ELEC 221 Final Exam Workbook ELEC 221 Workbook 1.1 Definitions and Equations Voltage: Amount of potential energy difference between two points. Energy required per unit charge for separation. Measured in volts. 푑푤 푣 = 푑푞 Where v = voltage (V), w = energy (J), q = charge (C) Current: Rate of charge flow. Measured in amps. 푑푞 푖 = 푑푡 Where i = current (A), q = charge (C), t = time (s) Power: Time rate of expending or absorbing energy. Measured in watts. 푑푤 푝 = 푑푡 Where p = power (W), w = energy (J), t = time (s) Conservation of Energy: Total power delivered = total power absorbed. Energy can only be transferred. ∑푝 = 0 Where p = power (W) 1.2 Basic Circuit Elements Resistor: A device having a designed resistance that electric current can flow through. R = Resistance, measured in 푣 = 푖 ∙ 푅 Ohms (Ω) 푝 = 푣 ∙ 푖 Switches: Two positions open and closed. Current cannot flow through an open circuit as there is no path. Open Circuit: i = 0, no current flow Short Circuit: Can have current flow, v = 0 due to no resistance 1 of 22 ELEC 221 Workbook Independent Sources: An idealized circuit component that fixes the voltage or current in a branch. Ideal Voltage Source Ideal Current Source Dependent Sources: A voltage or current source whose value depends on a voltage or current somewhere else in the network. Usually represented by a diamond shape. Dependent Voltage Source Dependent Current Source 1.3 Kirchhoff’s Laws Kirchhoff’s Current Law: The sum of the currents entering a node is zeros. (The incoming currents must equal the outgoing currents). For all general cases: For the diagram to the left: ∑푖푛표푑푒 = 0 푖푎 + 푖푏 − 푖푐 − 푖푑 = 0 Kirchhoff’s Voltage Law: The sum of voltage rises in a loop is zero. For all general cases: For the diagram to the left: ∑푣푑푟표푝푠 푖푛 푙표표푝 = 0 −푣푎 + 푣푏 − 푣푐 = 0 1.4 Basic Resistor Simplifications Resistors in Series: 푅푒푞 = 푅1 + 푅2 + 푅3 + 푅4 Resistors in Parallel: Any two elements that connect to form a loop are said to be in parallel. Parallel elements have the same voltage. 푅1 ∙ 푅2 푅푒푞 = 푅1 + 푅2 2 of 22 ELEC 221 Workbook 1.5 Divider Circuits Voltage Divider Circuit: Voltages are proportional to their corresponding resistance 푅1 푉1 = 푉푠 ∙ 푅1 + 푅2 푅2 푉2 = 푉푠 ∙ 푅1 + 푅2 Current Divider Circuit: Currents are reciprocal to their corresponding resistances 푅2 푖1 = 푖푠 ∙ 푅1 + 푅2 푅1 푖2 = 푖푠 ∙ 푅1 + 푅2 Example: Find the output voltage (v0) 2.1 Node-Voltage Analysis First make sure that the circuit schematic is neat and has no branches crossing over each other. After label, all the essential nodes on the circuit. Next choose an essential node as ground, this will be your reference node. This choice is arbitrary; however, it is recommended to ground the node with the most branches connected to it. Each node equation is generated by applying Kirchhoff’s current law at each node. Thus, the current entering each node minus the current leaving each node must equal zero. After all the nodes have their respective equations, it just comes down to re-arranging equations. 3 of 22 ELEC 221 Workbook i) Standard Example: Solve for the node voltages ii) With Dependent Source 4 of 22 ELEC 221 Workbook iii) Supernode Cutset A Supernode is when there is a voltage source between two essential nodes. The two nodes can be combined to form a Supernode. The voltage difference between the two nodes is equal to the voltage source between them, leading to easier calculations. 2.2 Mesh-Current Analysis A mesh is any loop without a smaller loop inside of it. Mesh-Current analysis applies Kirchhoff’s laws to write equations for each mesh. This method is okay, however for it to work you must know the voltage drop across every circuit element you pass through, thus it raises difficulties when going through current sources or dependent sources. i) Standard Example: Find mesh currents 5 of 22 ELEC 221 Workbook ii) Special Case 1: Current Source Between Two Meshes To create a Supermesh, the current source between the two meshes is simply avoided when writing current equations. Can then use this current source value to relate some mesh currents. 3.1 Delta-Wye Conversion This conversion is used to establish equivalent circuits between a delta (∆) and wye (Y) arrangement. This conversion can be used when three circuit elements are not sources and are connected in one of the proper arrangements seen below. Can convert between these two arrangement with the conversions seen below. 6 of 22 ELEC 221 Workbook Equations for the resistors can be seen below. Use the left side if going from Y to ∆. Use the right side if going from ∆ to Y. 푅푏푅푐 푅1푅2 + 푅2푅3 + 푅3푅1 푅1 = 푅푎 = 푅푎 + 푅푏 + 푅푐 푅1 푅푐푅푎 푅1푅2 + 푅2푅3 + 푅3푅1 푅2 = 푅푏 = 푅푎 + 푅푏 + 푅푐 푅2 푅푎푅푏 푅1푅2 + 푅2푅3 + 푅3푅1 푅3 = 푅푐 = 푅푎 + 푅푏 + 푅푐 푅3 3.2 Source Transformations Replace a voltage source in series with a resistor with a current source in parallel with a resistor. Can also do the reverse of the previous statement. Very useful in simplifying circuits, should be used immediately if one of the above situations is present in your circuit. where 푖푠 = 푉푠/푅 Note that the resistor value (R) stays the same for the source transformation. Example: Find the power from the 12V source. 7 of 22 ELEC 221 Workbook 4.1 Thevenin and Norton Equivalent Circuits Thevenin: Network can be replaced by equivalent circuit with an independent voltage source called Vth , in series with a resistor. The Thevenin equivalent is obtained by looking into a circuit from some terminals. Much more common than Norton and will be used in future courses. A Thevenin diagram can be seen below. Norton: Network can be replaced by equivalent circuit with an independent current source in parallel with a resistor. i) For a network with only independent sources: The first way to find a Thevenin is by simplifying the circuit to a voltage source in series with a resistor using source transformations seen in the previous example. The voltage source is the Vth, and the resistor is the Rth. However, for a network with only dependent sources, it may be more efficient to remove the sources. Simply short the voltage sources and open current sources. Then combine the resistors to find Rth. Find the Thenvein resistor for the question below by removing the sources. Redraw the circuit with the voltage source shorted and the current source open. Maximum Power Transfer: To maximize the power from a network to a load. 2 푉푡ℎ 푃푙 푚푎푥 = 4푅퐿 8 of 22 ELEC 221 Workbook 5.1 Superposition Principle When a system is excited by several independent sources or excitations, the total output is the sum of all individual responses. Solve for the desired value from each excitation then sum add them together. Total Output = Output1 + Output2 + … + Outputn To turn off the other sources: • Current Sources: Open • Voltage Sources: Short * Superposition can not be used to calculate power as power is a non-linear function* Find the current going through the 3KΩ resistor below, using superposition. 6.1 Non-Linear Circuit Elements in DC Conditions Inductor: An element which opposes change in current. Composed of a coil of wire wrapped around a core. 푑푖 푣 = 퐿 푙 푑푡 * INDUCTOR CURRENT CAN NOT JUMP, This is why you short circuit inductors at DC * Capacitor: An element which opposes change in voltage. Composed of two conductors separated by an insulator. 푑푣 푖 = 퐶 푐 푑푡 9 of 22 ELEC 221 Workbook * CAPACITOR VOLTAGE CAN NOT JUMP, This is why you open circuit capacitors at DC * Adding Inductors and Capacitors: Inductors add the same as resistors, capacitors add opposite of resistors (series = parallel, parallel = series). 7.1 First Order RL and RC Circuits, Second Order RLC Circuits in DC Conditions RL: Sources + Resistors + Inductor RC: Sources + Resistors + Capacitors **You assume that the initial condition of the switch has been there for a very long time.** 퐿 Time Constant: 휏 = = 푅퐶 푅 When the elapsed time exceeds five time constants, the current is less than 1% of the initial value. Thus considered past this point to be essentially zero. Natural Response: Response of a capacitive/inductive circuit to the initial conditions (no impulse). RL Circuit: i) Find initial current through the inductor, Io 퐿 ii) Find the time constant, 휏 = 푅 −푡 ⁄휏 iii) Use the equation 푖(푡) = 푖푓푖푛푎푙 + (푖표 − 푖푓푖푛푎푙)푒 , to get i(t) of the inductor Switch has been closed for a long time, opens at time 0. Find iL(t) for t ≥ 0, i0(t) for t ≥ 0+, vo(t) for t ≥ 0+. 10 of 22 ELEC 221 Workbook RC Circuit: i) Find initial voltage through the capacitor, Vo ii) Find the time constant, 휏 = 푅퐶 −푡/휏 iii) Use the equation 푣(푡) = 푣푓푖푛푎푙 + (푣0 − 푣푓푖푛푎푙)푒 , to get v(t) of the capacitor Switch has been in position x for a long time. At t = 0, switch instantly moves to position y. Find vc(t) for t ≥ 0, vo(t) for t ≥ 0+, io(t) for t ≥ 0+. 11 of 22 ELEC 221 Workbook Step Response: Response of a capacitive/inductive circuit to step inputs (abrupt changes in voltage or current, think applying a source). RL Circuit: Given the circuit below the step response current can be found using the equation below. 푅 −( )푡 푖(푡) = 푖퐹푖푛푎푙 + (푖0 − 푖퐹푖푛푎푙)푒 퐿 (same equation as we used for natural response… think about what I approaches as t goes to ∞) Ex: Find the expression for il(t) for t≥0.