Chapter 17 11/13/2014

Transcription

1 Chapter 17 Voltage / Current source conversions Mesh and Nodal analysis in an AC circuit Balance conditions and what elements are needed in a bridge network ECET 207 AC Circuit Analysis, PNC 2 1

2 Magnitude is independent of the network Displays terminal characteristics even if completely isolated FIG Independent sources. ECET 207 AC Circuit Analysis, PNC 4 2

3 Magnitude determined by the circuit it appears in Used in AC analysis of Bipolar Junction Transistors ECET 207 AC Circuit Analysis, PNC 5 FIG Source Conversion. ECET 207 AC Circuit Analysis, PNC 6 3

4 FIG Example ECET 207 AC Circuit Analysis, PNC 7 FIG Example ECET 207 AC Circuit Analysis, PNC 8 4

5 Used to find currents when multiple circuit loops with sources exist Accounts for contributions from other sources Matrix used to solve (Appendix C, pg. 1160) Calculator preferred method (Experiment 1) ECET 207 AC Circuit Analysis, PNC 9 1. Assign loops, all one arbitrary direction 2. Combine series elements 3. Convert I sources to E sources 4. Write out formula for each loop 1.Far right total sources in loop 2.Use 0 as place holder as needed I1( 1 + 2) 2( 2) = 1 1( 2) + 2( 2 + 3) = 2 ECET 207 AC Circuit Analysis, PNC 10 5

6 5. Form a matrix to solve each loop a) Substitute sources for that loop s values b) Base matrix without sources c) Only enter impedances into matrix Mat = 1 2( 2) 2 + 2( 2 + 3) Mat = 1( 1 + 2) 2( 2) 1( 2) + 2( 2 + 3) ECET 207 AC Circuit Analysis, PNC Values are det(mat Ix )/det(mat base ) I = E1 2 det 2 2 det Z ECET 207 AC Circuit Analysis, PNC 12 6

7 Treat source as open Forms one large I 1 loop Second equation is I + I = 0 Becomes in matrix FIG Applying mesh analysis to a network with an independent current source. ECET 207 AC Circuit Analysis, PNC 13 Find I 1 FIG Example ECET 207 AC Circuit Analysis, PNC 14 7

8 Find I 2 FIG Example ECET 207 AC Circuit Analysis, PNC 15 Form equations to find current through R 3 FIG Example ECET 207 AC Circuit Analysis, PNC 16 8

9 1. Combine all series impedances 2. Determine the number of nodes and label ECET 207 AC Circuit Analysis, PNC 18 9

10 3. Convert any voltage sources to current sources (in schematic or equations) 4. Form equations for each node, using 0 to fill spaces as needed (Y=1/Z) V1 2 V1( 1 Z1 + 1 Z2 ) 2( 1 2 ) = 1 1( 1 2 ) + 2( ) = 2 ECET 207 AC Circuit Analysis, PNC Form a matrix to solve each node a) Substitute sources for that loop s values b) Base matrix without sources c) Only enter impedances into matrix 6. Values are det(mat Ix )/det(mat base ) V = det 1 ( 1 2 ) 2 ( ) det ( 1 Z1 + 1 Z2 ) ( 1 2 ) ( 1 2 ) ( ) ECET 207 AC Circuit Analysis, PNC 20 10

11 Find V 1 FIG Example ECET 207 AC Circuit Analysis, PNC 21 Find Nodal Analysis formulas FIG Example ECET 207 AC Circuit Analysis, PNC 22 11

12 ECET 207 AC Circuit Analysis, PNC 23 ECET 207 AC Circuit Analysis, PNC 24 12

13 No matrix needed Short IES Form one supernode V1 and V2 become the same node Solve equation for V1 Solve for V2 using IES V = = ECET 207 AC Circuit Analysis, PNC 25 With components parallel to IES Same initial process Additional current source V1 + = 1 2 ECET 207 AC Circuit Analysis, PNC 26 13

14 Find voltage drop across the capacitor ECET 207 AC Circuit Analysis, PNC 27 Very common circuit used in sensing systems Two circuits in parallel with common source Adjustable point across from sensor for calibration Can use resistors, capacitors, or inductors Output measured as difference between arms ECET 207 AC Circuit Analysis, PNC 28 14

15 Balanced state critical Allows for zeroing output with no input Improves accuracy, dynamic range Output changes reflect sensor input Z Z = Z Z or Z Z = Z Z ECET 207 AC Circuit Analysis, PNC 29 Very common sensing circuit Two stabilizing resistors, equal value Nulling potentiometer Sensor (temp, pressure, etc) Potentiometer used to calibrate Known input used on sensor Potentiometer adjusted till zeroed ECET 207 AC Circuit Analysis, PNC 30 15

16 Used to measure unknown coil inductance R1 used to null output Adjustment amount reflects inductor value Q<10 Q>10 Maxwell Bridge Hay Bridge ECET 207 AC Circuit Analysis, PNC 31 This bridge is balanced. Find the value of L x ECET 207 AC Circuit Analysis, PNC 32 16

17 Effective voltage at 70.7% of amplitude More time near peak, better efficiency of energy delivered Voltage Ripple (VR) Percent of travel waveform amplitude translates though between peak and average ECET 207 AC Circuit Analysis, PNC 34 17

18 Single phase 100% VR Three phase Six pulses per cycle More consistent voltage Voltage never drops to zero 14 % VR High Frequency Most efficient <1% VR ECET 207 AC Circuit Analysis, PNC 35 Delta-WYE HV secondary used Delta provides 30 degree phase shift Pair of FWRs output to same circuit 4% voltage ripple ECET 207 AC Circuit Analysis, PNC 36 18

19 Convert Δ to Y Eq Convert Y to Δ Eq If all Z match Eq ALL PHASES MUST MATCH FIG Δ-Y configuration. Mirrored over Zc for Delta pointing up ECET 207 AC Circuit Analysis, PNC 37 FIG Converting the upper Δ of a bridge configuration to a Y. ECET 207 AC Circuit Analysis, PNC 38 19

20 FIG The network in Fig following the substitution of the Y configuration. ECET 207 AC Circuit Analysis, PNC 39 Convert to Y-Y configuration FIG Example ECET 207 AC Circuit Analysis, PNC 40 20

21 FIG Converting a Δ configuration to a Y configuration. ECET 207 AC Circuit Analysis, PNC 41 FIG Substituting the Y configuration in Fig into the network in Fig ECET 207 AC Circuit Analysis, PNC 42 21

22 FIG Converting the Y configuration in Fig to a Δ. ECET 207 AC Circuit Analysis, PNC 43 FIG Substituting the Δ configuration in Fig into the network in Fig ECET 207 AC Circuit Analysis, PNC 44 22

23 READ THE QUESTION Some only require equations Some only ask for value across an element Ensure you look at the correct element Ex - Resistor vs Inductor Q 30 The bridge is balanced. Prove that the current through the capacitor is zero by finding the loop currents that pass through it. ECET 207 AC Circuit Analysis, PNC 45 23