electronics fundamentals circuits, devices, and applications THOMAS L. FLOYD DAVID M. BUCHLA chapter 6
Identifying series-parallel relationships Most practical circuits have combinations of series and parallel components. Components that are connected in series will share a common path. Components that are connected in parallel will be connected across the same two nodes. 1 2
Summary Combination circuits Most practical circuits have various combinations of series and parallel components. You can frequently simplify analysis by combining series and parallel components. An important analysis method is to form an equivalent circuit. An equivalent circuit is one that has characteristics that are electrically the same as another circuit but is generally simpler.
Equivalent circuits For example: R 1 1.0 k 1.0 k is equivalent to R 1 2.0 k There are no electrical measurements that can distinguish the boxes.
Equivalent circuits Another example: is equivalent to R1 R2 1.0 k 1.0 k R 1,2 500 There are no electrical measurements that can distinguish the boxes.
is equivalent to R1 1.0 k 2.7 k R 3 4.7 k R 3.7 k 1,2 3 R 4.7 k is equivalent to R 1,2,3 2.07 k There are no electrical measurements that can distinguish between the three boxes.
Kirchhoff s voltage law and Kirchhoff s current law can be applied to any circuit, including combination circuits. For example, applying KVL, the path shown will have a sum of 0 V. V S 5.0 V 470 R 1 270 So will this path! R 4 100 R 3 330 R 5 100 R 6 Start/Finish Start/Finish 100
Kirchoff s current law can also be applied to the same circuit. What are the readings for node A? I + 26.5 ma A I + 8.0 ma V S 5.0 V + I + 18.5 ma 470 Ω R 4 100 Ω R 1 270 Ω R 3 330 Ω R 6 R 5 100 Ω 100 Ω
Combination circuits V S + 10 V 330 Ω R 1 270 Ω R 3 470 Ω Tabulating current, resistance, voltage and power is a useful way to summarize parameters. Solve for the unknown quantities in the circuit shown. I 1 = 21.6 ma R 1 = 270 Ω V 1 = 5.82 V P 1 = 126 mw I 2 = 12.7 ma = 330 Ω V 2 = 4.18 V P 2 = 53.1 mw I 3 = 8.9 ma R 3 = 470 Ω V 3 = 4.18 V P 3 = 37.2 mw I T = 21.6 ma R T = 464 Ω V S = 10 V P T = 216 mw
Kirchhoff s laws can be applied as a check on the answer. V S + 10 V 330 Ω R 1 270 Ω R 3 470 Ω Notice that the current in R 1 is equal to the sum of the branch currents in and R 3. The sum of the voltages around the outside loop is zero. I 1 = 21.6 ma R 1 = 270 Ω V 1 = 5.82 V P 1 = 126 mw I 2 = 12.7 ma = 330 Ω V 2 = 4.18 V P 2 = 53.1 mw I 3 = 8.9 ma R 3 = 470 Ω V 3 = 4.18 V P 3 = 37.2 mw I T = 21.6 ma R T = 464 Ω V S = 10 V P T = 216 mw
Loaded voltage divider The voltage-divider equation was developed for a series circuit. Recall that the output voltage is given by V R V 2 2 S RT A voltage-divider with a resistive load is a combinational circuit and the voltage divider is said to be loaded. The loading reduces the total resistance from node A to ground. + R 1 A R 3
Loaded voltage divider V S = +15 V R 1 330 Ω A What is the voltage across R 3? R 3 470 Ω 2.2 kω Form an equivalent series circuit by combining and R 3 ; then apply the voltage-divider formula to the equivalent circuit: R2,3 R2 R3 470 2.2 k = 387 R 2,3 387 V3 V2,3 VS 15 V R1 R 8.10 V 2,3 330 387
Stiff voltage divider V S R 1 A stiff voltage-divider is one in which the loaded voltage nearly the same as the no-load voltage. To accomplish this, the load current must be small compared R L to the bleeder current (or R L is large compared to the divider resistors). If R 1 = = 1.0 kω, what value of R L will make the divider a stiff voltage divider? What fraction of the unloaded voltage is the loaded voltage? R L > 10 ; R L should be 10 kω or greater. For a 10 kω load, R R 0.91 k V V V 0.476 V 2 L L S S S R1 R2 RL 1.0 k 0.91 k This is 95% of the unloaded voltage.
Loading effect of a voltmeter V S + 10 V Assume V S = 10 V, but the meter reads only 4.04 V when it is across either R 1 or. Can you explain what is happening? R 1 470 kω 470 kω All measurements affect the quantity being measured. A voltmeter has internal resistance, which can change the resistance of the circuit under test. In this case, a 1 MΩ internal resistance of the meter accounts for the readings. + 4.04 10 V + 4.04 V
Wheatstone bridge The Wheatstone bridge consists of a dc voltage source and four V S Output resistive arms forming two voltage dividers. The output is taken between the dividers. Frequently, one of the bridge resistors is adjustable. When the bridge is balanced, the output voltage is zero, and the products of resistances in the opposite diagonal arms are equal. + - R 1 R 3 R 4
Summary Wheatstone bridge Example: What is the value of if the bridge is balanced? 384 Ω V S 12 V + - R 1 R 3 470 Ω 330 Ω Output R 4 270 Ω
Thevenin s theorem Thevenin s theorem states that any two-terminal, resistive circuit can be replaced with a simple equivalent circuit when viewed from two output terminals. The equivalent circuit is: R TH V TH
Thevenin s theorem V TH is defined as the open circuit voltage between the two output terminals of a circuit. R TH is defined as the total resistance appearing between the two output terminals when all sources have been replaced by their internal resistances. R TH V TH
Thevenin s theorem What is the Thevenin voltage for the circuit? 8.76 V What is the Thevenin resistance for the circuit? 7.30 kω V S 12 V R 1 10 k 27 k Output terminals RL 68 k Remember, the load resistor has no affect on the Thevenin parameters.
Maximum power transfer The maximum power is transferred from a source to a load when the load resistance is equal to the internal source resistance. R S V S + R L The maximum power transfer theorem assumes the source voltage and resistance are fixed.
Maximum power transfer What is the power delivered to the matching load? The voltage to the load is 5.0 V. The power delivered is 2 V 2 P L R L 5.0 V = 0.5 W 50 V S + 10 V R S 50 Ω R L 50 Ω
Superposition theorem The superposition theorem is a way to determine currents and voltages in a linear circuit that has multiple sources by taking one source at a time and algebraically summing the results. What does the ammeter read for I 2? (See next slide for the method and the answer). V S1 12 V + - R 1 R 3 2.7 k 6.8 k + I 2 - V S2 18 V 6.8 k - +
What does the ammeter read for I 2? Set up a table of pertinent information and solve for each quantity listed: V VS1 S1 12 V + - R 1 R 3 2.7 kω k 6.8 kω k I 2 + 1.56 ma- 6.8 kω k V S2 18 V + - Source 1: R T(S1) = 6.10 kω I 1 = 1.97 ma I 2 = 0.98 ma Source 2: R T(S2) = 8.73 kω I 3 = 2.06 ma I 2 = 0.58 ma Both sources I 2 = 1.56 ma The total current is the algebraic sum.
Quiz 1. Two circuits that are equivalent have the same a. number of components b. response to an electrical stimulus c. internal power dissipation d. all of the above
Quiz 2. If a series equivalent circuit is drawn for a complex circuit, the equivalent circuit can be analyzed with a. the voltage divider theorem b. Kirchhoff s voltage law c. both of the above d. none of the above
Quiz 3. For the circuit shown, a. R 1 is in series with R 1 b. R 1 is in parallel with c. is in series with R 3 V S + - R 3 d. is in parallel with R 3
Quiz 4. For the circuit shown, a. R 1 is in series with b. R 4 is in parallel with R 1 R 1 R 4 c. is in parallel with R 3 V S + R 3 d. none of the above -
Quiz 5. A signal generator has an output voltage of 2.0 V with no load. When a 600 Ω load is connected to it, the output drops to 1.0 V. The Thevenin resistance of the generator is a. 300 Ω b. 600 Ω c. 900 Ω d. 1200 Ω.
Quiz 6. For the circuit shown, Kirchhoff's voltage law a. applies only to the outside loop b. applies only to the A junction. c. can be applied to any closed path. d. does not apply. V S + 10 V A R 1 270 Ω 330 Ω R 3 470 Ω
Quiz 7. The effect of changing a measured quantity due to connecting an instrument to a circuit is called a. loading b. clipping c. distortion d. loss of precision
Quiz 8. An unbalanced Wheatstone bridge has the voltages shown. The voltage across R 4 is a. 4.0 V b. 5.0 V c. 6.0 V V S 12 V + - R 1 R 3 7.0 V + R L - 1.0 V d. 7.0 V R 4
Quiz 9. Assume is adjusted until the Wheatstone bridge is balanced. At this point, the voltage across R 4 is measured and found to be 5.0 V. The voltage across R 1 will be a. 4.0 V b. 5.0 V c. 6.0 V d. 7.0 V V S 12 V + - R 1 R 3 R L + - R 4 5.0 V
Quiz 10. Maximum power is transferred from a fixed source when a. the load resistor is ½ the source resistance b. the load resistor is equal to the source resistance c. the load resistor is twice the source resistance d. none of the above
Quiz Answers: 1. b 6. c 2. c 7. a 3. d 8. a 4. d 9. d 5. b 10. b