Fall 2001ENGR201 Circuits I - Chapter 21 Chapter 2 – Resistive Circuits Read pages 14 – 50 Homework Problems - TBA Objectives: to learn about resistance and Ohm’s Law to learn how to apply Kirchhoff’s laws to resistive circuits to learn how to analyze circuits with series and/or parallel connections to learn how to analyze circuits that have wye or delta connections

Fall 2001ENGR201 Circuits I - Chapter 22 Resistance - Definition Resistance is an intrinsic property of matter and is a measure of how much a device impedes the flow of current. The greater the resistance of an object, the smaller the amount of current that will flow for a given applied voltage. The resistance of an object depends on the material used to construct the object (copper has less resistance than plastic), the geometry of the object (size and shape), and the temperature of the object. (R =  L/A)

Fall 2001ENGR201 Circuits I - Chapter 23 Resistance – Applications Sometimes we want to minimize the resistance of an object (in a conductor, for instance). Sometimes we want to maximize the resistance (in an insulator). Sometimes we to relate the resistance of the object to some physical parameter (such as a photoresistor or RTD). Sometimes we want to precisely control the resistance of an element in order to influence the behavior of a circuit such as an amplifier.

Fall 2001ENGR201 Circuits I - Chapter 24 Resistance - Sizing Resistors come in all shapes and sizes (see Figure 2.1 in your text). However, several common parameters are used to characterize resistors:  ohmic value (nominal) measured in Ohms (  ),  maximum power rating measured in Watts (W), and  precision (or tolerance) measured as a percentage of the ohmic value.

Fall 2001ENGR201 Circuits I - Chapter 25 Ohm’s Law - describes the relationship between the current through and the voltage across a resistor. Different devices connected to a power source demand different amounts of power from that source. That is, different devices present differing amounts of loading. The 6w bulb offers more resistance to the flow of current than the 12w bulb. I = 0.5A I = 1A 12V 6W 12W Ohm’s Law

Fall 2001ENGR201 Circuits I - Chapter 26 Rather than specify the load that a device represents in terms of its voltage/power rating, we can specify that load in terms of its resistance. The smaller the resistance the greater the load (the greater the power demand). Ohm’s Law – Mathematical Definition R = V/I I = V/R V = IR +V-+V- R I I = 0.5A I = 1A 12V 6W 12W R = 12V/0.5A = 24  R = 12V1A = 12 

Fall 2001ENGR201 Circuits I - Chapter 27 How much current will a 12V/12W lamp demand if 6V is applied to it? How much power is demanded? 6V 12V 12W Example A 12w/12v lamp will draw 1A of current: P = VI  12W = 12V  I  I = 1A V = IR (Ohm’s Law)  R = 12V/1A = 12  Therefore, if V = 6V  I = 6V/1A = 12  P = 6v  0.5A = 3W = 0.25  12W. Since both the voltage and current are halved, the power is cut by a factor of four.

Fall 2001ENGR201 Circuits I - Chapter 28 R is the resistance of the device, measured in ohms (  ). The greater the value of R, the smaller the value of I. R = V/I +V-+V- I = 0A 12V Open Circuit, R =  I = 0 regardless of the value of V (NO LOAD) (air, plastic, wood) Short & Open Circuits Short Circuit, R =0 V = 0 regardless of the value of I (wire) I =  12V V

Fall 2001ENGR201 Circuits I - Chapter 29 Ohms’ law relates the magnitude of the voltage with the magnitude of the current AND the polarity of the voltage to the direction of the current. Resistors always absorb power, so resistor current always flows through a voltage drop. +V-+V- I = V/R R Ohm’s Law – Voltage Polarity & Current Direction

Fall 2001ENGR201 Circuits I - Chapter 210 Ohms’ Law can be represented graphically – called a VI characteristic: +V-+V- R I = V/R I V m = Slope =  V/  I = R Ideal resistor, VI characteristic Ohm’s Law - Graphically

Fall 2001ENGR201 Circuits I - Chapter 211 V I Short circuit, slope = 0 (V = 0) Open circuit, slope =  (I = 0) V I Practical resistor VI characteristic P max Non-ideal Resistors

Fall 2001ENGR201 Circuits I - Chapter 212 Resistance is a measure of how much a device impedes the flow of current. Conductance is a measure of how little a device impedes the flow of current. Resistance and conductance are simply two different ways to describe the voltage-current characteristic of a device. At times, especially in electronic circuits, it is advantageous to work in terms of conductance rather than resistance Conductance

Fall 2001ENGR201 Circuits I - Chapter 213 Resistance: R = V/I,  (ohms) +V-+V- Conductance: G = I/V, S (seimens) +V-+V- (G = 1/R = R -1 ) Conductance - Units Old style symbol for conductance Old style units = mho 

Fall 2001ENGR201 Circuits I - Chapter 214 P = VI (any device) for a resistor: P = V(V/R) = V 2 /R or P = (IR)I = I 2 R P = VI (any device) In terms of conductance: P = V(VG) = V 2 G or P = (I/G)I = I 2 /G Resistance: R +V-+V- I V = IR = I/G Resistance – Power Equations P = VI P = V 2 /R P = I 2 R

Fall 2001ENGR201 Circuits I - Chapter 215 Kirchhoff’s Laws Kirchhoff’s Current Law (KCL) Kirchhoff’s Voltage Law (KVL) A node is a “point” in a circuit where two or elements are connected. R R R +-+- Node-A Kirchhoff’s Laws R R R Node-A +-+-

Fall 2001ENGR201 Circuits I - Chapter 216 Kirchhoff’s Current Law The algebraic sum of all currents at any node in a circuit is exactly zero. The sum of all currents entering = sum of all currents leaving We neither gain nor lose current at a node. I1 I2 I3 R R R +-+- Node-A I4 I1-I2+I3-I4 = 0 I1+I3 = I2+I4 KCL

Fall 2001ENGR201 Circuits I - Chapter 217 Kirchhoff’s Voltage Law (KVL) A loop is a closed path about a circuit that begins and ends at the same node. However, no element may be traversed more than once. A B C D E Five loops in the circuit shown are: A-C-B-A A-D-C-A C-D-E-C B-C-E-B A-D-E-B-A Are there more loops ? KVL

Fall 2001ENGR201 Circuits I - Chapter 218 The algebraic sum of all voltages about any loop in a circuit is exactly zero. The sum of all increases (rises) = sum of all voltage decreases (drops) We do not gain or lose voltage if we start and end at the same node. A B C D E + V1 - - V2 + V3 - + V4 - + V5 - + Vx - + V6 - + Vy - By KVL: V2 + V3 - V1 = 0 -V3 + V4 - Vx = 0 V1 + Vy - V6 = 0 Vx + V5 -Vy = 0 V2 + V4 + V5 - V6 = 0 KVL

Fall 2001ENGR201 Circuits I - Chapter 219 Two circuits are equivalent if, for any source connected to the circuits, they demand the same amount power. The two circuits “look” the same to the source I1I VsVs Device #1 P1 = V S  I 1 I2I VsVs Device #2 P2 = V S  I 2 P 1 = P 2  V S  I 1 = V S  I 2  I 1 = I 2 If the applied voltage is the same, two equivalent circuits will demand the same amount of current from the source. Equivalent Circuits

Fall 2001ENGR201 Circuits I - Chapter 220 R ab a b a b R1 R4 R3R2 R6R5 R ab  Series connection (all elements have the same current) a b R1 R4 R3R2 R6R5 V ab I I R ab V ab = I  R ab Series Resistance

Fall 2001ENGR201 Circuits I - Chapter 221 By KCL: I R1 = I R2 = … = I R6 = I By KVL: V ab = I  R1+ I  R2+ I  R3+ I  R4+ I  R5+ I  R6 V ab /I = (R1 + R2 + R3 + R4 + R5 + R6) = R ab a b R1 R4R4 R3R2 R6R5 V ab I I RabRab V ab = I  R ab The equivalent resistance of two or more series-connected resistors is the sum of the individual resistors. Series Resistance

Fall 2001ENGR201 Circuits I - Chapter 222 Parallel connection (all the elements have the same voltage) V ab /I = R ab Parallel Resistance V ab I R ab V ab I R1R2R3R4R5 A B by KCL: I = I1 + I2 + I3 + I4 + I5 I = V ab /R1 + V ab /R2 + V ab /R3 + V ab /R4 + V ab /R5 I = V ab [R R R R R5 -1 ] V ab /I = [R R R R R5 -1 ] -1

Fall 2001ENGR201 Circuits I - Chapter 223 V ab I R ab V ab I R1R2R3R4R5 V ab /I = R ab R ab = [R R R R R4 -1 ] -1 Since G = 1/R = R -1 R ab = [G1 + G2 + G3 + G4 + G5] -1 G ab = G1 + G2 + G3 + G4 + G5 Parallel Resistance

Fall 2001ENGR201 Circuits I - Chapter 224 The total voltage applied to a group of series-connected resistors will be divided among the resistors. The fraction of the total voltage across any single resistor depends on what fraction that resistor is of the total resistance. a b R1 R4 R3R2 R6R5 V ab I R TOTAL = R ab = R1 + R2 + R3 + R4 + R5 + R6 +V4-+V4- The Voltage Divider Rule (VDR)

Fall 2001ENGR201 Circuits I - Chapter 225 The total current applied to a group of resistors connected in parallel will be divided among the resistors. The fraction of the total current through any single resistor depends on what fraction that resistor is of the total conductance. G TOTAL = R R R R R5 -1 I5I5 V ab R1R2R3R4R5 I Total The Current Divider Rule (CDR)

Fall 2001ENGR201 Circuits I - Chapter 226 For two resistors: I total = I 1 + I 2 R1 R2 I1I1 I2I2 CDR – Two Resistors Observations: The smaller resistor will have the larger current. If R 1 = R 2, then I 1 = I 2 If R 1 = nR 2, then I 2 = nI 1