ECE 3301 General Electrical Engineering Presentation 01 Fundamental Facts of Circuit Theory

Why study circuit theory? Circuit theory is necessary for the understanding of all electrical systems. Circuit theory is an excellent exercise in system analysis. Circuit theory is an excellent exercise in problem solving. A course in circuit theory will improve your algebra and trigonometry skills.

Obstacles To Overcome Unfamiliarity. Words likes volts and amps and watts are vaguely familiar, but not understood. Phenomenon outside everyday experience. No intuitive concept of voltage and current. Lack of mathematical skill. Enough said. Student dodges. Trying to memorize an equation, only as needed to get by, rather than developing a deeper understanding.

Definition of Electricity Electricity (and magnetism) is a characteristic of the universe that may be used to transmit (and manipulate) energy and to transmit (and manipulate) information.

Physical Circuits and Models Resistance Model 12 V I 120 Ω Physical Circuit DC Voltage Source

Models and Physical Systems The model is never a perfect representation of the physical electrical system. However, if the idealized components in the model are chosen judiciously, the results gained from analyzing the model are sufficient to build a useful electrical system.

Models and Physical Systems The appropriate model must be chosen to reflect the realities of the physical system. Any idealized circuit will at best be a good approximation of a physical system.

Basic SI Units

SI Multipliers

SI Multipliers

Fundamental Quantities Charge: Electric charge is a characteristic of electrons and protons. In circuit theory, current is defined as the flow of charged particles. Each electron has an electric charge of - 1.6022  10 – 19 coulombs (C).

Electrical Charge Protons have a positive charge of the same magnitude. All charge exists in integer multiples of these values. Since each electron posses such a small value of charge, it takes about 6.24  10 18 electrons to accumulate –1 C of charge.

Electrical Charge As an extension of the law of the conservation of matter, charge can be neither created nor destroyed. It can only be transferred from one location in a circuit to another.

Fundamental Quantities Voltage: Voltage is a measure of the work (energy) required to move one coulomb of charge between two points (nodes) in a network. Voltage is a measure of the electrical potential energy difference between the two points.

Electrical Voltage The Voltage difference between points a and b is defined by: 𝑣 𝑎𝑏 = 𝑑𝑤 𝑑𝑞 joules per coulomb, J C Where 𝑣 𝑎𝑏 =voltage difference between points 𝑎 and 𝑏 volts, V 𝑤=energy joules, J 𝑞=charge (coulombs, C)

Electrical Voltage Voltage is always measured across a circuit element . a Circuit Element 𝑣 𝑎𝑏 = 𝑣 𝑎 − 𝑣 𝑏 b

Electrical Voltage The voltage at some point in a network is always measured with respect to the voltage at some other point in the network. This voltage is the measure of the potential difference between those two points. This may be measured across a single circuit element, or may be measured with respect to a reference point (ground.)

Electrical Voltage Each voltage in a network has a direction, or polarity associated with it. This is indicated by the “+” and “–” signs shown on the circuit diagram. a 𝑣 𝑎𝑏 b

Electrical Voltage The value of the voltage may be positive or negative. a 𝑣 𝑎𝑏 =3 V b a 𝑣 𝑎𝑏 =−3 V b Point a is 3 volts higher than point b. Point a is −3 volts higher than point b.

Fundamental Quantities Current: Current is a measure of the time rate of flow of electrical charge through a circuit element.

Electrical Current Current is defined by: 𝑖= 𝑑𝑞 𝑑𝑡 coulombs per second, C s Where 𝑖=current amperes, A 𝑞=charge coulombs, C 𝑡=time (seconds, s)

Electrical Current Current is measured through a circuit element. Each circuit element will have an electrical current flowing through it. Current is a measure of the electrons in motion due to the electric field that results from the potential difference (voltage) across the circuit element.

Electrical Current Positive Current Flow Electron Motion Voltage i - -

Electrical Current Current is defined as the flow of positive charges through a circuit element. Current flows in the opposite direction of electron flow. Current only flows in closed loops. This is an extension of the Law of Conservation of Matter.

Electrical Current Each current in a network has a direction associated with it. 𝑖 1 𝑖 2

Electrical Current In addition to a direction, the value of current may be positive or negative. 𝑖=3 A 𝑖=−3 A

Total Charge and Current Since the definition of electrical current is 𝑖= 𝑑𝑞 𝑑𝑡 The total charge transferred in a given time interval is 𝑞= 𝑡 0 𝑡 1 𝑖 𝑑𝑡

Fundamental Quantities Power: The time rate at which energy is dissipated in a circuit element.

Power Power is defined: 𝑝= 𝑑𝑤 𝑑𝑡 joules per second, J s Where 𝑝=power watts, W 𝑤=energy joules, J 𝑡=time (seconds, s)

Power Using the chain rule: 𝑝= 𝑑𝑤 𝑑𝑞 × 𝑑𝑞 𝑑𝑡 And the definitions of voltage and current 𝑝=𝑣𝑖 This describes the instantaneous power delivered to, dissipated in, absorbed by a circuit element.

Fundamental Quantities In each network, some devices will be sources of power, others will be sinks of power. In all instances, the total power delivered in the network will equal the total power absorbed. This is a statement of the Law of Conservation of Energy.

Fundamental Quantities Energy: The energy dissipated by a circuit element in a given time interval is given by 𝑤= 𝑡 0 𝑡 1 𝑝 𝑑𝑡 (joules)

Energy The electric utility company uses a more convenient measure, kilowatt-hours. 𝑤= 𝑡 0 𝑡 1 𝑝 kW 𝑑𝑡 (hours) (kWH) A 5000 Watt clothes dryer operated for 45 minutes consumes 5(.75) = 3.75 kWH At 10cents/kWH it costs about 38 cents.

Energy Batteries are rated in amp-hours (the voltage is constant). A 3.6 Volt, 800 mAH battery has 3.6(0.8)(60)(60) = 10368 Joules of energy

Fundamental Circuit Elements Resistance: The voltage across a resistance is directly proportional to the current through the resistance. The constant of proportionality is called the resistance R, measured in ohms. The relationship between the current through a resistance and the voltage across a resistance is expressed by Ohm’s Law.

Resistance v i R 𝑣=𝑖𝑅 𝑖= 𝑣 𝑅 Positive current enters the positive terminal of the resistance.

Resistance There is a difference between a “resistor” and a “resistance.” A “resistor” is a physical device with voltage, current and power and frequency limitations. It also has inductive, capacitive and thermal effects.

Resistance “Resistance” is the idealized model of a resistor. Provided a resistor is applied within its limitations, it can be very well modeled as an ideal resistance.

Resistance 𝑣=−𝑖𝑅 𝑖=− 𝑣 𝑅 v i 𝑣=−𝑖𝑅 𝑖=− 𝑣 𝑅 Positive current enters the negative terminal of the resistance. 40 40

Resistance The behavior of a resistor may be expressed graphically as follows. R V I

Resistance R = 5 Ω V = 10 V I = ? 𝐼= 𝑉 𝑅 A 5 ohm resistor with 10 V placed across it results in a current flow of 2 amps through the resistor.

Resistance R = 2 Ω V = ? I = 3 A 𝑉=𝐼𝑅 A 3 amp current through a 2 ohm resistor results in a voltage of 6 volts across the resistor.

Resistance R = ? V = 20 V I = 5 A 𝑅= 𝑉 𝐼 A 5 amp current through a resistor with a 20 V drop across it requires a 4 Ohm resistance.

Resistance Power dissipated in a resistance: The power dissipated in a resistor is given by the equation 𝑝=𝑣𝑖 recall 𝑣=𝑖𝑅 𝑝=𝑖𝑅𝑖 𝑝= 𝑖 2 𝑅

Resistance This is often referred to as the “i-squared-R” power loss. 𝑝= 𝑖 2 𝑅

Resistance Power dissipated by a resistance: The power dissipated in a resistor is given by the equation 𝑝=𝑣𝑖 recall 𝑖= 𝑣 𝑅 𝑝=𝑣 𝑣 𝑅 = 𝑣 2 𝑅

Conductance Conductance: On some occasions, instead of using resistance, the conductance of a device may be used, where 𝐺= 1 𝑅 siemens S (or mhos)

Fundamental Circuit Elements Inductance: The voltage across an inductance is proportional to the time rate-of-change of the current through it. The constant of proportionality is called the inductance L, measured in henries.

Inductance 𝑣=𝐿 𝑑𝑖 𝑑𝑡 𝑖= 1 𝐿 −∞ 𝑡 0 𝑣 𝑑𝑡 𝑖 𝑡 𝑣=𝐿 𝑑𝑖 𝑑𝑡 𝑖= 1 𝐿 −∞ 𝑡 0 𝑣 𝑑𝑡 L 𝑣 𝑡 The limits of integration are from –  to the present time. Thus the inductance has “memory.”

Inductance An “inductor” is a physical device that may be modeled as an “inductance.” An inductor also has voltage, current, power and frequency limitations because of resistive, capacitive and thermal effects.

Fundamental Circuit Elements Capacitance: The current through a capacitance is proportional to the time rate-of-change of the voltage across it. The constant of proportionality is call capacitance, C, measured in farads.

Capacitance 𝑖=𝐶 𝑑𝑣 𝑑𝑡 𝑣= 1 𝐶 −∞ 𝑡 0 𝑖 𝑑𝑡 𝑖 𝑡 𝑖=𝐶 𝑑𝑣 𝑑𝑡 𝑣= 1 𝐶 −∞ 𝑡 0 𝑖 𝑑𝑡 C 𝑣 𝑡 The limits of integration are from –  to the present time. Thus the capacitance has “memory.”

Capacitance A “capacitor” is a physical device that may be modeled as a “capacitance.” A capacitor also has voltage, current, power and frequency limitations because of resistive, inductive and thermal effects.

Fundamental Laws of Circuit Theory Kirchhoff’s Current Law: The algebraic sum of all currents entering any node equals zero. Consider the current into a node as positive, and the current leaving a node as negative. This Law is an extension of the Law of Conservation of Matter.

Kirchhoff’s Current Law Node 𝑖 1 𝑖 3 𝑖 4 𝑖 2 𝑖 5 𝑖 1 + 𝑖 2 − 𝑖 3 − 𝑖 4 − 𝑖 5 =0

Kirchhoff’s Current Law Node 𝑖 1 𝑖 3 𝑖 4 𝑖 2 𝑖 5 𝑖 1 + 𝑖 2 = 𝑖 3 + 𝑖 4 + 𝑖 5

Fundamental Laws of Circuit Theory Kirchhoff’s Voltage Law: The algebraic sum of voltage differences around any closed loop equals zero. Consider moving from a “–” to a “+” sign a positive voltage and moving form a “+” to a “–” sign a negative voltage. This Law is an extension of the Law of the Conservation of Energy.

Kirchhoff’s Voltage Law b c 𝑣 𝑎 𝑣 𝑏 𝑣 𝑐 𝑣 𝑏𝑎 𝑣 𝑐𝑏 𝑣 𝑐𝑎 + 𝑣 𝑎 + 𝑣 𝑏𝑎 − 𝑣 𝑏 =0

Kirchhoff’s Voltage Law b c 𝑣 𝑎 𝑣 𝑏 𝑣 𝑐 𝑣 𝑏𝑎 𝑣 𝑐𝑏 𝑣 𝑐𝑎 𝑣 𝑎 + 𝑣 𝑏𝑎 = 𝑣 𝑏

Kirchhoff’s Voltage Law 𝑣 𝑐𝑎 𝑣 𝑏𝑎 𝑣 𝑐𝑏 b a c 𝑣 𝑎 𝑣 𝑏 𝑣 𝑐 + 𝑣 𝑎 + 𝑣 𝑐𝑎 − 𝑣 𝑐𝑏 − 𝑣 𝑏 =0

Kirchhoff’s Voltage Law 𝑣 𝑐𝑎 𝑣 𝑏𝑎 𝑣 𝑐𝑏 b a c 𝑣 𝑎 𝑣 𝑏 𝑣 𝑐 𝑣 𝑎 + 𝑣 𝑐𝑎 = 𝑣 𝑐𝑏 + 𝑣 𝑏

Kirchhoff’s Voltage Law 𝑣 𝑐𝑎 𝑣 𝑏𝑎 𝑣 𝑐𝑏 b a c 𝑣 𝑎 𝑣 𝑏 𝑣 𝑐 𝑣 𝑎 + 𝑣 𝑐𝑎 = 𝑣 𝑐𝑏 + 𝑣 𝑏 = 𝑣 𝑐

Basic Assumptions Linear Elements: There is a linear relationship between the voltage across all circuit elements and the current through the circuit elements. This assumption is largely true for passive elements (resistors, inductors and capacitors) as long as they are operated within their voltage, current frequency and thermal limitations.

Basic Assumptions Time-Invariant Elements: The characteristics of the circuit elements do not change over time. That is, the resistance, inductance and capacitance are constants and not functions of time. In reality, all physical devices age and their electrical characteristics change.

Basic Assumptions Bi-lateral, Two-terminal Elements: The elements have two terminals. The electrical characteristics of the circuit elements are the same regardless of the polarity of the voltage across the element and regardless of the direction of the current through the circuit element. The same current flows through both terminals.

Basic Assumptions Lumped Elements: The circuit elements operate at a point within the electrical network. From another point of view, the electrical network is small compared to the wavelength of the highest frequency of interest. Since waveforms move through electrical networks at roughly the speed of light, the relevant dimensions of a network can be approximated.

Basic Assumptions Approximate Wavelengths at Various Frequencies. Frequency  /10 60 Hz 5106 m  3100 miles 310 miles 20 KHz 15103 m  9 miles .9 miles 100 MHz 3 m  10 feet 12 inches 600 MHz .5 m  20 inches 2 inches 1 GHz .3 m  12 inches 1.2 inches

Basic Assumptions Idealized Components: Idealized models are used for all circuit elements. These idealized models are accurate under many conditions. For other conditions a combination of idealized components may be used to represent real devices

Conventions for Variables Constant (time invariant) variables are denoted by upper case variables such as R, C, L, V, I. Variables that are time varying are designated using lower case variables v, i or explicitly v(t), i(t)

Conventions for Variables Complex variables (those with real and imaginary parts) will be indicated by upper case bold face. 𝑽, 𝑰 When hand-written, complex variables are written in upper case with a bar above the variable symbol. 𝑉 , 𝐼

Reference Node (Ground) The voltage at any point in an electrical network is not absolute. It is always measured with respect to the voltage at some other point in the universe. This demands we establish a reference point and assign that point a voltage. This reference point is traditionally called “ground” and is assigned the voltage of zero volts.

Reference Node (Ground) In some cases, this reference point involves a physical connection to the earth. In most instances, the reference point is the metallic case or chassis containing the circuit. The reference node in circuits is typically indicated by the “ground” symbol.

Other things you need to know Algebra Trigonometry Sinusoidal Functions Exponential Functions Complex Numbers Rectangular Form Polar Form Linear Algebra Matrices and determinates Solving sets of simultaneous equations Cramer’s Rule Differential and Integral calculus Differential Equations