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Lecture #6 EGR 260 – Circuit Analysis

Lecture #6 EGR 260 – Circuit Analysis. Reading Assignment: Chapter 3 in Electric Circuits, 9 th Edition by Nilsson . Current Division Applies to parallel circuits only. Current divides between parallel R’s with the smallest R getting the most current. Show that: . I 1.

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Lecture #6 EGR 260 – Circuit Analysis

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  1. Lecture #6 EGR 260 – Circuit Analysis Reading Assignment: Chapter 3 in Electric Circuits, 9th Edition by Nilsson • Current Division • Applies to parallel circuits only. • Current divides between parallel R’s with the smallest R getting the most current. • Show that:

  2. I1 Lecture #6 EGR 260 – Circuit Analysis Example:Find the current I1 using current division.

  3. I1 Lecture #6 EGR 260 – Circuit Analysis Current Division - Special Case: Two Resistors Only Show that for two resistors only the general form of current division can be expressed as follows. Example:Find the current I1 using current division (special form for two R’s only).

  4. 40 80 300 I1 200V 240 600 900 Lecture #6 EGR 260 – Circuit Analysis Example:Find the current I1 using repeated current division.

  5. + I1 I4 V2 _ + V6 + _ I3 V5 _ Lecture #6 EGR 260 – Circuit Analysis Example:Find I1, V2, I3, I4, V5, and V6 in the circuit shown below.

  6. R0 R1 R2 R5 Vs R3 R4 Lecture #6 EGR 260 – Circuit Analysis Bridge Circuits One type of resistive circuit that cannot be simplified through series and/or parallel combinations is the “bridge circuit.” A bridge circuit is shown below (drawn twice). Study the circuit to verify that there are no series resistors and no parallel resistors. R0 R2 R1 R5 Vs R4 R3

  7. Lecture #6 EGR 260 – Circuit Analysis Delta-to-Wye (-Y) and Wye-to-Delta (Y-) Transformations Bridge circuits contain resistors that are connected in delta () and wye (Y) configurations. One way to analyze this circuit is to use a -Y or a Y- transformation. Y and  connections of resistors are shown below:

  8. Lecture #6 EGR 260 – Circuit Analysis If the wye and delta circuits to be equivalent, then they should provide the same resistance between each pair of terminals (a-b, b-c, and c-a). Development: Determine the resistance seen at each set of terminals and equate them as follows: Ra-b (Delta) = Ra-b (Wye) Rb-c (Delta) = Rb-c (Wye) Rc-a (Delta) = Rc-a (Wye)

  9. Lecture #6 EGR 260 – Circuit Analysis Solving the equations on the previous page yields the following relationships: Note: The equations above must be used along with the circuit diagrams shown. The labeling of the resistors and nodes in the diagrams is critical.

  10. Lecture #6 EGR 260 – Circuit Analysis Example: Determine I in the circuit shown below using: a) -Y conversion

  11. Lecture #6 EGR 260 – Circuit Analysis Example: Determine I in the circuit shown below using: b) Y- conversion

  12. R0 R1 R2 R5 Vs R3 R4 Lecture #6 EGR 260 – Circuit Analysis The “balanced bridge” A bridge circuit is “balanced” when the following relationship exists: R1R4 = R2R3 When the bridge is balanced, it can be shown that I5 = 0 I5

  13. Lecture #6 EGR 260 – Circuit Analysis • Applications of bridge circuits • An unknown resistance can be determined using a bridge circuit as follows: • Replace R4 with an unknown resistor • Place an ammeter in series with R5 • Adjust R1 until the bridge is balanced (i.e., I5 = 0) • Solve for R4 A bridge circuit with resistive values, sometimes called a Wheatstone bridge, can be used to determine unknown resistor values. Other bridge circuits (such as the Scherring bridge) can be used to determine unknown values of capacitors and inductors in a similar manner.

  14. Lecture #6 EGR 260 – Circuit Analysis Impedance Bridge A bridge circuits used to find unknown values of resistance, inductance, or capacitance are sometimes called an impedance bridge. An impedance bridge is a common piece of test equipment and will be used in lab classes such as EGR 262 and EGR 270. A newer microprocessor-controlled impedance bridge. http://www.testequipmentconnection.com/products/914 An older style impedance bridge that involved adjusting knobs until the needle indicated that the bridge was balanced. Ref: http://www.testequipmentconnection.com/products/908

  15. Lecture #6 EGR 260 – Circuit Analysis Strain gauges Strain gauges are often attached to surfaces to measure forces as the surface moves (such as in the deflection of an airplane wing or a beam). The force can be determined from the amount of stretch in the wire by measuring the resistance of the wire with a bridge circuit. As a wire stretches note that resistance increases since Reference: http://www.allaboutcircuits.com/vol_1/chpt_9/7.html

  16. Lecture #6 EGR 260 – Circuit Analysis Strain gauge mounted on a vehicle suspension system component for fatigue and durability testing Ref: http://blog.prosig.com/2006/05/17/fatigue-durability-testing/

  17. Lecture #6 EGR 260 – Circuit Analysis Strain gauge mounted on a human bone to study walking and running impact Biomechanics lab at Iowa State University Ref: http://www.kin.hs.iastate.edu/research/labs/biomechanics_lab.htm

  18. Note: Test #2 material starts here. Lecture #6 EGR 260 – Circuit Analysis Reading Assignment: Sections 4.1-4.8 in Electric Circuits, 7th Edition by Nilsson • Chapter 4 – Methods of Analysis for Resistive Circuits • In Chapters 2 and 3 KVL and KCL were applied in a somewhat arbitrary manner. No systematic procedure was introduced as to where to write the equations and how many equations to write. This arbitrary approach would be difficult to use for larger, more complex circuits. • In this chapter we will develop methods for writing and solving simultaneous independent circuit equations. Two methods are introduced in this chapter and are commonly used throughout electrical engineering: 1. Node Equations (or Nodal Analysis) – result in a set of simultaneous, independent KCL equations 2. Mesh Equations (or Mesh Analysis) – result in a set of simultaneous, independent KVL equations. Systematic procedures will be introduced for writing these equations that will give a clear approaches that can be used even for very large circuits.

  19. Lecture #6 EGR 260 – Circuit Analysis Node Equations First, a couple of definitions: Ground ( or reference) – a node in the circuit used as a reference point for measuring node voltages. As far as node voltages are concerned, the ground node is the 0V point in the circuit. Ground symbols: Node voltage – the voltage at a node with respect to ground. For example, VA is the voltage at node A, meaning that the positive terminal is at A and the negative terminal is at the ground node. Note: Node voltages are relative measurements. The value of a node voltage changes if a different ground node is used.

  20. Lecture #6 EGR 260 – Circuit Analysis Illustration – The use of a ground essentially means that a common node will be used as the negative terminal for all voltages (node voltages) in the circuit. This would be like using a voltmeter to measure various voltages where the negative lead of the meter always stayed on the ground node and the positive lead then moved to various other nodes to measure “node voltages.” Case 2 – Using a voltmeter to measure node voltage VB Case 1 – Using a voltmeter to measure node voltage VA Note – In all cases the negative side of the meter is connected to the ground node in order to measure node voltages.

  21. Lecture #6 EGR 260 – Circuit Analysis Component Voltages Recall that node voltages are relative quantities. If a different reference is used, the node voltages change. Component voltages are not relative quantities. Component voltages can be determine from node voltages as illustrated below. Vx = VA - VB (prove that this is true for any ground)

  22. Voltage Node A used as ground Node B used as ground Node C used as ground Node D used as ground VA VB VC VD Vx = VA - VB Lecture #6 EGR 260 – Circuit Analysis • Example: The circuit below includes the value of the component voltages. • Determine the corresponding node voltages (fill out the table). • Show that Vx = VA - VB for each possible set of node voltages.

  23. Lecture #6 EGR 260 – Circuit Analysis • Node equations – Procedure • 1) Label each node. • 2) Select a node as the reference (or ground) node. • Note:It is generally easier to pick the ground adjacent to a voltage source. • 3) If the circuit has no voltage sources, skip to step 4. Otherwise: • A) For any voltage source or group of voltage sources adjacent to the ground, all node voltages adjacent to the sources can be determined so no KCL equation will be required. • B) For any voltage source or group of voltage sources not adjacent to the ground, a supernode is required (to be discussed later). • 4) Write a KCL equation at each node not adjacent to a voltage source and not at the ground node. (Also write a KCL equation for each supernode.) Express resistor currents in terms of node voltages. • 5) Solve the simultaneous KCL equations. In general, the number of equations required is: # Node Equations = #nodes - #voltage sources - 1

  24. _ V2 + 4  I1 2  8  4 A 2 A Lecture #6 EGR 260 – Circuit Analysis Example: Analyze the following circuit using node equations. Use the result to find I1, V2, and the power dissipated by the 8 ohm resistor.

  25. Lecture #6 EGR 260 – Circuit Analysis Example: Use node equations determine the current I. (Answer: I = 1.92 A.)

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