EECE 210 Lecture Circuit Elements, Spring 2012

Chapter 2 Circuit Elements Voltage and current sources  An electrical source is a device that is capable of converting nonelectric energy to electric energy and vice versa.  A discharging battery converts chemical energy to electric energy, whereas a battery being charged converts electric energy to chemical energy.  The important thing to remember about these sources is that they can either deliver or absorb electric power, generally maintaining either voltage or current.  An ideal voltage source is a circuit element that maintains a prescribed voltage across its terminals regardless of the current flowing in those terminals. Voltage and current sources  An ideal current source is a circuit element that maintains a prescribed current through its terminals regardless of the voltage across those terminals.  These circuit elements do not exist as practical devices they are idealized models of actual voltage and current sources Dependent or Independent Sources  Ideal voltage and current sources can be either independent sources or dependent sources.  An independent source establishes a voltage or current in a circuit without relying on voltages or currents elsewhere in the circuit. The value of the voltage or current supplied is specified by the value of the independent source alone.  A dependent source establishes a voltage or current whose value depends on the value of a voltage or current elsewhere in the circuit. You cannot specify the value of a dependent source unless you know the value of the voltage or current on which it depends. Ideal Independent Sources The circuit symbols for the ideal independent sources are as shown. Note that a circle is used to represent an independent source. To completely specify an ideal independent voltage source in a circuit, you must include the value of the supplied voltage and the reference polarity. Dependent Sources  As for the dependent sources, both the dependent current and voltage sources maybe controlled by either a current or a voltage elsewhere in the circuit, so there are a total of four variation. Dependent Sources a. Voltage-controlled voltage source b. Current-controlled voltage source c. Voltage-controlled current source d. Current-controlled current source Active and Passive Elements  An active element is one that models a device capable of generating electric energy.  Passive elements model physical devices that cannot generate electric energy. Resistors, inductors, and capacitors are examples of passive elements. Electrical Resistance (Ohm's Law)  Resistance is the capacity of materials to impede the flow of current or the flow of electric charge. The Circuit element used to model this behavior is the resistor. The figure shows the circuit symbol for the resistor, with R denoting the resistance value of the resistor.  In the course of the interaction of the moving electrons with the atomic structure of the materials, some amount of electric energy is converted to thermal energy and dissipated in the form of heat.  This effect may be undesirable. However, many useful electrical devices, take advantage of resistance heating, including stoves, toasters, irons, and space heaters. v R i + – v iR Conductance  The reciprocal of the resistance is referred to as conductance, is symbolized by the letter G, and is measured in siemens (S). Thus S v i R G  1  Ohm’s Law  For purposes of circuit analysis, we must reference the current in the resistor to the terminal voltage. The relationship between the voltage and the current is: v = R i Where v = the voltage in volts, i = the current in amperes, and R = the resistance in ohms (Ω). Power  We may calculate the power at the terminals of a resistor in several ways. The first approach is to use the defining equation and simply calculate the product of the terminal voltage and current. We write Pdissipated = vi  A second method of writing the power is in term of current and resistance: P= vi = (iR)i=  A third method of writing the power is in term of voltage and resistance: P 2 Ri 2 R  v Power dissipated or generated?  If the current direction through an element is going through a voltage drop, then Pdissipated = v.i  If the current direction through an element is going through a voltage rise, then Pdissipated = -v.i  If Pdissipated is negative, then that element is actually generating (not dissipating) power. Example: Find the Power dissipated by each element. V (V) I (A) Pdiss (W) V1=5 I1=2 V2=2 I1=2 V3=-3 I3=-3 V4=-1 I2=1 V5=4 I2=1 + V1 - + V5 - - V3 + + V2 - + V4 - I3 I1 I2 Example: Find the Power dissipated by each element. V (V) I (A) Pdiss (W) V1=5 I1=2 P1diss = -V1*I12=-5*2=-10 → P1gen = 10w V2=2 I1=2 P2diss=V2*I1=2*2=4 V3=-3 I3=-3 P3diss=V3*I3=(-3)(-3)=9 V4=-1 I2=1 P4diss=-V4*I2=-(-1)(1)=1 V5=4 I2=1 P5diss=-V5*I2=-(4)(1)=-4 → P5gen = 4w + V1 - + V5 - - V3 + + V2 - + V4 - I3 I1 I2 PTgenerated = 10 + 4 = 14W PTdissipated = 4 + 9 + 1 = 14W PTgenerated = PTdissipated Circuit Terminology  A circuit is an interconnection of circuit elements that provides one or more closed paths for the flow of current. A network is generally more complex than a circuit and can include several simpler circuits.  A node in a circuit is the junction of two or more circuit elements. An essential node is the junction of three or more circuit elements. The definition of a node includes that of an essential node, but not conversely. Normally, an essential node is indicated on circuit diagrams by a filled circle. Circuit Terminology  A path in a circuit is a set of one or more adjoining circuit elements that may be traversed in succession without passing through the same node more than once. A path generally has an initial node at its beginning and a final node at its end. If the initial and final nodes are the same, the path is closed and becomes a loop. A mesh is a loop that does not enclose any other loop.  A branch is a path that connects two consecutive nodes. An essential branch is a branch that connects two consecutive essential nodes (without passing through an essential node). The definition of a branch includes that of an essential branch, but not conversely. Example: Bridge circuit vSRC a b c d 1 2 3' 4 R1 R4 R3 R5 R2 – + – + Rsrc 5 3 1 R 3 R 4 R Nodes a, b, c, and d are essential nodes vSRC a b c d 1 2 3' 4 R1 R4 R3 R5 R2 – + – + Rsrc 5 3 1 R 3 R 4 R Nodes 1, 2 and 4 are not essential nodes vSRC a b c d 1 2 3' 4 R1 R4 R3 R5 R2 – + – + Rsrc 5 3 1 R 3 R 4 R Nodes 3 and b are one and the same because no circuit element is connected between them vSRC a b c d 1 2 3' 4 R1 R4 R3 R5 R2 – + – + Rsrc 5 3 1 R 3 R 4 R Nodes 5, d and 3’ are one and the same because no circuit element is connected between them vSRC a b c d 1 2 3' 4 R1 R4 R3 R5 R2 – + – + Rsrc 5 3 1 R 3 R 4 R Rsrc, vSRC, R’1, R1, R’4, R4, R’3, R3, R2 and R5 taken individually, are branches vSRC a b c d 1 2 3' 4 R1 R4 R3 R5 R2 – + – + Rsrc 5 3 1 R 3 R 4 R The combinations Rsrc-VSRC, R’1-R1, R’4-R4, and the individual branches R’3, R3, R2 and R5 are essential branches vSRC a b c d 1 2 3' 4 R1 R4 R3 R5 R2 – + – + Rsrc 5 3 1 R 3 R 4 R The closed paths d-5-1-a-b-d and d-5-1- a-c-b-d are loops vSRC a b c d 1 2 3' 4 R1 R4 R3 R5 R2 – + – + Rsrc 5 3 1 R 3 R 4 R The paths d-5-1-a-c-d, a-b-c-a, d-c-b-d (through R’3) and 3-3’-3 are meshes. vSRC a b c d 1 2 3' 4 R1 R4 R3 R5 R2 – + – + Rsrc 5 3 1 R 3 R 4 R Kirchhoff’s Laws There are two Kirchhoff’s laws: Kirchhoff’s current law (KCL) and Kirchhoff’s voltage law (KVL). These laws are basic circuit laws that must be satisfied under all conditions, because they derive from conservation of charge and conservation of energy Kirchhoff’s Current Law  KCL may be stated as follows: at any instant of time, the sum of currents entering a node is equal to the sum of currents leaving the node. At node N, for example, KCL gives: iA + iB = iC + iD. Alternatively, iA + iB – iC – iD = 0, where currents flowing towards a node have been arbitrarily assigned a positive sign, which means that currents flowing away from a node should be assigned a negative sign. N iA iC iB iD Kirchhoff’s Voltage Law KVL may be stated as follows: At any instant of time, the sum of the voltage rises around any loop is equal to the sum of the voltage drops around the loop v1 v5 v2 v4 v3 q + + + + + – – – – – + Kirchhoff’s Voltage Law An equivalent statement of KVL is: At any instant of time, the algebraic sum of the voltages around any loop is zero, since voltage drops and voltage rises are assigned opposite signs v1 v5 v2 v4 v3 q + + + + + – – – – – +  In the circuit shown, if the loop is traversed clockwise, then v1, v2, and v3 are voltage rises, whereas v4 and v5 are voltage drops. According to the first statement of KVL: v1 + v2 + v3 = v4 + v5, whereas according to the second statement v1 + v2 + v3 – v4 – v5 = 0, which is evidently an equivalent statement. v1 v5 v2 v4 v3 q + + + + + – – – – – + Example: KVL and KCL – – – – – – – + + + + + + + + 2 V 1 V 1 V 4 V 4.5 V 4.5 V 10 V 3 V 16 V 20  20  10  10  20  0.1 A 0.05 A 200  18  18  0.2 A 0.25 A 0.05 A – 0.3 A 1 2 3 4 5 6 7 8 – + – + 0.3 A Find io Solve Solve for i0 and i∆ Equivalence of Sources  Independent current sources in parallel add up (algebraically).  Independent voltage sources in series add up (algebraically). Equivalence of Sources  Current sources in series or voltage sources in parallel are forbidden unless the sources are pointing in the same direction and have exactly the same values invalid valid valid State whether the following connections are valid or not In each circuit, if the connection is valid, find the power dissipated (or generated) by each source. If the connection is not valid, explain why. Find v1 and vg if vo = 250mV (hint: start at vo and work back toward vg) P2.30 solution P2.31 Find i and vo, and show that power generated (or developed) equals power dissipated (or absorbed). Avoid writing KVL around a mesh or loop containing a current source (unless we need to find the voltage across that current source). Avoid writing KCL on a node connected to a branch containing only a voltage source (unless we need to solve for the current in that branch). P2.31 Find i and vo, and show that power generated (or developed) equals power dissipated (or absorbed). 50 + 20i -5i -40i =0; i = 2A P2.31 Find i and vo, and show that power generated (or developed) equals power dissipated (or absorbed). 50 + 20 i -18 i =0; i = 2A, i = 5A P2.31 Find i and vo, and show that power generated (or developed) equals power dissipated (or absorbed). 50 + 20 i -18 i =0; i = 2A, i = 5A ig i1 8i=40A, i1 = 2+40=42A, ig = 5+42=47A 40i -vd -20 =0; vd = 60V + vd - Power dissipated or generated?  If the current direction through an element is going through a voltage drop, then Pdissipated = v.i  If the current direction through an element is going through a voltage rise, then Pdissipated = -v.i  If Pdissipated is negative, then that element is actually generating (not dissipating) power. Pgenerated = Pdissipated element v i Pdissipated 50V V. source 50V 47A -50*47= -2350W  Pgenerated=2350W 20 i V. source 40V 47A -40*47= -1880W  Pgenerated=1880W 5 i V. source 10V 42A 10*42= 420 W 20V V. source 20V 40A 20*40= 800 W 8 i C. source 60V 40A 60*40= 2400W 18  Resistor 18*5=90V 5A 90*5 = 450W 40  Resistor 40*2=80V 2A 80*2 = 160W Pdissipated = 420+800+2400+450+160=4230W Pgenerated = 2350+1880 = 4230W Pdissipated = Pgenerated