# Series and Parallel Circuits

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1 Direct Current (DC) Direct current (DC) is the unidirectional flow of electric charge. The term DC is used to refer to power systems that use refer to the constant (not changing with time), mean (average) or zero-frequency voltage or current. For example, the voltage across a DC voltage source is constant as is the current through a DC current source. The DC solution of an electric circuit is the solution where all voltages and currents are constant. Series and Parallel Circuits Alternating Current (AC) Alternating current (AC) refers to the zero-mean time-varying voltage or current values, i.e., current or voltage signals whose magnitude vary with time around zero. AC is the form in which electric power is delivered to businesses and residences. The usual waveform of alternating current in most electric power circuits is a sine wave. In certain applications, different waveforms are used, such as triangular or square waves. The abbreviations AC and DC are often used to mean simply alternating and direct, as when they modify current or voltage. Voltage Sources A voltage source produces an electromotive force (e.m.f.) which causes a current to flow within a circuit unit of e.m.f. is the volt a volt is the potential difference between two points when a joule of energy is used to move one coulomb of charge from one point to the other Real voltage sources, such as batteries have resistance associated with them in analyzing circuits we use ideal voltage sources In this course, we will only deal with DC voltages and currents. Examples of voltage source symbols A voltage source has an internal source resistance, R s connected in series An ideal voltage source has zero source resistance, i.e. R s = : (AC voltage source) DC: Direct current AC: Alternating current (i.e., value does not change with time) (i.e., value changes with time) 1

2 Current Sources We also sometimes use the concept of an ideal current source unrealizable, but useful in circuit analysis can be a fixed current source while an ideal voltage source has zero output resistance, an ideal current source has infinite output resistance Examples of ideal current source symbols: A current source has an internal source resistance, R s connected in parallel An ideal current source has infinite source resistance, i.e. R s = : Independent Sources An independent voltage source is a voltage source whose value does not depend on a voltage or current somewhere else in the circuit. In other words, its value is not a function of any other current or voltage in the circuit. Dependent Sources A dependent source is a voltage source or a current source whose value depends on a voltage or current somewhere else in the circuit. Battery is an example of an independent voltage source. Example: V s = 5 V V s (t) = 5 V sin( t) (DC voltage source) (AC voltage source) An independent current source is a current source whose value does not depend on a voltage or current somewhere else in the circuit. In other words, its value is not a function of any other current or voltage in the circuit. Example: I s = 2 A I s (t) = 2 A sin( t) (DC current source) (AC current source) Dependent voltage and current sources generate power and supply it to a circuit only when there are other independent voltage or current sources in the circuit. These other independent sources produce a current to flow through or a voltage across the component that controls the magnitude of the voltage or current output from the dependent source. 2

3 Practical Dependent Sources - Transistors Bipolar Junction Transistors (BJTs) e.g., DC forward active model of BJT (current-controlled current source) Practical Dependent Sources (continued) - Amplifiers e.g., voltage-gain amplifier (voltage-controlled voltage source) e.g., current-gain amplifier (current-controlled current source) Metal-Oxide-Semiconductor Field Effect Transistors (MOSFETs) e.g., small signal saturation model of MOSFET (voltage-controlled current source) -Voltage and current regulators - Other devices include: Photodetectors, LEDs, and lasers Piezoelectric devices Thermocouples, thermovoltaic sources Series circuits All circuits have three common attributes. These are: 1. A source of voltage. 2. A load. Series circuit rule for current: Because there is only one path, the current everywhere is the same. For example, the reading on the first ammeter is, What do the other meters read? 3. A complete path. A series circuit is one that has only one current path. Series circuits The total resistance of resistors in series is the sum of the individual resistors. For example, the resistors in a series circuit are 680, 1.5 k, and 2.2 k. What is the total resistance? Tabulating current, resistance, voltage and power is a useful way to summarize parameters in a series circuit. Continuing with the previous example, complete the parameters listed in the Table k I 1 = 2.74 ma R 1 = 0.68 k V 1 = 1.86 V P 1 = 5.1 mw I 2 = 2.74 ma R 2 = 1.50 k V 2 = 4.11 V P 2 = 11.3 mw I 3 = 2.74 ma R 3 = 2.20 k V 3 = 6.03 V P 3 = 16.5 mw I T = 2.74 ma R T = 4.38 k V S = 12 V P T = 32.9 mw 3

4 Voltage sources in series Voltage sources in series add algebraically. For example, the total voltage of the sources shown is 27 V Power in Series Circuits Example: Use the voltage divider rule to find V 1 and V 2. Then find the power in R 1 and R 2 and P T. What is the total voltage if one battery is accidentally reversed? 9 V Solution: Applying the voltage divider rule: V 8.25 V The power dissipated by each resistor is: 0.29 W 0.21 W } P T = 0.5 W Voltage measurements Voltage is relative and is measured with respect to another point in the circuit. Ground reference is not always at the lowest point in a circuit. Assume the ground is moved to B as shown. Voltages that are given with respect to ground are shown with a single subscript. For example, V A means the voltage at point A with respect to ground (called reference ground). V B means the voltage at point B with respect to ground. V AB means the voltage between points A and B. What are V A, V B, and V C for the circuit? V A = 4 V V B = 0 V V C = 8 V What are V A, V B, and V AB for the circuit shown? V A = 12 V V B = 8 V V AB = 4 V Has V AB changed from the previous circuit? No, it is still 4 V Assume that R 2 is open. For this case, what are V A, V B, and V C for the circuit? Voltage Divider Rule The voltage drop across any given resistor in a series circuit is equal to the ratio of that resistor to the total resistance, multiplied by the total voltage. Answer: If R 2 is open, there is no current. Notice that V B = 0 V because it is ground and V A = 0 V because it has the same potential as V B. V C = 12 V because the source voltage is across the open. 4

5 Assume R 1 is twice the size of R 2. What is the voltage across R 1? 8 V Voltage dividers can be set up for a variable output using a potentiometer. In the circuit shown, the output voltage is variable. Example: What is the voltage across R 2? Solution: The total resistance is 25 k Applying the voltage divider formula: 8.0 V Notice that 40% of the source voltage is across R 2, which represents 40% of the total resistance. What is the largest output voltage available? 5.0 V Resistors in parallel Resistors that are connected to the same two points are said to be in parallel. Parallel circuits A parallel circuit is identified by the fact that it has more than one current path (branch) connected to a common voltage source. Parallel circuit rule for voltage Analysis of Two Resistors in Parallel Because all components are connected across the same voltage source, the voltage across each is the same. For example, the source voltage is 5.0 V. What will a voltmeter read if it is placed across each of the resistors? We can also write the equations in terms of conductances where 5

6 Parallel circuit rule for resistance Example: Resistance of two parallel resistors The total resistance of resistors in parallel is the reciprocal of the sum of the reciprocals of the individual resistors. Example: The resistors in a parallel circuit shown below are 680, 1.5 k, and 2.2 k. What is the total resistance? The resistance of two parallel resistors can be found by either: or What is the total resistance if R 1 = 27 k and R 2 = 56 k? 18.2 k Parallel circuit Current Divider When current enters a node (junction) it divides into currents with values that are inversely proportional to the resistance values. Tabulating current, resistance, voltage and power is a useful way to summarize parameters in a parallel circuit. Continuing with the previous example, complete the parameters listed in the Table. I 1 = 7.4 ma R 1 = 0.68 k V 1 = 5.0 V P 1 = 36.8 mw I 2 = 3.3 ma R 2 = 1.50 k V 2 = 5.0 V P 2 = 16.7 mw I 3 = 2.3 ma R 3 = 2.20 k V 3 = 5.0 V P 3 = 11.4 mw I T = 13.0 ma R T = 386 V S = 5.0 V P T = 64.8 mw The most widely used formula for the current divider is the two-resistor equation. For resistors R 1 and R 2, and Notice the subscripts. The resistor in the numerator is not the same as the one for which current is found. Example: Solution: Assume that R 1 is a 2.2 k resistor that is in parallel with R 2, which is 4.7 k. If the total current into the resistors is 8.0 ma, what is the current in each resistor? Power in parallel circuits Power in each resistor can be calculated with any of the standard power formulas. Most of the time, the voltage is known, so the equation is most convenient ma 2.55 ma Notice that the larger resistor has the smaller current. As in the series case, the total power is the sum of the powers dissipated in each resistor. What is the total power if 10 V is applied to the parallel combination of R 1 = 270 and R 2 = 150? 1.04 W 6

7 Assume there are 8 resistive wires that form a rear window defroster for an automobile. (a) If the defroster dissipates 90 W when connected to a 12.6 V source, what power is dissipated by each resistive wire? (b) What is the total resistance of the defroster? Identifying series-parallel relationships Most practical circuits have combinations of series and parallel components. Answer: (a) Each of the 8 wires will dissipate 1/8 of the total power or Components that are connected in series will share a common path. (b) The total resistance is Components that are connected in parallel will be connected across the same two nodes. 1 2 Follow up: What is the resistance of each wire? 1.76 x 8 = 14.1 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: There are no electrical measurements that can distinguish the boxes. Another example: There are no electrical measurements that can distinguish the boxes. There are no electrical measurements that can distinguish between the three boxes. 7

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