Physics
Voltage Sources in Parallel
When voltage sources are connected in parallel, their positive terminals are connected together and their negative terminals are connected together. This configuration allows the sources to share the load, providing a combined voltage output equal to the voltage of any single source. The total current supplied by the parallel sources is the sum of the individual source currents.
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6 Key excerpts on "Voltage Sources in Parallel"
eBook - PDFEssentials of Advanced Circuit Analysis
A Systems Approach
- Djafar K. Mynbaev(Author)
- 2024(Publication Date)
- Wiley(Publisher)
It must be pointed out that the above discussion refers to the ideal sources. 1.5.2 Series and Parallel Sources, Ideal and Real Sources, and Source Transformation Three more points are worth consideration. 1) First, the voltage sources can be connected in series, and the current sources can be connected in parallel. These configurations enable us to increase the output of the sources. Figures 1.23a Figure 1.22 Current sources: a) Independent source; b) voltage-controlled current source; c) current- controlled current source. (Sources are shown in Multisim notations.) 1.5 Sources (Inputs, Excitations, or Drives) 57 and 1.23b illustrate the concept. Two voltage sources connected in series supply the sum of their voltages to the load circuit, as shown in Figure 1.23a. Two current sources connected in parallel feed the load circuit with the sum of their currents, as illustrated in Figure 1.23b. However, these summation rules are valid only for sources; series and parallel connections of the passive components (resistors, capacitors, and inductors) obey different regulations, as dis- cussed in Section 1.3. Another particularity of the source connections is that the voltage sources must be never connected in parallel, and the current sources cannot be wired in series. Question Can the controlled (dependent) voltage sources be connected in series? Controlled current sources in parallel? 2. The second point is about the difference between an ideal and a real (practical) source. The ideal sources are shown in Figures 1.20 through 1.23 and described in the preceding part of this section. In reality, however, every source includes its internal resistance, R int ( ) Ω . Figure 1.24a shows that an internal resistor, R int 1 , is connected in series in a practical (real) voltage source. Figure 1.24b demonstrates that resistor R int 2 is wired in parallel in a practical (real) current source. The load circuit’s viewpoint at the sources is shown in Figure 1.24c.
- Frank R. Spellman, Nancy E. Whiting(Authors)
- 2013(Publication Date)
- CRC Press(Publisher)
E 2 + – E 2 + + – – E 1 + – R 1 R 1 R 2 Series Aiding R 2 Series Opposing FIGURE 11.36 Series aiding and opposing sources. R 1 40 ohms E b2 40 v A E b1 140 v + + – – R 2 20 ohms + + – + – – I E b3 20 v FIGURE 11.37 Solving for circuit current in a multiple-source circuit. 275 Fundamental Engineering Concepts 11.7.7.1 Parallel Circuit Characteristics A parallel circuit is defined as a circuit having two or more components connected across the same voltage source (see Figure 11.38). Recall that a series circuit has only one path for current flow. As additional loads (resistors, etc.) are added to the circuit, the total resistance increases and the total current decreases. This is not the case in a parallel circuit. In a parallel circuit, each load (or branch) is connected directly across the voltage source. In Figure 11.38, commencing at the voltage source ( E b ) and tracing counterclockwise around the circuit, two complete and separate paths can be identified in which current can flow. One path is traced from the source through resistance R 1 and back to the source, the other from the source through resistance R 2 and back to the source. 11.7.7.2 Voltage in Parallel Circuits Recall that in a series circuit the source voltage divides proportionately across each resistor in the circuit. In a parallel circuit (see Figure 11.38), the same voltage is present across all of the resistors of a parallel group. This voltage is equal to the applied voltage ( E b ) and can be expressed in equa-tion form as E b = E R 1 = E R 2 = E Rn (11.31) We can verify Equation 11.31 by taking voltage measurements across the resistors of a parallel cir-cuit, as illustrated in Figure 11.39. Notice that each voltmeter indicates the same amount of voltage; that is, the voltage across each resistor is the same as the applied voltage. Note: In a parallel circuit, the voltage remains the same throughout the circuit.
eBook - PDF- Giovanni Saggio(Author)
- 2014(Publication Date)
- CRC Press(Publisher)
Ideal current sources cannot be connected in series. 4.5.3 Non-Ideal Voltage and Current Sources The model given from ideal voltage and current sources, although used often, does not exactly match those that are the real ones. In fact, ideal sources are mathemati-cal abstractions, and their model can also lead to certain incongruities. For example, an ideal voltage source can provide infinite current with non-zero voltage difference, and an ideal current source can provide infinite voltage difference with non-zero current, so either device is theoretically capable of delivering infinite power! We know that for two devices in a parallel topology, the voltage across them is the same, but then what happens if those two devices are two ideal voltage sources, each of them providing a voltage of different value? We will have the paradox that the voltage should be the same, but each of the sources can impose a different one. An analogous paradox is for two current sources in series, each of them pro-viding a different current value. What will be the real amount of current flowing through them? These incongruities find solutions when considering the independent sources as a real model that takes into account their unwanted but unavoidable internal electrical resistance (as defined in Section 3.9 of Chapter 3). Since these devices I S (a) I S (b) V I (c) FIGURE 4.41 (a), (b) Current source symbols, (c) I-V characteristic of an ideal current source. i s (a) i s (a) (b) i s (c) FIGURE 4.42 Some commonly adopted symbols for the variable current source. Two-Terminal Components 119 are made of materials with non-zero resistive value, their internal resistance can-not be null. So, this internal resistance will be such to distinguish the ideal model from the real one. For the voltage source this means that it may not always provide a voltage of constant value, at any time and for every load.
- Frank R. Spellman(Author)
- 2013(Publication Date)
- CRC Press(Publisher)
10.14.9 S ERIES A IDING AND O PPOSING S OURCES Sources of voltage that cause current to flow in the same direction are considered to be series aiding, and their volt-ages are added. Sources of voltage that would tend to force current in opposite directions are said to be series opposing , and the effective source voltage is the difference between the opposing voltages. When two opposing sources are inserted into a circuit, current flow would be in a direction determined by the larger source. Examples of series aiding and opposing sources are shown in Figure 10.39. 10.14.10 K IRCHHOFF ’ S L AW AND M ULTIPLE S OURCE S OLUTIONS Kirchhoff’s law can be used to solve multiple source circuit problems. When applying this method, the exact same pro-cedure is used for multiple-source circuits as was used for single-source circuits. This is demonstrated by the following example. ■ EXAMPLE 10.18 Problem: Find the amount of current in the circuit shown in Figure 10.40. Solution: Start at point A. Basic equation: E a + E b + E c + … + E n = 0 From the circuit: E b 2 + E 1 – E b 1 + E b 3 + E 2 = 0 40 + 40 I – 140 + 20 + 20 I = 0 Combining like terms, we obtain 60 I – 80 = 0 60 I = 80 I = 1.33 amps 10.15 GROUND The term ground is used to denote a common electrical point of zero potential. The reference point of a circuit is always considered to be at zero potential. The earth (ground) is said to be at zero potential. In Figure 10.41, point A is the zero reference or ground and is symbolized as such. Point C is 60 volts positive and point B is 20 volts positive with respect to ground. The common ground for much electrical/electronics equipment is the metal chassis. The value of ground is noted R 1 R 2 Series aiding E 1 E 2 + + – – R 1 R 2 Series opposing E 2 + + – – FIGURE 10.39 Aiding and opposing sources. I E b 1 140 V E b 2 140 v E b 3 20 V R 2 20 ohms + + + – + – + A – – – R 1 40 ohms FIGURE 10.40 Solving for circuit current in a multiple source circuit.
eBook - PDF- A. Henderson(Author)
- 2014(Publication Date)
- Arnold(Publisher)
In that case, however, the source currents cannot be calculated. Two current sources in series are forbidden if the sum of the source strengths is not zero (for in this case KirchhofF s current law for the node between the two sources) is not valid. Two current sources in series are not forbidden if the sum of the source strengths is zero. In that case the source voltages cannot be calculated. In network theory it often happens that there is analogy between two formulas, between two elements or between two circuits. For instance, one Kirchhoff's law turns into the other if one substitutes voltage for current and vice versa. We therefore say that the current law is the dual of the voltage law and vice versa. The dual character is also found in • voltage - current • open nodes - short circuit • resistance - conductance In the following chapters we shall often meet this phenomenon of duality. 1.3 Energy and power The voltage V A B between two points A and B is defined as the work needed to move a unit charge (1 C concentrated in a point) from point B to point A. If the charge is Aq the work is therefore AW = (V A - V B )Aq = V AB Aq, (1.14) in which VA and VB are the potentials of the points A and B. If VAB is constant (d.c.) and if the work is done in a time At, the average power is p _ A W _ Aq r At v At ' in which V = VAB-For At -> 0 we obtain P = VI. (1.15) So the power, in the case of d.c, is the product of voltage and current. Energy is expressed in joule (J), power in watt (W). Power can be consumed or supplied. If a current I flows through a network N with two terminals (also called a one-port) and if the polarity of the voltage V is such that I flows from + to - the power consumed is positive (Figure 1.11). 18 d.c. currents and d.c. voltages The voltage source transports (positive) charge from minus to plus and so delivers electrical energy to N (this energy is supplied by the chemical or mechanical system outside the network).
eBook - PDFCircuits, Signals, and Systems for Bioengineers
A MATLAB-Based Introduction
- John Semmlow(Author)
- 2005(Publication Date)
- Academic Press(Publisher)
A similar apparent contradiction applies to current sources: They are open circuits with respect to voltage yet produce a specified current. In this section, voltage and current sources are described in terms of fixed values (i.e., DC sources), but the basic arguments do not change if V S or I S are time varying. This generalization will hold true for the other arguments presented below. 7.2.2 Real Voltage Sources: The Thévenin Source Unlike ideal sources, real voltage sources are not immune to the current flowing through them, nor are real current sources immune to the voltage falling across them. In real voltage sources, the source voltage drops as more current is drawn from the source. This gives rise to a v -i plot such as shown in Figure 7.11, where the line is no longer horizontal but decreases with increasing current. The decrease indicates the presence of an internal, nonzero, resistance having a value equal to the negative slope of the line. The slope is negative because the voltage drop across the internal resistance is opposite in sign to V s and hence subtracts from the values of V s . 7.2 IDEAL AND REAL SOURCES 255 Figure 7.10 A v-i plot of an ideal current source. This plot shows that the resistor-like properties of a current source have an infi-nite value. Real sources, then, are simply ideal sources with some nonzero resistance. They can be represented as an ideal source in series with a resistor as shown in Figure 7.12. This configuration is also known as a Thévenin source, named after an engi-neer who developed a network reduction theory described below. Finding values for V T and R T given a physical (and therefore real ) source is straightforward. The value of the internal ideal source, V T , is just the voltage that would be measured at the output, v out , if no current was flowing through the circuit; that is, if the output was 256 CHAPTER 7 RELATIONSHIPS BETWEEN ANALOG ELEMENTS Figure 7.11 The v-i plot of a real voltage source ( solid line ).
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