hail Electric Circuits EE 202 Professor / Mohamed A H Eleiwa

Summary Example-1 Scientific and Engineering Notation Very large and very small numbers are represented with scientific and engineering notation. Example-1 47,000,000 = 4.7 x 107 (Scientific Notation) = 47 x 106 (Engineering Notation)

Summary Example-2 Example-3 Scientific and Engineering Notation 0.000 027 = 2.7 x 10-5 (Scientific Notation) = 27 x 10-6 (Engineering Notation) Example-3 0.605 = 6.05 x 10-1 (Scientific Notation) = 605 x 10-3 (Engineering Notation)

Summary Metric Conversions Numbers in scientific notation can be entered in a scientific calculator using the EE key. Most scientific calculators can be placed in a mode that will automatically convert any decimal number entered into scientific notation or engineering notation.

Summary SI Fundamental Units Quantity Unit Symbol Length Meter m Kilogram kg Second s Ampere A Kelvin K Candela cd Mole mol Mass Time Electric current Temperature Luminous intensity Amount of substance

Summary Some Important Electrical Units Except for current, all electrical and magnetic units are derived from the fundamental units. Current is a fundamental unit. Quantity Unit Symbol Current Ampere A Coulomb C Volt V Ohm W Watt W Charge These derived units are based on fundamental units from the meter-kilogram-second system, hence are called mks units. Voltage Resistance Power

Summary Large Engineering Metric Prefixes P T G M k peta tera giga mega kilo 1015 1012 109 106 103 Can you name the prefixes and their meaning?

Summary Small Engineering Metric Prefixes m n p f milli micro nano pico femto 10-3 10-6 10-9 10-12 10-15 Can you name the prefixes and their meaning?

Summary Example-1 Metric Conversions Smaller unit 0.47 MW = 470 kW When converting from a larger unit to a smaller unit, move the decimal point to the right. Remember, a smaller unit means the number must be larger. Smaller unit Example-1 0.47 MW = 470 kW Larger number

Summary Example-2 Metric Conversions Larger unit 10,000 pF = 0.01 mF When converting from a smaller unit to a larger unit, move the decimal point to the left. Remember, a larger unit means the number must be smaller. Larger unit Example-2 10,000 pF = 0.01 mF Smaller number

Summary Example-1 Metric Arithmetic 10,000 W + 22 kW = When adding or subtracting numbers with a metric prefix, convert them to the same prefix first. Example-1 10,000 W + 22 kW = 10,000 W + 22,000 W = 32,000 W Alternatively, 10 kW + 22 kW = 32 kW

Summary Example-2 Metric Arithmetic 200 mA + 1.0 mA = When adding or subtracting numbers with a metric prefix, convert them to the same prefix first. Example-2 200 mA + 1.0 mA = 200 mA + 1,000 mA = 12,000 mA Alternatively, 0.200 mA + 1.0 mA = 1.2 mA

Summary Voltage The defining equation for voltage is One volt is the potential difference (voltage) between two points when one joule of energy is used to move one coulomb of charge from one point to the other.

Summary Question: Current Current (I) is the amount of charge (Q) that flows past a point in a unit of time (t). The defining equation is: One ampere is a number of electrons having a total charge of 1 C moving through a given cross section in 1 s. Question: What is the current if 2 C passes a point in 5 s? 0.4 A

Summary Resistance Resistance is the opposition to current. One ohm (1 W) is the resistance if one ampere (1 A) is in a material when one volt (1 V) is applied. Conductance is the reciprocal of resistance. Components designed to have a specific amount of resistance are called resistors.

Summary Resistance color-code Resistance value, first three bands: First band – 1st digit Second band – 2nd digit *Third band – Multiplier (number of zeros following second digit) Fourth band - tolerance * For resistance values less than 10 W, the third band is either gold or silver. Gold is for a multiplier of 0.1 and silver is for a multiplier of 0.01.

Summary Question What is the resistance and tolerance of each of the four-band resistors? 5.1 kW ± 5% 820 kW ± 5% 47 W ± 10% 1.0 W ± 5%

Summary Alphanumeric Labeling Two or three digits, and one of the letters R, K, or M are used to identify a resistance value. The letter is used to indicate the multiplier, and its position is used to indicate decimal point position.

Summary Variable resistors Variable resistors include the potentiometer and rheostat. The center terminal of a variable resistor is connected to the wiper. R Variable resistor (potentiometer) R To connect a potentiometer as a rheostat, one of the outside terminals is connected to the wiper. Variable resistor (rheostat)

Summary The electric circuit Circuits are described pictorially with schematics. For example, the flashlight can be represented by Switch Battery (2 cells) Lamp

Summary The DMM VW The DMM (Digital Multimeter) is an important multipurpose instrument which can measure voltage, current, and resistance. Many include other measurement options.

Summary Analog meters An analog multimeter is also called a VOM (volt-ohm-milliammeter). Analog meters measure voltage, current, and resistance. The user must choose the range and read the proper scale. Photo courtesy of Triplett Corporation

Summary Review of V, I, and R the amount of energy per charge available to move electrons from one point to another in a circuit and is measured in volts. Voltage is the rate of charge flow and is measured in amperes. Current is the opposition to current and is measured in ohms. Resistance is

Summary Question: Ohm’s law The most important fundamental law in electronics is Ohm’s law, which relates voltage, current, and resistance. Georg Simon Ohm (1787-1854) formulated the equation that bears his name: Question: What is the current in a circuit with a 12 V source if the resistance is 10 W? 1.2 A

Summary Question: Ohm’s law If you need to solve for voltage, Ohm’s law is: Question: What is the voltage across a 680 W resistor if the current is 26.5 mA? 18 V

Summary Question: Ohm’s law If you need to solve for resistance, Ohm’s law is: What is the (hot) resistance of the bulb? Question: 115 V 132 W

Summary Energy and Power In electrical work, the rate energy is dissipated can be determined from any of three forms of the power formula. Together, the three forms are called Watt’s law.

Summary Energy and Power Example-1: Solution: What power is dissipated in a 27 W resistor is the current is 0.135 A? Solution: Given that you know the resistance and current, substitute the values into P =I 2R.

Summary Summary Series circuits All circuits have three common attributes. These are: 1. A source of voltage. 2. A load. 3. A complete path.

Summary Summary A series circuit is one that has Series circuits A series circuit is one that has only one current path. R1 R1 R2 VS R2 VS R1 R2 R3 VS R3 R3

Summary Summary Series circuit rule for current: Because there is only one path, the current everywhere is the same. 2.0 mA For example, the reading on the first ammeter is 2.0 mA, What do the other meters read? 2.0 mA 2.0 mA 2.0 mA

Summary Summary The total resistance of resistors in series is Series circuits The total resistance of resistors in series is the sum of the individual resistors. For example, the resistors in a series circuit are 680 W, 1.5 kW, and 2.2 kW. What is the total resistance? 4.38 kW

Summary Summary Summary Series circuit Tabulating current, resistance, voltage and power is a useful way to summarize parameters in a series circuit. Continuing with the previous example, complete the parameters listed in the Table. I1= R1= 0.68 kW V1= P1= I2= R2= 1.50 kW V2= P2= I3= R3= 2.20 kW V3= P3= IT= RT= 4.38 kW VS= 12 V PT= 2.74 mA 1.86 V 5.1 mW 2.74 mA 4.11 V 11.3 mW 2.74 mA 6.03 V 16.5 mW 2.74 mA 32.9 mW

Summary Summary Voltage sources in series Voltage sources in series add algebraically. For example, the total voltage of the sources shown is 27 V What is the total voltage if one battery is accidentally reversed? Question: 9 V

Summary Summary Kirchhoff’s voltage law Kirchhoff’s voltage law (KVL) is generally stated as: The sum of all the voltage drops around a single closed path in a circuit is equal to the total source voltage in that closed path. KVL applies to all circuits, but you must apply it to only one closed path. In a series circuit, this is (of course) the entire circuit. A mathematical shorthand way of writing KVL is

Summary Summary Kirchhoff’s voltage law Notice in the series example given earlier that the sum of the resistor voltages is equal to the source voltage. I1= R1= 0.68 kW V1= P1= 2.74 mA 1.86 V 5.1 mW I2= R2= 1.50 kW V2= P2= 2.74 mA 4.11 V 11.3 mW I3= R3= 2.20 kW V3= P3= 2.74 mA 6.03 V 16.5 mW IT= RT= 4.38 kW VS= 12 V PT= 2.74 mA 32.9 mW

Summary Summary Voltage divider rule The voltage drop across any given resistor in a series circuit is equal to the ratio of that resistor to the total resistance, multiplied by source voltage. Assume R1 is twice the size of R2. What is the voltage across R1? Question: 8 V

Summary Summary Voltage divider What is the voltage across R2? Example: What is the voltage across R2? Solution: The total resistance is 25 kW. Notice that 40% of the source voltage is across R2, which represents 40% of the total resistance. Applying the voltage divider formula: 8.0 V

Summary Summary Resistors in parallel Resistors that are connected to the same two points are said to be in parallel.

Summary Summary Parallel circuits A parallel circuit is identified by the fact that it has more than one current path (branch) connected to a common voltage source.

Summary Summary Parallel circuit rule for voltage Because all components are connected across the same voltage source, the voltage across each is the same. For example, the source voltage is 5.0 V. What will a volt- meter read if it is placed across each of the resistors?

Summary Summary Parallel circuit rule for resistance The total resistance of resistors in parallel is the reciprocal of the sum of the reciprocals of the individual resistors. For example, the resistors in a parallel circuit are 680 W, 1.5 kW, and 2.2 kW. What is the total resistance? 386 W

Summary Summary Special case for resistance of two parallel resistors The resistance of two parallel resistors can be found by either: or Question: What is the total resistance if R1 = 27 kW and R2 = 56 kW? 18.2 kW

Summary Summary Parallel circuit Tabulating current, resistance, voltage and power is a useful way to summarize parameters in a parallel circuit. Continuing with the previous example, complete the parameters listed in the Table. I1= R1= 0.68 kW V1= P1= I2= R2= 1.50 kW V2= P2= I3= R3= 2.20 kW V3= P3= IT= RT= 386 W VS= 5.0 V PT= 7.4 mA 5.0 V 36.8 mW 3.3 mA 5.0 V 16.7 mW 2.3 mA 5.0 V 11.4 mW 13.0 mA 64.8 mW

Summary Summary Kirchhoff’s current law Kirchhoff’s current law (KCL) is generally stated as: The sum of the currents entering a node is equal to the sum of the currents leaving the node. Notice in the previous example that the current from the source is equal to the sum of the branch currents. I1= R1= 0.68 kW V1= P1= I2= R2= 1.50 kW V2= P2= I3= R3= 2.20 kW V3= P3= IT= RT= 386 W VS= 5.0 V PT= 5.0 V 13.0 mA 2.3 mA 3.3 mA 7.4 mA 36.8 mW 16.7 mW 11.4 mW 64.8 mW

Summary Summary Current divider When current enters a node (junction) it divides into currents with values that are inversely proportional to the resistance values. and The most widely used formula for the current divider is the two-resistor equation. For resistors R1 and R2, Notice the subscripts. The resistor in the numerator is not the same as the one for which current is found.

Summary Summary Example Solution Current divider Assume that R1is a 2.2 kW resistor that is in parallel with R2, which is 4.7 kW. If the total current into the resistors is 8.0 mA, what is the current in each resistor? Solution 5.45 mA 2.55 mA Notice that the larger resistor has the smaller current.

Summary Summary Power in parallel circuits Question: Power in each resistor can be calculated with any of the standard power formulas. Most of the time, the voltage is known, so the equation is most convenient. As in the series case, the total power is the sum of the powers dissipated in each resistor. Question: What is the total power if 10 V is applied to the parallel combination of R1 = 270 W and R2 = 150 W? 1.04 W

Summary Summary Question: Answer: Follow up: Assume there are 8 resistive wires that form a rear window defroster for an automobile. (a) If the defroster dissipates 90 W when connected to a 12.6 V source, what power is dissipated by each resistive wire? (b) What is the total resistance of the defroster? (a) Each of the 8 wires will dissipate 1/8 of the total power or Answer: (b) The total resistance is Follow up: What is the resistance of each wire? 1.76 W x 8 = 14.1 W

Summary Summary Combination circuits Most practical circuits have various combinations of series and parallel components. You can frequently simplify analysis by combining series and parallel components. An important analysis method is to form an equivalent circuit. An equivalent circuit is one that has characteristics that are electrically the same as another circuit but is generally simpler.

Summary Summary Kirchhoff’s voltage law and Kirchhoff’s current law can be applied to any circuit, including combination circuits. So will this path! For example, applying KVL, the path shown will have a sum of 0 V.

Summary Summary Kirchoff’s current law can also be applied to the same circuit. What are the readings for node A?

Summary Summary Combination circuits Tabulating current, resistance, voltage and power is a useful way to summarize parameters. Solve for the unknown quantities in the circuit shown. I1= R1= 270 W V1= P1= 21.6 mA 5.82 V 126 mW I2= R2= 330 W V2= P2= 12.7 mA 4.18 V 53.1 mW I3= R3= 470 W V3= P3= 8.9 mA 4.18 V 37.2 mW IT= RT= VS= 10 V PT= 21.6 mA 464 W 216 mW

Summary Summary Kirchhoff’s laws can be applied as a check on the answer. equal to the sum of the branch currents in R2 and R3. Notice that the current in R1 is The sum of the voltages around the outside loop is zero. I1= R1= 270 W V1= P1= 21.6 mA 5.82 V 126 mW I2= R2= 330 W V2= P2= 12.7 mA 4.18 V 53.1 mW I3= R3= 470 W V3= P3= 8.9 mA 4.18 V 37.2 mW IT= RT= VS= 10 V PT= 21.6 mA 464 W 216 mW

Summary Summary Thevenin’s theorem Thevenin’s theorem states that any two-terminal, resistive circuit can be replaced with a simple equivalent circuit when viewed from two output terminals. The equivalent circuit is:

Summary Summary Thevenin’s theorem the open circuit voltage between the two output terminals of a circuit. VTH is defined as the total resistance appearing between the two output terminals when all sources have been replaced by their internal resistances. RTH is defined as

Summary Summary Thevenin’s theorem What is the Thevenin voltage for the circuit? 8.76 V What is the Thevenin resistance for the circuit? 7.30 kW Output terminals Remember, the load resistor has no affect on the Thevenin parameters.

Summary Summary Maximum power transfer The maximum power is transferred from a source to a load when the load resistance is equal to the internal source resistance. The maximum power transfer theorem assumes the source voltage and resistance are fixed.

Summary Summary Example: Solution: Maximum power transfer What is the power delivered to the matching load? Solution: The voltage to the load is 5.0 V. The power delivered is

Summary Summary Superposition theorem Example: The superposition theorem is a way to determine currents and voltages in a linear circuit that has multiple sources by taking one source at a time and algebraically summing the results. Example: What does the ammeter read for I2? (See next slide for the method and the answer).

Summary Summary What does the ammeter read for I2? Source 1: RT(S1)= I1= I2= Source 2: RT(S2)= I3= I2= Both sources I2= Set up a table of pertinent information and solve for each quantity listed: 6.10 kW 1.97 mA 0.98 mA 8.73 kW 2.06 mA 0.58 mA 1.56 mA The total current is the algebraic sum.

Node Voltage Method (Nodal Analysis)

Summary The Basic Capacitor Capacitors are one of the fundamental passive components. In its most basic form, it is composed of two conductive plates separated by an insulating dielectric. The ability to store charge is the definition of capacitance. Conductors Dielectric

Summary The Basic Capacitor Initially uncharged Source removed Fully charged Charging The charging process… A capacitor with stored charge can act as a temporary battery.

Example Capacitance Capacitance is the ratio of charge to voltage Rearranging, the amount of charge on a capacitor is determined by the size of the capacitor (C) and the voltage (V). Example If a 22 mF capacitor is connected to a 10 V source, the charge is 220 mC

Capacitance A capacitor stores energy in the form of an electric field that is established by the opposite charges on the two plates. The energy of a charged capacitor is given by the equation where W = the energy in joules C = the capacitance in farads V = the voltage in volts

Summary Capacitance The capacitance of a capacitor depends on three physical characteristics. C is directly proportional to the relative dielectric constant and the plate area. C is inversely proportional to the distance between the plates

Summary Example Capacitance Find the capacitance of a 4.0 cm diameter sensor immersed in oil if the plates are separated by 0.25 mm. Example The plate area is The distance between the plates is 178 pF

Summary Series capacitors When capacitors are connected in series, the total capacitance is smaller than the smallest one. The general equation for capacitors in series is The total capacitance of two capacitors is …or you can use the product-over-sum rule

Summary Example Series capacitors If a 0.001 mF capacitor is connected in series with an 800 pF capacitor, the total capacitance is 444 pF

Summary Example Parallel capacitors When capacitors are connected in parallel, the total capacitance is the sum of the individual capacitors. The general equation for capacitors in parallel is Example If a 0.001 mF capacitor is connected in parallel with an 800 pF capacitor, the total capacitance is 1800 pF

Summary The Basic Inductor One henry is the inductance of a coil when a current, changing at a rate of one ampere per second, induces one volt across the coil. Most coils are much smaller than 1 H. The effect of inductance is greatly magnified by adding turns and winding them on a magnetic material. Large inductors and transformers are wound on a core to increase the inductance. Magnetic core

Summary Factors affecting inductance Four factors affect the amount of inductance for a coil. The equation for the inductance of a coil is where L = inductance in henries N = number of turns of wire m = permeability in H/m (same as Wb/At-m) l = coil length on meters

Summary Example What is the inductance of a 2 cm long, 150 turn coil wrapped on an low carbon steel core that is 0.5 cm diameter? The permeability of low carbon steel is 2.5 x10-4 H/m (Wb/At-m). 22 mH

Summary Example Series inductors When inductors are connected in series, the total inductance is the sum of the individual inductors. The general equation for inductors in series is Example If a 1.5 mH inductor is connected in series with an 680 mH inductor, the total inductance is 2.18 mH

Summary Parallel inductors When inductors are connected in parallel, the total inductance is smaller than the smallest one. The general equation for inductors in parallel is The total inductance of two inductors is …or you can use the product-over-sum rule.

Summary Example Parallel inductors If a 1.5 mH inductor is connected in parallel with an 680 mH inductor, the total inductance is 468 mH

OBJECTIVES Become familiar with the characteristics of a sinusoidal waveform, including its general format, average value, and effective value. Be able to determine the phase relationship between two sinusoidal waveforms of the same frequency. Understand how to calculate the average and effective values of any waveform. Become familiar with the use of instruments designed to measure ac quantities.

SINUSOIDAL ac VOLTAGE CHARACTERISTICS AND DEFINITIONS Generation Sinusoidal ac voltages are available from a variety of sources. The most common source is the typical home outlet, which provides an ac voltage that originates at a power plant. Most power plants are fueled by water power, oil, gas, or nuclear fusion.

SINUSOIDAL ac VOLTAGE CHARACTERISTICS AND DEFINITIONS Definitions FIG. 13.3 Important parameters for a sinusoidal voltage.

AVERAGE POWER AND POWER FACTOR Resistor Inductor Capacitor Power Factor

AVERAGE POWER AND POWER FACTOR FIG. 14.34 Purely inductive load with Fp = 0. FIG. 14.33 Purely resistive load with Fp = 1.

COMPLEX NUMBERS A complex number represents a point in a two-dimensional plane located with reference to two distinct axes. This point can also determine a radius vector drawn from the origin to the point. The horizontal axis is called the real axis, while the vertical axis is called the imaginary axis.

COMPLEX NUMBERS FIG. 14.38 Defining the real and imaginary axes of a complex plane.

RECTANGULAR FORM The format for the rectangular form is:

POLAR FORM The format for the polar form is:

MATHEMATICAL OPERATIONS WITH COMPLEX NUMBERS Complex Conjugate Reciprocal Addition Subtraction Multiplication Division

IMPEDANCE AND THE PHASOR DIAGRAM Resistive Elements FIG. 15.1 Resistive ac circuit.

IMPEDANCE AND THE PHASOR DIAGRAM Resistive Elements In phasor form,

IMPEDANCE AND THE PHASOR DIAGRAM Resistive Elements FIG. 15.2 Example 15.1.

IMPEDANCE AND THE PHASOR DIAGRAM Capacitive Reactance We learned in Chapter 13 that for the pure capacitor in Fig. 15.13, the current leads the voltage by 90° and that the reactance of the capacitor XC is determined by 1/ψC. We have

SERIES CONFIGURATION R-L-C FIG. 15.36 Applying phasor notation to the circuit in Fig. 15.35.

VOLTAGE DIVIDER RULE FIG. 15.41 Example 15.10.

FREQUENCY RESPONSE FOR SERIES ac CIRCUITS Series R-C ac Circuit FIG. 15.47 Determining the frequency response of a series R-C circuit.

ILLUSTRATIVE EXAMPLES FIG. 16.1 Example 16.1.

ILLUSTRATIVE EXAMPLES FIG. 16.2 Network in Fig. 16.1 after assigning the block impedances.

ILLUSTRATIVE EXAMPLES FIG. 16.4 Network in Fig. 16.3 after assigning the block impedances. FIG. 16.3 Example 16.2.

Δ-Y, Y-Δ CONVERSIONS The Δ-Y, Y-Δ (or p-T, T-p as defined in Section 8.12) conversions for ac circuits are not derived here since the development corresponds exactly with that for dc circuits. FIG. 17.45 Δ-Y configuration.

Δ-Y, Y-Δ CONVERSIONS FIG. 17.46 The T and π configurations.

Δ-Y, Y-Δ CONVERSIONS FIG. 17.47 Converting the upper Δ of a bridge configuration to a Y.

Δ-Y, Y-Δ CONVERSIONS FIG. 17.48 The network in Fig. 17.47 following the substitution of the Y configuration.

Δ-Y, Y-Δ CONVERSIONS FIG. 17.49 Example 17.21.

Δ-Y, Y-Δ CONVERSIONS FIG. 17.50 Converting a Δ configuration to a Y configuration.

Δ-Y, Y-Δ CONVERSIONS FIG. 17.51 Substituting the Y configuration in Fig. 17.50 into the network in Fig. 17.49.

Δ-Y, Y-Δ CONVERSIONS FIG. 17.52 Converting the Y configuration in Fig. 17.49 to a Δ.

Δ-Y, Y-Δ CONVERSIONS FIG. 17.53 Substituting the Δ configuration in Fig. 17.54 into the network in Fig. 17.49.