Module 1 – Part 3 Circuit Elements

1. Module 1 – Part 3Circuit Elements Filename: DPKC_Mod01_Part03.ppt

2. Overview of this Part In this part of the module, we will cover the following topics: • What a circuit element is • Independent voltage sources and current sources • Dependent voltage and current sources • Resistors and Ohm’s Law Note: Some of these topics will be review for some students, particularly those who have had some exposure to circuits before. However, it would be wise to skim through this material quickly, to make sure that we are using terms in a way that is familiar to you.

3. Textbook Coverage This material is covered in your textbook in the following sections: • Circuits by Carlson: Sections 1.2, 1.3, 2.3 • Electric Circuits 6th Ed. by Nilsson and Riedel: Sections 2.1, 2.2, 2.3 • Basic Engineering Circuit Analysis 6th Ed. by Irwin and Wu: Sections 1.3, 2.1 • Fundamentals of Electric Circuits by Alexander and Sadiku: Section 1.6, 2.2 • Introduction to Electric Circuits 2nd Ed. by Dorf: Sections 2-2, 2-3, 2-4

4. Circuit Elements • In circuits, we think about basic circuit elements that are the basic “building blocks” of our circuits. This is similar to what we do in Chemistry with chemical elements like oxygen or nitrogen. • A circuit element cannot be broken down or subdivided into other circuit elements. • A circuit element can be defined in terms of the behavior of the voltage and current at its terminals.

5. The 5 Basic Circuit Elements There are 5 basic circuit elements: • Voltage sources • Current sources • Resistors • Inductors • Capacitors We are going to define the first three here in this module. We will not introduce inductors or capacitors until later.

6. Voltage Sources • A voltage source is a two terminal circuit element that maintains a voltage across its terminals. • The value of the voltage is the defining characteristic of a voltage source. • Any current can go through the voltage source, in any direction. It can also be zero. The voltage source does not “care about” current. It “cares” only about voltage.

7. Voltage Sources – Ideal and Practical • A voltage source maintains a voltage across its terminals no matter what you connect to those terminals. • We often think of a battery as being a voltage source. For many situations, this is fine. Other times it is not a good model. A real battery will have different voltages across its terminals in some cases, such as when it is supplying a large amount of current. As we have said, a voltage source should not change its voltage as the current changes. • We sometimes use the term ideal voltage source for our circuit elements, and the term practical voltage source for things like batteries. We will find that a more accurate model for a battery is an ideal voltage source in series with a resistor. More on that later.

8. Voltage Sources – 2 kinds There are 2 kinds of voltage sources: • Independent voltage sources • Dependent voltage sources, of which there are 2 forms: • Voltage-dependent voltage sources • Current-dependent voltage sources

9. Voltage Sources – Schematic Symbol for Independent Sources The schematic symbol that we use for independent voltage sources is shown here. This is intended to indicate that the schematic symbol can be labeled either with a variable, like vS, or a value, with some number, and units. An example might be 1.5[V]. It could also be labeled with both.

10. Voltage Sources – Schematic Symbols for Dependent Voltage Sources The schematic symbols that we use for dependent voltage sources are shown here, of which there are 2 forms: • Voltage-dependent voltage sources • Current-dependent voltage sources

11. Notes on Schematic Symbols for Dependent Voltage Sources The symbol m is the coefficient of the voltage vX. It is dimensionless. For example, it might be 4.3 vX. The vX is a voltage somewhere in the circuit. The schematic symbols that we use for dependent voltage sources are shown here, of which there are 2 forms: • Voltage-dependent voltage sources • Current-dependent voltage sources The symbol r is the coefficient of the current iX. It has dimensions of [voltage/current]. For example, it might be 4.3[V/A] iX. The iX is a current somewhere in the circuit.

12. Current Sources • A current source is a two terminal circuit element that maintains a current through its terminals. • The value of the current is the defining characteristic of the current source. • Any voltage can be across the current source, in either polarity. It can also be zero. The current source does not “care about” voltage. It “cares” only about current.

13. Current Sources - Ideal • A current source maintains a current through its terminals no matter what you connect to those terminals. • While there will be devices that reasonably model current sources, these devices are not as familiar as batteries. • We sometimes use the term ideal current source for our circuit elements, and the term practical current source for actual devices. We will find that a good model for these devices is an ideal current source in parallel with a resistor. More on that later.

14. Current Sources – 2 kinds There are 2 kinds of current sources: • Independent current sources • Dependent current sources, of which there are 2 forms: • Voltage-dependent current sources • Current-dependent current sources

15. Current Sources – Schematic Symbol for Independent Sources The schematic symbols that we use for current sources are shown here. • Independent current sources This is intended to indicate that the schematic symbol can be labeled either with a variable, like iS, or a value, with some number, and units. An example might be 0.2[A]. It could also be labeled with both.

16. Current Sources – Schematic Symbols for Dependent Current Sources The schematic symbols that we use for dependent current sources are shown here, of which there are 2 forms: • Voltage-dependent current sources • Current-dependent current sources

17. Notes on Schematic Symbols for Dependent Current Sources The symbol g is the coefficient of the voltage vX. It has dimensions of [current/voltage]. For example, it might be 16[A/V] vX. The vX is a voltage somewhere in the circuit. The schematic symbols that we use for dependent current sources are shown here, of which there are 2 forms: • Voltage-dependent current sources • Current-dependent current sources The symbol b is the coefficient of the current iX. It is dimensionless. For example, it might be 53.7 iX. The iX is a current somewhere in the circuit.

18. Voltage and Current Polarities • Previously, we have emphasized the important of reference polarities of currents and voltages. • Notice that the schematic symbols for the voltage sources and current sources indicate these polarities. • The voltage sources have a “+” and a “–” to show the voltage reference polarity. The current sources have an arrow to show the current reference polarity.

19. Dependent Voltage and Current Sources – Units of Coefficients • Some textbooks use symbols other than the ones we have used here (m, b, r, and g). There are no firm standards. We hope this is not confusing. • Perhaps more important is that the different textbooks take different approaches to the use of units with the coefficients r and g. There seems to be two approaches: • Assume that r always has units of [V/A], which is the same thing as Ohms [W]. Assume that g always has units of [A/V], which is the same thing as Siemens [S]. The values for these coefficients are always shown without units. • Always show units for the coefficients r and g, somewhere in a given problem. Most textbooks follow Approach 1. However, for these modules, we will follow Approach 2, and always show units. This seems to be the clearest thing to do when the students may or may not have already made an assumption about the units that will be used. As always, when in doubt, show units.

20. Showing Units of Coefficients In these modules, we will always show units for the values of the coefficients r and g, somewhere in a given problem. General practice in electrical engineering is that variables should not have units. Rather, when we substitute in a value for a variable, the units must be given with that value. • For example, all of these expressions are fine: • vX = 120[V] • iQ= 35[A] • pabs = 24.5[kW] • pdel = vQ(13[A]) • pabs = vXiX • For example, there are missing units in the following expressions: • vX = 1.5 • pdel = 25iQ • iX = 15

21. Why do we have these dependent sources? • Students who are new to circuits often question why dependent sources are included. Some students find these to be confusing, and they do add to the complexity of our solution techniques. • However, there is no way around them. We need dependent sources to be able to model amplifiers, and amplifier-like device. Amplifiers are crucial in electronics. Therefore, we simply need to understand and be able to work with dependent sources. Go back to Overview slide.

22. Resistors • A resistor is a two terminal circuit element that has a constant ratio of the voltage across its terminals to the current through its terminals. • The value of the ratio of voltage to current is the defining characteristic of the resistor. In many cases a light bulb can be modeled with a resistor.

23. Resistors – Definition and Units • A resistor obeys the expression where vR is the voltage across the resistor, and iR is the current through the resistor, and R is called the resistance. • In addition, it works both ways. If something obeys this expression, we can think of it, and model it, as a resistor. • This expression is called Ohm’s Law. The unit ([Ohm] or [W]) is named for Ohm, and is equal to a [Volt/Ampere]. • It is very important to use Ohm’s Law only on resistors. It does not hold for sources. To a first-order approximation, the body can modeled as a resistor. Our goal will be to avoid applying large voltages across our bodies, because it results in large currents through our body. This is not good.

24. Schematic Symbol for Resistors The schematic symbols that we use for resistors are shown here. This is intended to indicate that the schematic symbol can be labeled either with a variable, like RX, or a value, with some number, and units. An example might be 390[W]. It could also be labeled with both.

25. Resistor Polarities • Previously, we have emphasized the important of reference polarities of current sources and voltages sources. There is no corresponding polarity to a resistor. You can flip it end-for-end, and it will behave the same way. • However, even in a resistor, direction matters in one sense; we need to have defined the voltage and current in the passive sign convention to use the Ohm’s Law equation the way we have it listed here.

26. Getting the Sign Right with Ohm’s Law If we use the Active Sign Convention to write our reference polarities for the voltage and current, then we have If we use the Passive Sign Convention to write our reference polarities for the voltage and current, then we have

27. Why do we have to worry about the sign in Ohm’s Law? • It is reasonable to ask why the sign in Ohm’s Law matters. We may be used to thinking that resistance is always positive. • Unfortunately, this is not true. The resistors we use, particularly the electronic components we call resistors, will always have positive resistances. However, we will have cases where a device will have a constant ratio of voltage to current, but the value of the ratio is negative when the passive sign convention is used. These devices have negative resistance. They provide positive power. This can be done using dependent sources. Go back to Overview slide.

28. Why do we have to worry about the sign in Everything? • This is one of the central themes in circuit analysis. The polarity, and the sign that goes with that polarity, matters. The key is to find a way to get the sign correct every time. • This is why we need to define reference polarities for every voltage and current. • This is why we need to take care about what relationship we have used to assign reference polarities (passive sign convention and active sign convention). An analogy: Suppose I was going to give you $10,000. This would probably be fine with you. However, it will matter a great deal which direction the money flows. You will care a great deal about the sign of the $10,000 in this transaction. If I give you -$10,000, it means that you are giving $10,000 to me. This would probably not be fine with you! Go back to Overview slide.