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Week 14 Sinusoidal Steady State Analysis Chapter Objectives Week 14 Sinusoidal Steady State Analysis Chapter Objectives: Apply previously learn circuit techniques to sinusoidal steady-state analysis. Learn how to apply nodal and mesh analysis in the frequency domain. Learn how to apply superposition, Thevenin’s and Norton’s theorems in the frequency domain. KNU/EECS/CT1 Dr. Kalyana Veluvolu 1 1 20cos(5t − 30 ° ) A 1 Ω H 5 10 2 F KNU/EECS/CT1 Dr. Kalyana Veluvolu 2 Find V 0 in the circuit shown below Z1 KNU/EECS/CT1 Dr. Kalyana Veluvolu 3 Problem KNU/EECS/CT1 Dr. Kalyana Veluvolu 4 Chapter 9, Problem 51. KNU/EECS/CT1 Dr. Kalyana Veluvolu 5 ° Problem 9.46 If is = 5 cos (10 t + 40 ) A in the circuit in the Figure, find io. KNU/EECS/CT1 Dr. Kalyana Veluvolu 6 Steps to Analyze AC Circuits Transform the circuit to the Phasor Domain. Solve the problem using circuit techniques listed below 1) Nodal Analysis 2) Mesh Analysis 3) Superposition 4) Source transformation 5) Thevenin or Norton Equivalents Transform the resulting circuit back to time domain. KNU/EECS/CT1 Dr. Kalyana Veluvolu 7 Nodal Analysis Since KCL is valid for phasors, we can analyze AC circuits by NODAL analysis. Determine the number of nodes within the network. Pick a reference node and label each remaining node with a subscripted value of voltage: V 1, V 2 and so on. Apply Kirchhoff’s current law at each node except the reference. Assume that all unknown currents leave the node for each application of Kirhhoff’s current law. Solve the resulting equations for the nodal voltages. For dependent current sources: Treat each dependent current source like an independent source when Kirchhoff’s current law is applied to each defined node. However, once the equations are established, substitute the equation for the controlling quantity to ensure that the unknowns are limited solely to the chosen nodal voltages. KNU/EECS/CT1 Dr. Kalyana Veluvolu 8 Nodal Analysis Since KCL is valid for phasors, we can analyze AC circuits by NODAL analysis. Practice Problem 10.1: Find v1 and v2 using nodal analysis KNU/EECS/CT1 Dr. Kalyana Veluvolu 9 Nodal Analysis Practice Problem 10.1 KNU/EECS/CT1 Dr. Kalyana Veluvolu 10 Nodal Analysis Practice Problem 10.1 KNU/EECS/CT1 Dr. Kalyana Veluvolu 11 Mesh Analysis Since KVL is valid for phasors, we can analyze AC circuits by MESH analysis. Practice Problem 10.4: Calculate the current Io Meshes 2 and 3 form a supermesh as shown in the circuit below. KNU/EECS/CT1 Dr. Kalyana Veluvolu 12 Mesh Analysis KNU/EECS/CT1 Dr. Kalyana Veluvolu 13 Mesh Analysis KNU/EECS/CT1 Dr. Kalyana Veluvolu 14 Superposition Theorem The superposition theorem eliminates the need for solving simultaneous linear equations by considering the effect on each source independently. To consider the effects of each source we remove the remaining sources; by setting the voltage sources to zero (short-circuit representation) and current sources to zero (open-circuit representation). The current through, or voltage across, a portion of the network produced by each source is then added algebraically to find the total solution for current or voltage. The only variation in applying the superposition theorem to AC networks with independent sources is that we will be working with impedances and phasors instead of just resistors and real numbers. KNU/EECS/CT1 Dr. Kalyana Veluvolu 15 Superposition Theorem Superposition Theorem applies to AC circuits as well. For sources having different frequencies, separate phasor circuit for each frequency must be solved independently, and the total response must be obtained by adding individual responses in time domain. Exp. 10.6 Superposition Technique for sources having different frequencies KNU/EECS/CT1 Dr. Kalyana Veluvolu 16 Superposition Theorem a) All sources except DC 5-V set to zero b) All sources except 10cos(10 t) set to zero c) All sources except 2 sin 5t set to zero vo= v1+ v2+ v3 KNU/EECS/CT1 Dr. Kalyana Veluvolu 17 Superposition Theorem P.P.10.6 Superposition Technique for sources having different Frequencies KNU/EECS/CT1 Dr. Kalyana Veluvolu 18 Superposition Theorem KNU/EECS/CT1 Dr. Kalyana Veluvolu 19 Superposition Theorem KNU/EECS/CT1 Dr. Kalyana Veluvolu 20 Source Transformation Transform a voltage source in series with an impedance to a current source in parallel with an impedance for simplification or vice versa. KNU/EECS/CT1 Dr. Kalyana Veluvolu 21 Source Transformation Practice Problem 10.4: Calculate the current Io If we transform the current source to a voltage source, we obtain the circuit shown in Fig. (a). KNU/EECS/CT1 Dr. Kalyana Veluvolu 22 Source Transformation Practice Problem 10.4: Calculate the current Io KNU/EECS/CT1 Dr. Kalyana Veluvolu 23 Thevenin Equivalent Circuit Thévenin’s theorem, as stated for sinusoidal AC circuits, is changed only to include the term impedance instead of resistance. Any two-terminal linear ac network can be replaced with an equivalent circuit consisting of a voltage source and an impedance in series. VTh is the Open circuit voltage between the terminals a-b. ZTh is the impedance seen from the terminals when the independent sources are set to zero. KNU/EECS/CT1 Dr. Kalyana Veluvolu 24 Norton Equivalent Circuit The linear circuit is replaced by a current source in parallel with an impedance. IN is the Short circuit current flowing between the terminals a-b when the terminals are short circuited. Thevenin and Norton equivalents are related by: = = VTh ZI NN Z Th Z N KNU/EECS/CT1 Dr. Kalyana Veluvolu 25 Thevenin Equivalent Circuit P.P.10.8 Thevenin Equivalent At terminals a-b KNU/EECS/CT1 Dr. Kalyana Veluvolu 26 Thevenin Equivalent Circuit P.P.10.9 Thevenin and Norton Equivalent for Circuits with Dependent Sources To find Vth , consider the circuit in Fig. (a). KNU/EECS/CT1 Dr. Kalyana Veluvolu 27 Thevenin Equivalent Circuit P.P.10.9 Thevenin and Norton Equivalent for Circuits with Dependent Sources KNU/EECS/CT1 Dr. Kalyana Veluvolu 28 Thevenin Equivalent Circuit P.P.10.9 Thevenin and Norton Equivalent for Circuits with Dependent Sources KNU/EECS/CT1 Dr. Kalyana Veluvolu 29 Thevenin Equivalent Circuit P.P.10.9 Thevenin and Norton Equivalent for Circuits with Dependent Sources Since there is a dependent source, we can find the impedance by inserting a voltage source and calculating the current supplied by the source from the terminals a-b. KNU/EECS/CT1 Dr. Kalyana Veluvolu 30 End of Circuit Theory class……… Best of Luck for you Final Exam Thanks for attending my Lectures ☺ Grades will be posted on my Website, Check after 24-June. KNU/EECS/CT1 Dr. Kalyana Veluvolu 31.
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