# ES250: Electrical Science. HW1: Electric Circuit Variables, Elements and Kirchhoff s Laws

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1 ES250: Electrical Science HW1: Electric Circuit Variables, Elements and Kirchhoff s Laws

2 Introduction Engineers use electric circuits to solve problems that are important to modern society, such as: 1. the generation, transmission, and consumption of electric power and energy 2. the encoding, decoding, storage, retrieval, transmission, and processing of information In Chapter 1, we will do the following: 1. Represent the current and voltage of an electric circuit element, paying attention to the reference current and voltage directions 2. Use the passive convention to determine whether the power or energy is supplied or received by a circuit element 3. Use scientific notation to represent electrical quantities within a wide range of magnitudes

3 Electric Circuits and Current An electric circuit (or network) is an interconnection of electric elements linked together in a closed path so that an electric current may flow continuously through it Consider a simple circuit consisting of two well known electrical elements: a battery and a resistor In this case, each element (or device) is represented by a two terminal t l( (or node) element as shown

4 Electric Circuits and Current Charge is the intrinsic property of matter responsible for electric phenomena: 1. The quantity of charge q can be expressed in terms of the charge of one electron which is coulombs 2. Charge may flow in an electric circuit and current is the time rate of change of charge past a given point, expressed as 3. The unit of current is the ampere (A); an ampere is 1 3 eu o cu e s e a pe e( ); a a pe e s coulomb per second

5 Electric Circuits and Current A complete description of current requires both a value, which can be positive or negative, and a direction indicated by an arrow; thus we say that a current exists in or through an element Note, while the flow of current is historically represented as a flow of positive charges we realize that current flow in circuits is typically due to the flow of negatively charged electrons (this is not always the case in devices such as semiconductors)

6 Electric Circuits and Current If the current flowing through an element is constant, we represent it by the constant I, as shown below, and refer to it as a direct current (dc): Atime varying current i(t)cantakemanyforms suchasa A time varying current i(t) can take many forms, such as a ramp, a sinusoid called an alternating current (ac), or an exponential, as shown below:

7 Electric Circuits and Current If the charge q is known, the current i is readily found using Example Calculating Current from Charge: Find the current in an element when the charge entering the element is where t is the time in seconds Solution: Since the unit of charge is coulombs, C, the current is given by where the unit of current is amperes, A

8 Electric Circuits and Current If the current i is known, the charge q is readily found using where q(0) is the charge at t =0 Example Calculating Charge from Current : Find the Example Calculating Charge from Current : Find the charge that has entered the terminal of an element from t =0s to t = 3 s when the current entering the element is as shown:

9

10 Systems of Units The fundamental or base units of SI are:

11 Derived SI Units

12 SI Prefixes The great advantage of the SI system is that it incorporates a decimal system for relating larger or smaller quantities to the basic unit. An example of the common use of a prefix is the centimeter (cm), which is meter. The standard prefixes given below:

13 Voltage Voltage describes the energy required to cause charge to flow and we say that voltage exists across an element The voltage notation (or references) consist of a value and an assigned direction given by its polarities (+, ) When considering v ab, terminal a is called the + terminal and terminal b is called the terminal Note, voltages v ab and v ba have the same magnitude but opposite sign (polarity) v ab canbereadas the as voltage at terminal a with respect to terminal b; alternatively, we say that v ab is the voltage drop from terminal a to terminal b

14 Voltage The unit of voltage is the volt, V, and the equation for the voltage across the element is where v is voltage, w is energy (or work), and q is charge Note, a charge of 1 coulomb delivers an energy of 1 joule as it moves through a voltage of 1 volt

15 Power and Energy Power is the time rate of supplying or absorbing energy: where p is power in watts, and w is energy in joules The power associated with the current flow through h an element is: The power and energy delivered to an element are of great importance. For example, the useful output of an electric lightbulb can be expressed in terms of power. We know that a 300 watt bulb delivers more light than a 100 watt bulb.

16 The Passive Convention

17 Power and Energy When the element voltage and current adhere to the passive convention, the energy absorbed by an element can be determined from: Example Electrical Power and Energy: Consider the Example Electrical Power and Energy: Consider the element shown labled using the passive convention with v = 4 V and i = 10 A. Find the power absorbed by the element and the energy absorbed over a 10 s interval

18 Power and Energy

19 Resistors Resistance is the physical property of an element or device that impedes the flow of current; it is represented by the symbol R For the passive convention, Ohm's law relates voltage and current: Schematic symbol for a resistor with resistance of R ohms What is the expression for Ohm s law that adheres to the What is the expression for Ohm s law that adheres to the passive convention?

20 Resistors The power delivered to a resistor (when the passive convention is used) is: Alternatively, since v = ir, we can write: The energy delivered to a resistor is: Si h b b d i l i h Since the energy absorbed is always nonnegative, the resistor is referred to as a passive element

21 Example Power Dissipated by a Resistor

22 Independent Sources Sources are intended to supply energy to a circuit; there are two types: voltage sources and current sources: The voltage v(t) of a voltage source is specified but its current is determined by the rest of the circuit; while the converse is true for a current source

23 Short and Open Circuits A short circuit is an ideal voltage source having v(t) = 0 with the current determined by the rest of the circuit, while an open circuit is an ideal current source having i(t) = 0 with the voltage across an open circuit is determined by the rest of the circuit Note, the power absorbed by these devices is zero

24 Voltmeters and Ammeters A meter is connected to a circuit using terminals called probes to measure voltage or current Probes are color coded to indicate the reference direction of the variable being measured, e.g, red for the positive terminal and black for the negative terminal

25 Voltmeters and Ammeters Note, the voltmeter is connected in parallel with the voltage to be measured using the passive convention. The current flow into an ideal voltmeter is zero, meaning that the ideal voltmeter dissipates zero power

26 Voltmeters and Ammeters Note, the ammeter is connected in series with the current to be measured using the passive convention. The voltage across an ideal ammeter is zero, meaning that the ideal ammeter dissipates zero power

27 Voltmeter and Ammeter Examples Circuit measurements: Equivalent circuit:

28 Voltmeter and Ammeter Examples

29 Voltmeter and Ammeter Examples

30 Switches Switches have two distinct states: open and closed Ideally, a switch acts as a short circuit when it is closed and as an open circuit when it is open We ll consider two types of switches: single pole, single throw (SPST) and single pole, double throw (SPDT):

31 Switches Switches have two distinct states: open and closed Ideally, a switch acts as a short circuit when it is closed and as an open circuit when it is open We ll consider two types of switches: single pole, single throw (SPST) and single pole, double throw (SPDT):

32 Example 2.9 1: Switches Based on the schematic in red, when do the others take place? (If needed, see text for solution)

33 Exercise 2.9 2: Make before break SPDT Switch What is the value of the voltage v at time t =4s? At t =6s? Solution: IR = ( 2m )( 3k ) = 6V, for t < 5s vt () = 0V, for t > 5s

34 Kirchhoff's Laws An electric circuit consists of circuit elements connected together by nodes It is common practice to draw circuits using straight lines positioning the elements horizontally or vertically These equivalent circuits have six elements (1 6) and fours nodes (a d)

35 Example 3.2 1: Different Drawings of the Same Circuit The same circuit can be drawn in several different looking ways, but drawings represent the same circuit if corresponding elements are connected to corresponding nodes A th t i it i l t? Are these two circuits equivalent? If so, why?

36 Kirchhoff s Current Law (KCL) Kirchhoff s current law states that the algebraic sum of the currents entering any node is identically zero for all instants of time The phrase algebraic sum indicates that we must take reference directions into account as we add up the currents of elements connected to a particular node one way to take reference directions into account is to use a plus sign when the current is directed d away from a node and a minus sign when the current is directed toward a node, or currents in equal to currents out

37 Kirchhoff s Current Law (KCL) Write the KCL equation for node a:

38 Kirchhoff s Voltage Law (KVL) Kirchhoff s voltage law states that the algebraic sum of the voltages around any loop in a circuit is identically zero for all time A loop is a closed path through a circuit that does not encounter any intermediate node more than once How many loops does this circuit contain?

39 Kirchhoff s Voltage Law (KVL) The phrase algebraic sum indicates that we must take polarity into account as we add up the voltages of elements that comprise a loop one way to do this is to write the voltage with a plus sign when we encounter the + of the polarity before the and a minus sign when we encounter the of the polarity before the + Write the KVL eqn. associated with the loop containing elements 3 6 of the circuit:

40 Example 3.2 3: Ohm's and Kirchhoff's Laws Consider the circuit below noting the passive convention was used to assign references and find all currents and voltages when R 1 =8Ω, v 2 = 10 V, i 3 = 2 A, and R 3 =1Ω. Also, determine the resistance R 2

41

42 Example Ohm's and Kirchhoff's Laws Determine the value of the current, in amps, measured by the ammeter: Equivalent circuit, replacing the ammeter with a short circuit with an appropriate reference direction:

43

44 Dependent Sources Dependent sources model the situation in which the voltage or current of one circuit element is proportional to the voltage or current of the second circuit element used to model electronic devices such as transistors or amplifiers, e.g., the output voltage of an amplifier is proportional to the voltage input of that amplifier Each hdependent d source consists of ftwo parts: the controlling part and the controlled part the controlling part is either an open circuit or a short circuit the controlled part is either a voltage source or a current source There are four types of dependent source that correspond to the four ways of choosing a controlling part and a controlled part, called: voltage controlled voltage sources (VCVS) current controlled voltage controlled voltage sources (VCVS), current controlled voltage sources (CCVS), voltage controlled current sources (VCCS), and current controlled current sources (CCCS)

45 Types of Dependent Sources

46 Types of Dependent Sources

47 Transistor Example

48 Exercise Find the power absorbed by the CCCS: Since the CCCS references use the passive convention, the power absorbed b is given by: p = i v = 4i v = = 115.2W ( ) ( ) d d c d This implies that the CCCS is actually supplying energy to the circuit!

49 Transducers Transducers are devices that convert physical quantities to electrical quantities: potentiometers convert position to resistance temperature sensors convert temperature to current The symbol and A model for the potentiometer

50 Example Potentiometer Circuit Suppose R p = 10 kω and I = 1 ma, an angle of 163 would cause an output of v m = 4.53 V. Alternately, a meter reading of 7.83 V would indicate that Θ= 282

51 Questions?

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