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DC Circuits

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Published in: Electrical | Electronics
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Presentation on DC Circuits

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  1. CHAPTER 1 DC CIRCUITS
  2. DEFINATIONS Linear elements : In an electric circuit, a linear element is an electrical element with a linear relationship between current and voltage. Resistors are the most common example of a linear element; other examples include capacitors, inductors, and transformers. Nonlinear Elements : A nonlinear element is one which does not have a linear input/output relation. In a diode, for example, the current is a non-linear function of the voltage.Most semiconductor devices have non-linear characteristics. Active Elements : The elements which generates or produces electrical energy are called active elements. Some of the examples are batteries, generators,transistors,operational amplifiers,vacuum tubes etc. Passive Elements : All elements which consume rather than produce energy are called passive elements, like resistors,inductors and capacitors.
  3. In unilateral element, voltage — current relation is not same for both the direction. Example: Diode, Transistors. In bilateral element, voltage — current relation is same for both the direction. Example: Resistor The voltage generated by the source does not vary with any circuit quantity. It is only a function of time. Such a source is called an ideal voltage Source. The current generated by the source does not vary with any circuit quantity. It is only a function of time. Such a source is called as an ideal current source. Resistance : It is the property of a substance which opposes the flow of current through it. The resistance of element is denoted by the symbol "R". It is measured in Ohms. R — PL / A Q
  4. Ohm's Law: The current flowing through the electric circuit is directly proportional to the potential difference across the circuit and inversely proportional to the resistance of the circuit, provided the temperature remains constant. i(t) i(t) v(t) = v(t) = Ri(t) - Ri(t) (2.1) (2.2)
  5. Basic Laws of Circuits Ohm's Law: Directly proportional means a straight line relationship. v(t) i(t) v(t) = Ri(t) The resistor is a model and will not produce a straight line for all conditions of operation.
  6. Basic Laws of Circuits Ohm's Law: About Resistors: The unit of resistance is ohms( Q). A mathematical expression for resistance is 1 I : The length of the conductor (meters) A : The cross — sectional area (meters ) p : The resistivity (Q • m) (2.3)
  7. Basic Laws of Circuits Ohm's Law: About Resistors: We remember that resistance has units of ohms. The reciprocal of resistance is conductance. At one time, conductance commonly had units of mhos (resistance spelled backwards). In recent years the units of conductance has been established as seimans (S). Thus, we express the relationship between conductance and resistance as 1 (2.4) We will see later than when resistors are in parallel, it is convenient to use Equation (2.4) to calculate the equivalent resistance.
  8. Basic Laws of Circuits Ohm's Law: Ohm's Law: Example 2.1. Consider the following circuit. 115 V RMS V (ac) (100 Watt light bulb) Determine the resistance of the 100 Watt bulb. P=VI- P = 12R ohms 100 (2.5) A suggested assignment is to measure the resistance of a 100 watt light bulb with an ohmmeter. Debate the two answers.
  9. Circuit Definitions ' Node — any point where 2 or more circuit elements are connected together — Wires usually have negligible resistance — Each node has one voltage (w.r.t. ground) Branch — a circuit element between two nodes Loop — a collection of branches that form a closed path returning to the same node without going through any other nodes or branches twice
  10. Example How many nodes, branches & loops? + VS Vo
  11. Three nodes Is + VS Vo
  12. 5 Branches Is
  13. Three Loops, if starting at node A + VS Vo
  14. Basic Laws of Circuits Kirchhoff's Current Law As a consequence of the Law of the conservation of charge, we have: , The sum of the current entering a node (junction point) equal to the sum of the currents leaving. la, 1b, I c, and Id can each be either a positive or negative number.
  15. Basic Laws of Circuits Kirchhoff's Current Law The algebraic sum of the currents entering a node equal to zero. la, 1b, Ic, and Id can each be either a positive or negative number.
  16. Basic Laws of Circuits Kirchhoff's Current Law The algebraic sum of the currents leaving a node equal to zero. la, 1b, Ic, and Id can each be either a positive or negative number.
  17. Basic Laws of Circuits Kirchhoff's Current Law: Find the current I x' Ans: 14 Example 2.2. Highlight the box then use bring to front to see answer.
  18. Basic Laws of Circuits Kirchhoff's Current Law: Example 2.3 Find the currents I 12 A
  19. Basic Laws of Circuits Kirchhoff's Current Law Kirchhoff's current law can be generalized to include a surface. We assume the elements within the surface are interconnected. A closed 3D surface We can now apply Kirchhoff's current law in the 3 forms we discussed with a node. The appearance might be as follows: Currents entering and leaving a closed surface that contains interconnected circuit elements
  20. Kirchoff's Voltage Law (KVL) The algebraic sum of voltages around each loop is zero Beginning with one node, add voltages across each branch in the loop (if you encounter a + sign first) and subtract voltages (if you encounter a — sign first) E voltage drops - E voltage rises = 0 Or E voltage drops = E voltage rises
  21. Circuit Analysis When given a circuit with sources and resistors having fixed values, you can use Kirchoffs two laws and Ohm's law to determine all branch voltages and currents + VAB - 70 12 v C 30
  22. Circuit Analysis , By Ohm's law: v AB = 1-70 and VB c = , By KVL: v AB 12v=o , Substituting: 1-70 + -12 v = o ' Solving: 1= 1 AB 70 12 v 30 BC
  23. Example Circuit iC)D Solve for the currents through each resistor And the voltages across each resistor
  24. Example Circuit iC)D Using Ohm's law, add polarities and expressions for each resistor voltage
  25. Example Circuit Write 1st Kirchoff's voltage law equation -50 v + + 12080 = o
  26. Example Circuit iC)D Write 2nd Kirchoff's voltage law equation or 12 - 13 = 1.25 13
  27. Example Circuit We now have 3 equations in 3 unknowns, so we can solve for the currents through each resistor, that are used to find the voltage across each resistor , Since 1 0 Il = 12 +13 ' Substituting into the 1st K VL equation -50 v + (12 + 13)' IOC) + 12' 80 = O or Q + 10 Q = 50 volts
  28. Example Circuit ' But from the 2nd K VL equation, 12 1.25013 ' Substituting into 1st K VL equation: (1.25 13>18 Q + 13 • 10 Q = 50 volts 13 22.5 Q+ 13 , 10 Q = 50 volts Or: 32.5 Q = 50 volts Or: 13 = 50 volts/32.5 Q Or: : 13 = 1.538 amps Or
  29. Example Circuit , Since I 3 = 1.538 amps 12 — . • 3 = 1.923 amps 1251 , Since 11 — 12 + 13 Il = 3.461 amps The voltages across the resistors: 34.61 volts •IOC) = 15.38 volts 12 •6Q = 9.23 volts 13 •4Q = 6.15 volts 13
  30. Star Delta Transformation We can now solve simple series, parallel or bridge type resistive networks using Kirchoff's Circuit Laws, mesh current analysis or nodal voltage analysis techniques but in a balanced 3-phase circuit we can use different mathematical techniques to simplify the analysis of the circuit and thereby reduce the amount of math's involved which in itself is a good thing. Standard 3-phase circuits or networks take on two major forms with names that represent the way in which the resistances are connected, a Star connected network which has the symbol of the letter, Y (wye) and a Delta connected network which has the symbol of a triangle, A (delta). If a 3-phase, 3-wire supply or even a 3-phase load is connected in one type of configuration, it can be easily transformed or changed it into an equivalent configuration of the other type by using either the Star Delta Transformation or Delta Star Transformation process. A resistive network consisting of three impedances can be connected together to form a T or "Tee" configuration but the network can also be redrawn to form a Star or Y type network as shown below
  31. Delta Star Transformation To convert a delta network to an equivalent star network we need to derive a transformation formula for equating the various resistors to each other between the various terminals. Consider the circuit below. Delta to Star Network. 3 2 3 p 2 Compare the resistances between terminals 1 and 2. P + Q = A in parallel with (B + C) .EQI Resistance between the terminals 2 and 3 Q + R = C in parallel with (A + B)
  32. EQ2 Resistance detuveen the terminals 1 and 3 P + R = B in parallel with (A + C) .EQ3 This now gives us three equations and taking equation 3 from equation 2 gives EQ3 - EQ2 Delta to Star Transformations Equations AB AC BC
  33. Star Delta Transformation We have seen above that when converting from a delta network to an equivalent star network that the resistor connected to one terminal is the product of the two delta resistances connected to the same terminal, for example resistor P is the product of resistors A and B connected to terminal 1. By rewriting the previous formulas a little we can also find the transformation formulas for converting a resistive star network to an equivalent delta network giving us a way of producing a star delta transformation as shown below.
  34. Star to Delta Network, Star Delta Transformation Equations
  35. Star-DeIta Transformation z? 2 2 3 3 Zl-ZaZb/(Za+Zb+Zc) Z2-ZaZc/(Za+Zb+Zc) Z3-ThZc/(Za+Zb+Zc) za—z1+z2+fz?mz3) STARDELTA TRANSFORNIATIONS
  36. D.C. Transient response The storage elements deliver their energy to the resistances, hence the response changes with time, gets saturated after sometime, and is referred to the transient response.
  37. The Differential Equation vr(t) vs (t) KVL around the loop: Lect12 EEE 202 vc(t) vs(t) 37
  38. RC Differential Equation(s) t 1 i(x)dx = vs (t) From KVL: Ri(t) + di(t) dvs (t) Multiply by C; RC take derivative dvr (t) dvs (t) Multiply by R; + vr (t) = RC RC note VFR, i Lect12 EEE 202 dt 38
  39. LR Series Circuit An LR Series Circuit consists basically of an inductor of inductance L connected in series with a resistor of resistance R. The resistance R is the DC resistive value of the wire turns or loops that goes into making up the inductors coil. Consider the LR series circuit below. The above LR series circuit is connected across a constant voltage source, (the battery) and a switch. Assume that the switch, S is open until it is closed at a time t = 0, and then remains permanently closed producing a "step response' type voltage input. The current, i begins to flow through the circuit but does not rise rapidly to its maximum value of Imax as determined by the ratio of V / R (Ohms Law).
  40. This limiting factor is due to the presence of the self induced emf within the inductor as a result of the growth of magnetic flux, (Lenz's Law). After a time the voltage source neutralizes the effect of the self induced emf, the current flow becomes constant and the induced current and field are reduced to zero. we can use Kirchoffs Voltage Law, (KVL) to define the individual voltage drops that exist around the circuit and then hopefully use it to give us an expression for the flow of current. Kirchoffs voltage law gives us:
  41. Kirchaffs voltage law gives us: R L The voltage drop across the resistor, R is IR (Ohms Law). The voltage drop across the inductor, L is Oy now our familiar expression L = di/dt v di dt Then the final expression far the individual voltage drags around the LR series circuit can Oe given as: (t) di dt We can see that the voltage drop across the resistor depends upon the current, i, while the voltage drop across the inductor depends upon the rate of change of the current, di/dt. When the current is equal to zero, ( i = 0 ) at time t = 0 the above expression, which is also a first order differential equation, can be rewritten to give the value of the current at any instant of time
  42. Expression for the Current in an LR Series Circuit -RUL 1 The L/R term in the above equation is known commonly as the Time Constant, ( ) of the LR series circuit and V/R also represents the final steady state current value in the circuit. Once the current reaches this maximum steady state value at 51, the inductance of the coil has reduced to zero acting more like a short circuit and effectively removing it from the circuit. Therefore the current flowing through the coil is limited only by the resistive element in Ohms of the coils windings. A graphical representation of the current growth representing the voltage/time characteristics of the circuit can be presented as.
  43. Transient Curves for an LR Series Circuit Steady State Value Y (I—e Rt/L) 63% -Rt/L Transient Tim e
  44. Time constant of RC and RL The time taken to reach 36.8% of initial current in an RC circuit is called the time constant of RC circuit. Time constant (t) = RC. The time taken to reach 63.2% of final value in a RL Circuit is called the time constant of RL circuit. Time constant (t) = L / R
  45. Important Concepts The differential equation for the circuit Forced (particular) and natural (complementary) solutions Transient and steady-state responses ' 1st order circuits: the time constant (c) ' 2nd order circuits: natural frequency (00) and the damping ratio (C)
  46. Differential Equation Solution The total solution to any differential equation consists of two parts: x(t) = xp(t) + xc(t) Particular (forced) solution is x (t) — Response particular to a given source Complementary (natural) solution is xc(t) — Response common to all sources, that is, due to the "passive" circuit elements

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