Some Important Electrical Units Quantity Unit Symbol Current Charge Voltage Resistance Power Ampere Coulomb Volt Ohm Watt A C V W W These derived units are based on fundamental units from the meterkilogram-second system, hence are called mks units. Except for current, all electrical and magnetic units are derived from the fundamental units. SI Fundamental Units Current is a fundamental unit.
Unit Symbols Metric Prefixes
Error, Accuracy, and Precision Experimental uncertainty is part of all measurements. Error is the difference between the true or best accepted value and the measured value. Accuracy ( 정확도 ) is an indication of the range of error in a measurement. Precision ( 정밀도 ) is a measure of repeatability or consistency. } Error Precise, but not accurate. 교정 (correction) 이필요함 Tolerance ( 허용오차 ) ~ 1 %, 5 %, 10 %
Significant Digits ( 유효숫자 ) When reporting a measured value, one uncertain digit may be retained but other uncertain digits should be discarded. Normally this is the same number of digits as in the original measurement. 1. Nonzero digits are always considered to be significant. Example: 23.92 has four nonzero digits they are all significant. 2. Zeros to the left of the first nonzero digit are never significant. Example: 0.00276 has three zeros to the left of the first nonzero digit. There are only three significant digits. 3. Zeros between nonzero digits are always significant. Example: 806 has three significant digits. 4. Zeros to the right of the decimal point for a decimal number are significant. Example: 9.00 has three significant digits. 5. Zeros to the left of the decimal point with a whole number may or may not be significant depending on the measurement. Example: 4000 does not have a clear number of significant digits.
Electrical Safety Safety is always a concern with electrical circuits. Knowing the rules and maintaining a safe environment is everyone s job. Do not work alone, or when you are drowsy. Do not wear conductive jewelry. Know the potential hazards of the equipment you are working on; check equipment and power cords frequently. Avoid all contact with energized circuits; even low voltage circuits. Maintain a clean workspace. Know the location of power shutoff and fire extinguishers. Don t have food or drinks in the laboratory or work area. ( 컴퓨터옆에둔커피잔은반드시엎어진다 ) Body resistance ~ 20 kw
Current Current (I) is the amount of charge (Q) that flows past a point in a unit of time (t). The defining equation is: I Q t Current sources are not as common as voltage sources, but they are useful for production testing. The units shown here include current sources as well as measurement instruments.
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. 1 l Conductance is the reciprocal of resistance. G R R A Components designed to have a specific amount of resistance are called resistors.
Resistance color-code Color Digit Multiplier Tolerance Resistance value, first three bands: First band 1 st digit Second band 2 nd digit *Third band Multiplier (number of zeros following second digit) Black Brown Red Orange Yellow Green Blue Violet Gray White 0 1 2 3 4 5 6 7 8 9 10 0 10 1 10 2 10 3 10 4 10 5 10 6 10 7 10 8 10 9 1% (five band) 2% (five band) 흑갈빨주노초파보회흰 Fourth band - tolerance Gold Silver ±5% ± 10% 10-1 10-2 5% (four band) 10% (four band) No band ± 20% * 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. 5.1 kw ± 5% 47 W ± 10%
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. Variable resistors include the potentiometer and rheostat. The center terminal of a variable resistor is connected to the wiper. R R Variable resistor (potentiometer) 분압기 Variable resistor (rheostat) 가감저항기
Switches
Voltage The defining equation for voltage is V W Q 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. Ideally, a voltage source can provide a constant voltage for any current required by a circuit. The IV curve for an ideal voltage source has a constant voltage for all current. In practice, ideal sources do not exist, but they can be closely approximated by actual sources.
A basic power supply
The DMM The DMM (Digital Multimeter) is an important multipurpose instrument which can measure voltage, current, and resistance. Many include other measurement options. Voltmeter connection to measure voltage in a simple circuit.
31 2-digit DMM illustrates how the resolution changes with the number of digits in use A typical analog multimeter
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: If you need to solve for voltage, Ohm s law is: V IR If you need to solve for resistance, Ohm s law is: R V I I V R What is the (hot) resistance of the bulb? 115 V OFF V Hz V mv 10 A A Range Autorange Touch/Hold 1 s 1 s V 40 ma COM Fused
Graph of Current versus Voltage Current (ma) Notice that the plot of current versus voltage for a fixed resistor is a line with a positive slope. What is the resistance indicated by the graph? 2.7 kw What is its conductance? 0.37 ms 10 8.0 6.0 4.0 2.0 0 0 10 20 30 Voltage (V) + DC Ammeter - Power Supply Gnd 5 V 2A V A +15 V - + - + The resistor is green-blue brown-gold. What should the ammeter read? 초 - 파 - 갈 ~ 560 W 흑갈빨주노초파보
Energy and Power Energy is closely related to work. Energy is the ability to do work. It is measured in the same units as work, namely the newton-meter (N-m) or joule (J). What amount of energy is converted to heat in sliding a box along a floor for 5 meters if the force to move it is 400 N? W = Fd = (400 N)(5 m) = 2000 N-m = 2000 J Power is the rate of doing work. Because it is a rate, a time unit is required. The unit is the joule per second (J/s), which defines a watt (W). P W t What power is developed if the box in the previous example is moved in 10 s? W 2000 J P 200 W t 10 s
Electrical Energy and Power In electrical work, the rate energy is dissipated can be determined from any of three forms of the power formula. P 2 I R P VI P V R 2 Together, the three forms are called Watt s law. What power is dissipated in a 27 W resistor is the current is 0.135 A? Given that you know the resistance and current, substitute the values into P =I 2 R. P 2 I R 2 (0.135 A) 27 0.49 W W
Resistor failures Relative sizes of metal-film resistors with standard power ratings of 1 8 W, 1 4 W, 1 2 W, and 1 W. Typical resistors with high power ratings A triple output power supply
Series circuits A series circuit is one that has only one current path. If you break one of the resistors in series circuit, current does not flow at all. The current everywhere is the same, because there is only one path. The total resistance of resistors in series is the sum of the individual resistors.
Kirchhoff s voltage law 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 n i1 V i 0 V S Sum of n voltage drops equals the source voltage.
Voltage divider R 1 15 kw V S + R 2 20 V 10 kw What is the voltage across R 2? V R 10 kw V 20 V 25 kw 2 2 S RT 8.0 V Notice that 40% of the source voltage is across R 2, which represents 40% of the total resistance. V S + 15 V R 1 20 kw R 2 10 kw V OUT What is the largest output voltage available? 5.0 V Voltage dividers can be set up for a variable output using a potentiometer. In the circuit shown, the output voltage is variable.
Power in Series Circuits R 1 470 W V S + R 2 20 V 330 W Use the voltage divider rule to find V 1 and V 2. Then find the power in R 1 and R 2 and P T. Applying the voltage divider rule: V 1 470 800 W 20 V W V 2 330 800 W 20 V W The power dissipated by each resistor is: P 1 11.75 V 2 470 W P 2 8.25 V 2 330 W P T = P 1 + P 2
Voltage measurements Voltage is relative and is measured with respect to another point in the circuit. V S + 12 V A R 1 5.0 kw B R 2 10 kw C For example, V A means the voltage at point A with respect to ground (called reference ground). V B means the voltage at point B with respect to ground. V AB means the voltage between points A and B. Q1. What are V A, V B, and V AB for the circuit shown? V A = 12 V V B = 8 V V AB = 4 V V S + 12 V A R 1 5.0 kw B R 2 10 kw C Q2. What are V A, V B, and V C for the circuit? V A = 4 V V B = 0 V V C = -8 V Q3. Has V AB changed from the previous circuit?
Resistors in parallel Resistors that are connected to the same two points are said to be in parallel. ( 병렬저항 ) A parallel circuit is identified by the fact that it has more than one current path (branch) connected to a common voltage source.
Voltage across parallel branches is the same
Parallel circuit rules Because all components are connected across the same voltage source, the voltage across each is the same. +5.0 V + - +5.0 V + - +5.0 V + - +5.0 V + - V S R 1 R 2 R 3 +5.0 V 680 W 1.5 kw 2.2 kw The total resistance of resistors in parallel is the reciprocal of the sum of the reciprocals of the individual resistors. RT R 1 R 2 1 1 1 R R 1 2 or R T RR 1 2 R R 1 2
Connecting resistors in parallel reduces total resistance and increases total current. Circuit with n resistors in parallel.
Kirchhoff s current law The sum of the currents entering a node is equal to the sum of the currents leaving the node. 5.0 V + V S R 1 R 2 R 3 680 W 1.5 kw 2.2 kw Tabulating current, resistance, voltage and power is a useful way to summarize parameters in a parallel circuit. I T = I 1 +I 2 +I 3 =V S /R T
Current divider When current enters a node (junction) it divides into currents with values that are inversely proportional to the resistance values. The most widely used formula for the current divider is the two-resistor equation. For resistors R 1 and R 2, R 1 R 2 Notice the subscripts. The resistor in the numerator is not the same as the one for which current is found. Notice that the larger resistor has the smaller current.
Power in parallel circuits 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. P V R 2 As in the series case, the total power is the sum of the powers dissipated in each resistor.
Combination circuits Most practical circuits have various combinations of series and parallel components. You can frequently simplify analysis by combining series and parallel components.
Equivalent circuits 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
Kirchhoff s Law Kirchhoff s voltage law and Kirchhoff s current law can be applied to any circuit, including combination circuits. V S + 10 V R 1 270 W R 2 330 W R 3 470 W Tabulating current, resistance, voltage and power is a useful way to summarize parameters. Solve for the unknown quantities in the circuit shown. Kirchhoff s laws can be applied as a check on the answer. Notice that the current in R 1 is equal to the sum of the branch currents in R 2 and R 3. The sum of the voltages around the outside loop is zero.
Loaded voltage divider + R 1 A R 2 R 3 The voltage-divider equation was developed for a series circuit. Recall that the output voltage is given by V R V R V 2 2 2 S S RT R1 R2 A voltage-divider with a resistive load is a combinational circuit and the voltage divider is said to be loaded. The loading reduces the total resistance from node A to ground. + R 1 A Form an equivalent series circuit by combining R 2 and R 3 ; then apply the voltage-divider formula to the equivalent circuit: R 2 R 3 V V R V, R R RR 2,3 2 3 2 3 S 2,3 R1 R 2,3 R2 R3 3 R2 R 2 R 3 R 2 R 2 2 S S S R3 R2 R3 R RR 1 2 1 R 2 R1 R 2 R1R2 R2 R 3 R 3 R3 V V V V
Stiff voltage divider V S R 1 A stiff voltage-divider is one in which the loaded voltage nearly the same as the no-load voltage. R 2 R L To accomplish this, the load current must be small compared to the bleeder current (or R L is large compared to the divider resistors). If R 1 = R 2 = 1.0 kw, what value of R L will make the divider a stiff voltage divider? What fraction of the unloaded voltage is the loaded voltage? R L > 10 R 2 ; R L should be 10 kw or greater. For a 10 kw load, R R R V V V 0.476 V 2 L 2 L S S S R1 R2 RL RR 1 2 R1R2 R3 This is 95% of the unloaded voltage.
Loading effect of a voltmeter Assume V S = 10 V, but the meter reads only 4.04 V when it is across either R 1 or R 2. Can you explain what is happening? V S + 10 V R 1 470 kw + 4.04 10 V V R 2 470 kw + 4.04 V All measurements affect the quantity being measured. A voltmeter has internal resistance, which can change the resistance of the circuit under test. In this case, a 1 MW internal resistance of the meter accounts for the readings.
- + Wheatstone bridge V S R 1 R 3 Output R 2 R 4 The Wheatstone bridge consists of a dc voltage source and four resistive arms forming two voltage dividers. The output is taken between the dividers. Frequently, one of the bridge resistors is adjustable. When the bridge is balanced, the output voltage is zero, and the products of resistances in the opposite diagonal arms are equal. How to prove it: R R R R 1 4 2 3
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. V TH R TH Step 0: The original circuit. Step 1: Calculating the equivalent output voltage. Assume an infinite load between A and B Step 2: Calculating the equivalent resistance. Assume current flow between A and B Thevenin s equivalent circuit
Superposition theorem + - + - 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. V S1 12 V R 1 R 3 2.7 kw 6.8 kw + I 2 - R 2 6.8 kw V S2 18 V What does the ammeter read for I 2? Source 1: R T(S1) = 6.10 kw I 1 = 1.97 ma I 2 = 0.98 ma Source 2: R T(S2) = 8.73 kw I 3 = 2.06 ma I 2 = 0.58 ma Both sources I 2 = 1.56 ma The total current is the algebraic sum.
Capacitor ++++ Capacitor is composed of two conductive plates separated by an insulating dielectric. The ability to store charge is the definition of capacitance. Conductors Dielectric Capacitance is the ratio of charge to voltage ~ C Q V [Farad] = [Coulomb]/[Volt] 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 W 1 CV 2 2 C is directly proportional to the relative dielectric constant and the plate area. C is inversely proportional to the distance between the plates Foil Mica Foil Mica Foil Mica Foil.022 47 MF VTT VTT ++++ 47 MF VTT VTT C 12 r 8.85 10 F/m A - d.022 Mica capacitor Electrolytic capacitors Variable capacitors
Percent of final value When capacitors are connected in series, the total capacitance is smaller than the smallest one. C T 1 1 1 1 1... C C C C 1 2 3 T When capacitors are connected in parallel, the total capacitance C C C C C is the sum of the individual capacitors. T 1 2 3... n For an RC circuit, the time constant for charging or discharging is t = RC [seconds] ~ exponential curve 100% 80% 86% 95% 98% 99% 60% 63% 40% 37% R R 20% 14% 5% 2% 1% C C 0 0 1t 2t 3t 4t 5t Number of time constants
Capacitive reactance Capacitive reactance is the opposition to ac by a capacitor. X C 1 1 2πfC C The reactance of a 0.047 mf capacitor when a frequency of 15 khz is applied is 226 W When capacitors are in series, the total reactance is X the sum of the individual reactances. C( tot ) XC1 XC2 XC3 XC n When capacitors are in parallel, the total reactance is the reciprocal of the sum of the reciprocals of the individual reactances. X C( tot ) 1 1 1 1 1 X X X X C1 C2 C3 Cn Capacitive Voltage Divider 1.0 V f = 33 khz C 1 1000 pf C 2 0.01 µf V out X C 1 X C 2 1 1 4.82 k 2πfC 2π 33 khz 1000 pf W 1 2 1 1 482 2πfC 2π 33 khz 0.01 μf W XC( tot ) XC1 XC2 5.30 kw V out X V 482 W C 2 s X C( tot ) 5.30 kw 1.0 V
Low-Pass Filters V out (V) When a signal is applied to an RC circuit, and the output is taken across the capacitor as shown, the circuit acts as a low-pass filter. As the frequency increases, the output amplitude decreases. 8.46 1.57 0.79 9.98 9 8 7 6 5 4 3 2 1 0.1 1 10 20 100 f (khz) V in V out 10 V dc 0 10 V dc 100 W 1 mf 10 V dc 0 10 V rms 100 W ƒ = 1 khz 1 mf 8.46 V rms 10 V rms 100 W ƒ = 10 khz 1 mf 1.57 V rms 10 V rms 100 W ƒ = 20 khz 1 mf 0.79 V rms
High-Pass Filters Reversing the components, and taking the output across the resistor as shown, the circuit acts as a high-pass filter. As the frequency increases, the output amplitude also increases. V in 9.87 5.32 0.63 V out (V) 10 9 8 7 6 5 4 3 2 1 0 0.01 0.1 1 10 f (khz) V out 10 V dc 0 1 mf 10 V dc 100 W 0 V dc 10 V rms 1 mf ƒ = 100 Hz 100 W 0.63 V rms 10 V rms 1 mf ƒ = 1 khz 100 W 5.32 V rms 10 V rms 9.87 V rms 1 mf ƒ = 10 khz 100 W
Inductor When a length of wire is formed into a coil, it becomes a basic inductor. When there is current in the inductor, a three-dimensional magnetic field is created. [henry] B m ni mni / total 0 NBA Li NBA i 2 L NA m0n m0n A Faraday s law The amount of voltage induced in a coil is directly proportional to the rate of change of the magnetic field with respect to the coil. Lenz s law When the current through a coil changes and an induced voltage is created as a result of the changing magnetic field, the direction of the induced voltage is such that it always opposes the change in the current. S N - V+ Practical inductors Actual inductors have winding resistance (R W ) due to the resistance of the wire and winding capacitance (C W ) between turns. An equivalent circuit for a practical inductor including these effects is shown: R W C W L
Series inductors When inductors are connected in series, the total inductance is the sum of the individual inductors. LT L1 L2 L3... L n Parallel inductors When inductors are connected in parallel, the total inductance is smaller than the smallest one. L T 1 1 1 1 1... L L L L 1 2 3 T For an RL circuit, the time constant is τ i =I F + (I i - I F )e -Rt/L I F = final value of current I i = initial value of current i = instantaneous value of current L R R L V initial 0 t Inductor voltage after switch closure Inductive reactance is the opposition to ac by an inductor. The equation for inductive reactance is I final 0 Current after switch closure t
Inductive reactance Inductive reactance is the opposition to ac by an inductor. X 2πfL L L When inductors are in series, the total reactance is the sum of the individual reactances. That is, X X X X X L( tot ) L1 L2 L3 Ln When inductors are in parallel, the total reactance is the reciprocal of the sum of the reciprocals of the individual reactances. X L( tot ) 1 1 1 1 1 X X X X L1 L2 L3 Ln