Review: Lecture 9. Instantaneous and Average Power. Effective or RMS Value. Apparent Power and Power Factor. Complex Power. Conservation of AC Power

Transcription

1 Review: Lecture 9 Instantaneous and Average Power p( t) VmI m cos ( v i ) VmI m cos ( t v i ) Maximum Average Power Transfer Z L R L jx Effective or RMS Value I rms I m L R P * TH Apparent Power and Power Factor eff TH jx V m TH I m Z cos(θ v θ i R ) L RTH X TH ZTH V rms I rms cos(θ v θ i ) P max V TH 8 R TH P V rms I rms cos(θ Complex Power S V I V rms I rms θ v θ i Conservation of AC Power v θ i ) S cos(θ v θ i ) S V I * V ( I * I * ) V I * V I * S S

2 Review: Inductors Magnetic flux Ampere s law d dt Faraday s law N: turns Current i di dt v N d dt Three-dimensional magnetic field Oppose the change Inductance: i vs. v v N d dt L N d di di dt Len s law

3 Lecture 0 AC Circuits Magnetically Coupled Circuit

4 Contents Mutual Inductance Energy in a Coupled Circuit What is a transformer? Linear Transformers Ideal Transformers Applications 4

5 Mutual Inductance Voltage induced Magnetic flux Links coil only d v N dt Magnetically coupled Magnetic flux link both d v N dt Express v in terms of the change of i : M d d di v N N dt di dt Mutual inductance of coil with respect to coil

6 Mutual Inductance It is the ability of one inductor to induce a voltage across a neighboring inductor, measured in henrys (H). di M dt di M dt v v The open-circuit mutual voltage across coil The open-circuit mutual voltage across coil 6

7 M and M? The Upper Limit? The coupling coefficient, k, is a measure of the magnetic coupling between two coils; 0 k. M k L L The instantaneous energy stored in the circuit is given by w L i L i MI I 7

8 Mutual Inductance: Dot convention If a current enters the dotted terminal of one coil, the reference polarity of the mutual voltage in the second coil is positive at the dotted terminal of the second coil. Illustration of the dot convention. 8

9 Examples of the dot convention Dot convention for coils in series; the sign indicates the polarity of the mutual voltage; (a) series-aiding connection, (b) series-opposing connection. L L L M (series - aiding connection) L L L M (series - opposing connection) 9

10 Mutual Inductance: Time and Frequency domains KVL Time-domain analysis of a circuit containing coupled coils. di di di di ir L M v 0 ir L M v 0 dt dt dt dt Current leaves the dot Frequency-domain analysis of a circuit containing coupled coils jl I j I V 0 Z I jl I jmi 0 ZI M L 0

11 Mutual Inductance: Example Example Calculate the phasor currents I and I in the circuit shown below. Ans: I A; I A.

12 Energy in a Coupled Circuit: Example Example Consider the circuit below. Determine the coupling coefficient. Calculate the energy stored in the coupled inductors at time t = s if v=60cos(4t +30 ) V. Ans: k=0.56; w()=0.73j

13 What is a transformer? It is an electrical device designed on the basis of the concept of magnetic coupling It uses magnetically coupled coils to transfer energy from one circuit to another It is the key circuit elements for stepping up or stepping down ac voltages or currents, impedance matching, isolation, etc. 3

14 Transformer It is generally a four-terminal device comprising two (or more) magnetically coupled coils Linear: wound on a magnetically linear materials, air-core radio/tv Connected to the source Connected to the load Z in V I R jl Z R, Z R R M jl Z L is reflected impedance 4

15 Linear Transformer: Example Example 3 In the circuit below, calculate the input impedance and current I. Take Z =60-j00Ω, Z =30+j40Ω, and Z L =80+j60Ω. Ans: Zin ; I 0.53.A 5

16 Ideal Transformer An ideal transformer is a unity-coupled, lossless transformer in which the primary and secondary coils have infinite self-inductances. V V N N n I N I N n Ideal Transformer (a) and Circuit symbol (b) V > V step-up transformer V < V step-down transformer 6

17 Ideal Transformer: Example Example 4 An ideal transformer is rated at 400/0V, 9.6 kva, and has 50 turns on the secondary side. Calculate: (a) the turns ratio, (b) the number of turns on the primary side, and (c) the current ratings for the primary and secondary windings. Ans: (a) (b) (c) This is a step-down transformer, n=0.05 N = 000 turns I = 4A and I = 80A 7

18 Applications: Isolation device Transformer as an Isolation Device to isolate ac supply from a rectifier 8

19 Applications: Matching device Transformer as a Matching Device Using an ideal transformer to match the speaker to the amplifier Equivalent circuit 9

20 Applications: Example Example 5 Calculate the turns ratio of an ideal transformer required to match a 00Ω load to a source with internal impedance of.5kω. Find the load voltage when the source voltage is 30V. Ans: n = 0.; V L = 3V 0

21 Lecture AC Circuits Frequency Response

22 Appendix

23 Applications: Power distribution A typical power distribution system 3