Ch.2 Resistive Circuits Resistive Circuits 학습목표 • Ohm’s Law • Kirchhoff’s Laws - Kirchhoff Current Law (KCL) - Kirchhoff Voltage Law (KVL) • Series/Parallel Resistor • Wye-Delta 변환 • Dependent Source가 있는 회로 Resistors Conductance A linear resistor obeys OHM’s Law v(t ) i (t ) 𝑖 𝑡 𝑣 𝑡 𝑣 𝑡 𝑅 𝑅𝑖 𝑡 1 𝑅 𝐺 : Conductance unit of conductance : Siemens 𝑖 𝑡 𝐺𝑣 𝑡 𝑖 Linear approximation 𝑣 Linear range Actual v-I relationship Some practical resistors Symbol Sign convention Two special resistor values i v R Circuit Represent ation v0 i0 Short Circuit 𝑅 𝐺 0 ∞ Open Circuit 𝑅 𝐺 ∞ 0 OHM′s Law Given current and resistance Find the voltage I 2A R 5 Given Voltage and Resistance Compute Current 𝑉 10 𝑉 R 3 12[V ] 𝐼 4𝐴 RESISTORS AND ELECTRIC POWER Given 𝑖, 𝑅 Given 𝑣, 𝑅 Determine current and power absorbed by the resistor. 𝑃 𝑃 𝑉𝐼 12 𝑉 𝐼 𝑅 𝑉 𝑅 6 𝑚𝐴 𝟕𝟐 𝒎𝑾 Vs = ?, I = ? 𝑃 𝑉 𝑅 𝑉 6𝑉 Vs = ?, P = ? 𝑉 𝐼 𝐺 𝐼𝑅 ⇒ 𝑉 𝑃 𝐼 𝑅 𝐼 𝐺 Kirchhoff’s Laws NODES, BRANCHES, LOOPS Node: point where two or more elements are joined (for example, node 1) Loop: A closed path that never goes twice over a node Branch: Component connected between two nodes (for example, component R4) Kirchhoff’s Current Law “CHARGE CANNOT BE CREATED NOR DESTROYED” KIRCHHOFF CURRENT LAW (KCL) KCL equation for this node or 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 0 하나의 노드로 들어가는 전류의 합은 해당 노드에서 나오는 전류의 합과 같다. ->하나의 노드로 들어가는(or 나가는) 전류의 대수적인 합은 0이다. SUM OF CURRENTS INTO NODE IS ZERO 𝐼 𝐼 𝐼 𝐼 d c a -3A 2A 4A b Ibe = ? 2𝐴, 3𝐴 4𝐴 ? e 𝐼 4𝐴 3𝐴 2𝐴 0 Example 2.5 Write KCL for the circuit below. Linearly independent 𝑖𝟐 𝑡 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 0 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 0 𝑖𝟒 𝑡 𝑖𝟓 𝑡 𝑖𝟕 𝑡 0 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 0 𝑖𝟔 𝑡 𝑖𝟕 𝑡 𝑖𝟖 𝑡 0 Linearly dependent WRITE ALL KCL EQUATIONS 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 0 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 0 0 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 0 Adding the above 4 eqs. linearly independent 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 0 Supernode = Generalized Node Sum of currents leaving node 2 and 3 is ZERO. Supernode에서 KCL 식을 세우면 내부의 전류 i4는 자동으로 사라짐. Leaving 2: Leaving 3: Adding 2 & 3: 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 𝑖 0 0 𝑖 𝑖 𝑖 0 Example 2.6 Find the unknown current. 𝐼 0.07 0.03 0 𝐼 𝐼 𝐼 0 0.07 𝐼 𝐼 0.05 0 0.03 𝐼 0.04 0 𝐼 100 mA, 𝐼 90mA, 𝐼 70mA, 𝐼 70 mA 30 mA 50 mA 40 mA 10mA Example 2.7 (Dependent Source) Write KCL equations. Again linearly Dependent!! Kirchhoff’s Voltage Law THIS IS A CONSERVATION OF ENERGY PRINCIPLE “ENERGY CANNOT BE CREATE NOR DESTROYED” KIRCHHOFF VOLTAGE LAW (KVL) KVL says that the algebraic sum of voltages around any loop is zero. 𝑉 𝑉 𝑉 𝑉 0 “THOUGHT EXPERIMENT” B VB 𝒒𝑽𝑨𝑩 AB C V V B 𝜟𝑾 A Positive charge gains energy as it moves to a point with higher voltage and releases energy as it moves to a point with lower voltage. q 𝜟𝑾 VC VA VCA 𝜟𝑾 𝒒𝑽𝑩𝑪 If the charge comes back to the initial points, the net energy gain must be zero. 𝒒𝑽𝑪𝑨 𝑞 𝑉 𝑉 KVL: Sum of voltage drops around any loop must be zero. 𝑉 0 Find VR3 Example 2.9 𝑉 𝑉 18𝑉 loop a-b-c-d-e-f-a 𝑉 5 𝑉 𝑉 15 20 𝑉 𝑉 30 0 12𝑉 Example 2.10: Linearly Independent/dependent In KCL, not all possible KCL equations are independent. The same situation arises in KVL. The third equation is the SUM of the other two! Example 2.11 Find Vae, Vec. Left loop 𝑉 10 24 0 Right loop 16 12 4 Either one 𝑉 가장 simple 한 loop 사용. 4 𝑉 6 6 0 𝑉 14 𝑉 𝑉 10 𝑉 0 Example 2.12 𝑉 𝑉 Write KVL eqs for the two closed loops. 𝑉 0 20𝑉 𝑉 𝑉 0 Single Loop Circuits Basic Voltage Division Applying 1. KCL 2. KVL 3. Ohm’s Law 𝑣 𝑣 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑣 𝑡 𝑣 𝑡 Example 2.14 Find the power delivered to the load and the power loss in the transmission line. 𝑷𝒍𝒐𝒂𝒅 𝑽𝒍𝒐𝒂𝒅 𝟏𝟖𝟑. 𝟓 𝟒𝟎𝟎𝒌 𝟏𝟖𝟑. 𝟓 𝟏𝟔. 𝟓 𝑷𝒍𝒊𝒏𝒆 𝑷𝒊𝒏 𝑷𝒍𝒐𝒂𝒅 𝑰𝟐 𝑹𝒍𝒊𝒏𝒆 𝑰𝟐 𝑹𝒍𝒐𝒂𝒅 𝟑𝟔𝟔. 𝟐𝟒 𝒌𝑽 𝟔𝟔 𝑴𝑾 LOSS!! How can one reduce the losses? 𝟕𝟑𝟒 𝑴𝑾 Collect all sources on one side Multiple Sources KVL 𝑣 𝑣 𝒆𝒒 𝑣 𝑣 𝑣 𝑣 𝑣 𝑣 𝑣 𝑣 𝑣 𝑣 𝑣 𝑣 0 Multiple Resistors Single resistor로 Resistors in Series ( 직렬 저항) Find I, Vbd, Vbc and power delivered to 30k resistor. Example 2.15 10k𝐼 10kI 𝑉 20k𝐼 𝑉 12 60k𝐼 𝐼 30k𝐼 30kI 6 20k 20k 40k Power 30kΩ 6 𝐼 𝑅 6 0 6 0.1 mA 0 ⟹ 𝑉 10 V 2𝑉 10 𝐴 30 ∗ 10 Ω 0.3mW Example 2.16 Find 1)Vs 2)power loss in the transmission line. 30 330 Ω 𝑉 330 30 458.3 330 𝐼 458.3 k 330 𝑃 𝐼 𝑅 500kV 1.389 kA 57.88 MW Single Node-Pair Circuit Resistors in Parallel ⟹ 𝟏 𝑹𝒑 𝟏 𝟏 + 𝑹𝟏 𝑹𝟐 𝑹𝒑 𝑹𝟏 𝑹𝟐 𝑹𝟏 𝑹𝟐 THE CURRENT DIVISION 𝒊𝟏 𝒕 𝑹𝟐 𝑹𝟏 𝑹𝟐 𝒊 𝒕 𝒊𝟐 𝒕 𝑹𝟏 𝑹𝟏 𝑹𝟐 𝒊 𝒕 Find I , I , V Example 2.17 80𝑘 ∗ 𝐼 = 24[V] 𝐼 I 40k 80k 0.9 60k 40k 80k 0.6 mA 60k 60k 40k 80k 0.3 mA 0.9 10 10 Example 2.18 Car stereo and circuit model 등가 회로 6Ω 450mA 225mA Power per each speaker 𝑃 𝐼 𝑅 225x10 6 303.75 mW 225mA 6Ω Multiple Sources 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 𝑖 𝑡 Equivalent single current source 𝑖 𝑡 𝑖 𝑡 0 Multiple Resistors 𝑖 𝑡 𝑖 𝑡 1 𝑅 𝑖 𝑡 𝑖 𝑡 ⋯ 𝑖 𝑡 1 1 ⋯ 𝑣 𝑡 𝑅 𝑅 𝑣 𝑡 𝑅 Ohm’s law at every resistor Example 2.19 Find IL. 1 𝑅 1 18k 𝑅 𝐼 1 9k 1 12k 4kΩ 4k 1 10 4k 12k 0.25 mA Series/Parallel Resistor Combinations SERIES COMBINATIONS PARALLEL COMBINATION 1 𝑅 1 𝑅 1 𝑅 𝐺 1 𝑅 ⋯ 𝐺 𝐺 ... 𝐺 Example 2.20 추가 예제 Show that 𝑹𝒂𝒃 Homework a 𝟏𝟏. 𝟐𝛀 Example 2.21 Build 0.0667 Ω when only 0.1 Ωresistors are available. Tolerance for resistors are typically 5%, and 10%. E12 series 𝟏 𝟏𝟐 Example 2.22 Effect of Resistor Tolerance Nominal Resistor Value: 3.3 kΩ Tolerance : 10% RANGES FOR CURRENT AND POWER? Nominal Current: 𝑰 10 3.3 3.03mA Minimum Current: 𝐼 Maximum Current: 𝐼 Nominal Power: 𝑃 10 1.1 3.3 10 0.9 3.3 Minimum Power V ∗ Imin : 27.5mW Maximum Power V∗ Imax : 33.7mW 2.75mA 3.37mA 10 3.3 30.3mW Circuits with Serial/Parallel Resistors Example 2.24 Find all the currents and voltages. 𝐼 12𝑉 12𝑘 𝑉 3 3 9 12 Wye-Delta Transformation This circuit has no resistor in series or parallel 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 || 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 || 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 𝑅 Example 2.27 Compute Is. Circuits with Dependent Sources FIND 𝑉 Example 2.29 10 V KVL: 4 kΩ 10 𝑉 2k ∗ 𝐼 𝐼 2𝑚𝐴 3k ∗ 𝐼 𝑉 4k ∗ 𝐼 𝑉 0 4𝑘 ∗ 𝐼 8 𝑉 FIND 𝑉 Example 2.30 10 10 1𝑘 𝑉 𝑉 10 𝑉 6𝑘 2𝑘 𝑉 3𝑘 4𝑘 4𝑘 4𝑘 2𝑘 4𝑉 3𝑘 𝑉 𝑉 3𝑘 4𝐼 0⟹𝑉 2 12 𝑉 3 0⟹𝐼 12 8𝑉 𝑉 3𝑘 Example 2.32 FIND 𝐺 KVL 𝑣 𝑡 𝑣 𝑡 𝑣 𝑡 𝑖 𝑡 𝑅 𝑣 𝑡 KCL 𝑣 𝑡 𝑖 𝑡 𝑅 𝑔 𝑣 𝑡 𝑣 𝑡 𝑅 𝑔 𝑣 𝑡 𝑅 𝐆𝐚𝐢𝐧 𝑣 𝑡 𝑣 𝑡 𝑅 𝑅 𝑅 𝑅 𝑣 𝑡 0 𝑔 𝑅 𝑅 𝑣 𝑡 𝑅 𝑅 𝑔 𝑅 𝑅 𝑅 𝑅 Resistor Technologies Chip resistor and SMT/SMD SMT = Surface Mount Technology SMD = Surface Mount Device Thick-film Resistor • • • • a resistive layer with a thickness of 100 μm on a ceramic base. for general applications with no high requirements lower prices usually packaged as a SMD chip resistor Market Share Thin-film Resistor • • • • thin film with a thickness in the order of 0.1 μm photo etching is used to create patterns in the film to increase/calibrated the resistive path/value used for precision applications. more expensive than thick-film resistor Silicon-Diffused Resistor
