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4 | DIRECT CURRENT SERIES & PARALLEL CIRCUITS “ You will never always be motivated. You have to learn to be DISCIPLINED. DC-CURRENT SERIES CIRCUITS A series circuit is a circuit in which there is only one path for current to flow along. The current I is the same in all parts of the circuits. I 𝑰𝑺𝑶𝑼𝑹𝑪𝑬 = 𝑰𝑹𝟏 = 𝑰𝑹𝟐 = 𝑰𝑹𝟑 The total resistance in the circuit is equal to the sum of the resistance of all the parts of the circuit. 𝑹𝒕𝒐𝒕𝒂𝒍 𝑹𝑻𝒐𝒕𝒂𝒍 = 𝑹𝟏 + 𝑹𝟐 + 𝑹𝟑 𝑽𝒕𝒐𝒕𝒂𝒍 The total voltage across a series circuit is equal to the sum of the voltages across each resistance of the circuit. 𝑽𝑻𝒐𝒕𝒂𝒍 = 𝑽𝟏 + 𝑽𝟐 + 𝑽𝟑 𝑷𝒕𝒐𝒕𝒂𝒍 The total power across a series circuit is equal to the sum of the power across each resistance of the circuit. 𝑷𝑻𝒐𝒕𝒂𝒍 = 𝑷𝟏 + 𝑷𝟐 + 𝑷𝟑 PROBLEM 1 A series circuit has a 50-Ω, a 75Ω, and a 100- Ω resistor in series shown on the figure below. Find the total resistance of the circuit. PROBLEM 2 A series circuit has a 6V across R1, 30V across R2, and 54V across R3. What is the total voltage across the circuit? PROBLEM 3 A resistor of 45-Ω and a bell of 60- Ω are connected in series shown below. What voltage is required across this combination to produce a current of 0.3A? PROBLEM 4 A 95V battery is connected in series with three resistors: 20-Ω, 50-Ω, and 120- Ω. Find the voltage across each resistor. PROBLEM 5 In the circuit shown below, find the total power Pt and power dissipated by R1, R2 and R3. POLARITY OF VOLTAGE DROPS The polarity of the voltage drop is determined by the direction of conventional current from positive to a more negative potential. The nearer to the positive terminal of the voltage source is more positive; while the point nearer to the negative terminal of the applied voltage is more negative. V1 = 10V A • + V1, V2, V3 are voltage drops. B • + V2 = 25V - •D - V3 = 60V + •C Va = 95V Vb = 85V Vc = 60V Vd = 0V CONDUCTORS • Conductor is a material having many free electrons. • Generally most metals are good conductors but top 3 good electrical conductors are COPPER, SILVER, and ALUMINUM. • Conductors have very low resistance. Copper wire has a typical resistance of <1Ω for 10feet. • Function of a wire conductor is to connect a source of applied voltage to a load resistance with minimum IR voltage drop in the conductor. WIRE MEASUREMENT RESISTIVITY TEMPERATURE COEFFICIENT WIRE MEASUREMENT • Standard wire sizes corresponds to the American Wire Gauge (AWG). • Gauge numbers specify the size of round wire in terms of its diameter and cross-sectional circular area. Cross-sectional area of round wire is measure in CIRCULAR MILS (CMIL or CM) where 1 mil is one-thousandth of an inch (0.001 inch). cmil = CM = 𝒅𝟐 COPPER WIRE TABLE WIRE MEASUREMENT 1. Gage numbers increase (1 to 40), the diameter & circular area decrease. Gauge# 5 Gauge# 12 Gauge# 15 2. The circular area doubles for every 3 gauge sizes. 3. The higher the gauge number and the smaller the wire, the greater the resistance of the wire for any given length. RESISTIVITY • Resistance of a given length (of any conductor) depends upon the resistivity of the material, length of the wire, and the cross-sectional area of the wire. R= 𝒍 ρ 𝑨 R = resistance of the conductor, Ω l = length of the wire, ft A = cross-sectional area of the wire, CM Ρ = specific resistance or resistivity, CM.Ω/𝑓𝑡 2 PROBLEM 5 What is the resistance of 500-ft of No.20 and No.23 copper wires? TEMPERATURE COEFFICIENT • The temperature coefficient of resistance, , indicates how much the resistance changes for a change in temperature. • A positive value means R increases with temperature; a negative means R decreases; and a zero means R is constant (not varying with changes in temperature. 𝑹𝒕 = 𝑹𝒐 + 𝑹𝒐 ( T) R = resistance of the conductor, Ω l = length of the wire, ft A = cross-sectional area of the wire, CM Ρ = specific resistance or resistivity, CM.Ω/𝑓𝑡 2 PROBLEM 5 A tungsten wire has a 10-Ω resistance at 20ºC. Find its resistance at 120ºC. DC-CURRENT PARALLEL CIRCUITS A parallel circuit is a circuit in which two or more components are connected across the same voltage source. The voltage V is the same in all parts of the circuits. V1 V2 V3 𝑽𝑺𝑶𝑼𝑹𝑪𝑬 = 𝑽𝑹𝟏 = 𝑽𝑹𝟐 = 𝑽𝑹𝟑 I I1 I2 I3 The total current 𝑰𝑻 is equal to the sum of all branch currents. 𝑰𝑺𝑶𝑼𝑹𝑪𝑬 = 𝑰𝑹𝟏 +𝑰𝑹𝟐 + 𝑰𝑹𝟑 𝑷𝟏 𝑷𝑻 𝑷𝟐 𝑷𝟑 The total power 𝑃𝑇 equals the sum of the individual values of power in each branch. 𝑷𝑻 = 𝑷𝟏 +𝑷𝟐 +𝑷𝟑 PROBLEM 6 Two lamps each drawing 2A and third lamp drawing 1A are connected in parallel across a 10-V line. What is the total current? Two branches 𝑅1 and 𝑅2 across a 110-V power line draw a total line current of 20A. Branch 𝑅1 takes 12A. What is the current 𝐼2 in branch 𝑅2 ? PROBLEM 7 I 𝐼1 𝐼2 PROBLEM 8 A parallel circuit consists of a coffee maker, a toaster, and a frying pan plugged into a kitchen appliance circuit on a 120-V line. What currents will flow in each branch of the circuit and what is the total current drawn by all appliances? 𝑹𝑻 = ? The total resistance in parallel is given by the general reciprocal formula: 𝟏 𝑹𝑻 = 𝟏 𝑹𝟏 + 𝟏 𝑹𝟐 + 𝟏 𝑹𝟑 𝑹𝑻 = ? Total resistance of EQUAL resistors in parallel is equal to the resistance of 1 resistor divided by the number of resistors. 𝑹 𝑹𝑻 = 𝑵 SIMPLIFIED FORMULAS 𝑹𝑻 = ? Total resistance of NONEQUAL resistors in parallel is equal to: 𝑹𝟏 𝑹𝟐 𝑹𝑻 = 𝑹𝟏 + 𝑹𝟐 SIMPLIFIED FORMULAS OPEN CIRCUIT X Open in main line OPEN CIRCUIT X Open in parallel branch SHORT CIRCUIT Open in parallel branch PROBLEM 9 Find the current in each parallel branch below. If the resistor in the second branch burns out, causing an open circuit, find the new branch currents. X CONDUCTANCES IN PARALLEL • Conductance is the opposite of resistance. • The less the resistance, the higher the conductance. • The symbol for conductance is G and its units is SIEMENS (S). 𝟏 𝑮= 𝑹 𝑮𝑻 = 𝑮𝟏 + 𝑮𝟐 + 𝑮𝟑 + ⋯ . +𝑮𝒏 Find 𝐼1 and 𝐼2 expressing it with 𝐼𝑇 , 𝑅1 and 𝑅2 . PROBLEM 10 𝐼𝑇 𝐼1 𝐼2 PROBLEM 11 Find 𝑅𝑇 and the 𝑉3 of the circuit? BOARD WORK 1 Find 𝑅𝑇 and 𝐼4 of the circuit? 𝐼𝑇 = 10A PROBLEM 12 Find 𝑅𝑇 and the 𝑉5 of the circuit? PROBLEM 13 Find 𝑅𝑇 of the circuit? 𝐼𝑇 = 10A END