Lecture 01 Introduction, circuit analysis, and simple DC...

CA2627 Building Science

Instructor: Jiayu Chen Ph.D.

Lecture 01 Introduction, circuit analysis, and simple DC circuits

© Jiayu Chen, Ph.D. 2

Introduction

Name: Jiayu Chen Course Instructor Office: Y6621 Email: [email protected] Course Wiki: ca2627.wikispaces.com

mailto:[email protected]


Course Requirements

Grading = 50% Coursework +50% Examination


Course Requirements

Introduction to Course Wiki ca2627.wikispaces.com

1. First login requirements account registration and membership request

2. You can edit the content of the wiki

3. I will post the course contents and some sample questions for you to prepare your exams


• Identify the principal elements of electrical circuits: nodes, loops, meshes, branches, and voltage and current sources.

• Apply Kirchhoff’s Laws to simple electrical circuits and derive the basic circuit equations.

• Apply the passive sign convention and compute power dissipated by circuit elements.

• Apply the voltage and current divider laws to calculate unknown variables in simple series, parallel and series-parallel circuits.

• Understand the rules for connecting electrical measuring instruments to electrical circuits for the measurement of voltage, current, and power.

Learning Objectives – Lecture 01


Basic elements in electric circuits 1. Charge

Charge is the quantity of electricity responsible for electric phenomena.

The smallest charge (unit charge) is the charge of an electron

The atomic structure of matter consists of a nucleus—neutrons and protons—surrounded by electrons

C is coulomb (C), named after Charles Coulomb.

𝒒𝒒𝒆𝒆 = −𝟏𝟏.𝟔𝟔𝟔𝟔𝟔𝟔 × 𝟏𝟏𝟔𝟔−𝟏𝟏𝟏𝟏𝑪𝑪


Basic elements in electric circuits

dtdqi =

2. Current Current is the time rate of flow of electric charge past a given point.

DC: 𝐼𝐼 = 𝑄𝑄/𝑡𝑡 𝑄𝑄 = 𝐼𝐼𝑡𝑡

q(0)τdiτdiq(t)t

0

t+⋅=⋅= ∫∫ ∞−

*Note: A current has a magnitude and a sign, denoting its direction.

AC:


Basic elements in electric circuits 2. Current Direct Current vs. Alternating Current When a current is constant with time, we say that we have direct currents (DC). On the other hand, a current that varies with time, reversing direction periodically, is called alternating current, abbreviated as AC. DC are constant with current, whereas AC vary with time.


Basic elements in electric circuits 3. Voltage Voltage is the energy required to cause charge to flow. Voltage is given by a value (or a variable name) and an assigned direction. We say a voltage exists across a circuit element. Voltage is proportional to the work required to move a positive charge from point a to point b. The voltage across an element is the work (energy) required to move a unit positive charge from the - terminal to the + terminal. The unit of voltage is the volt. V.

𝑉𝑉𝑎𝑎𝑎𝑎 = 𝑉𝑉𝑎𝑎

− 𝑉𝑉𝑎𝑎

𝑉𝑉𝑎𝑎𝑎𝑎 = 𝑉𝑉𝑎𝑎

– 𝑉𝑉𝑎𝑎


Basic elements in electric circuits 3. Voltage


Basic elements in electric circuits 3. Voltage EMF (electromotive force) = active voltage, supplies energy to a circuit PD (potential difference) = associated with a passive component


Basic elements in electric circuits 4. Power

Power is the time rate of using or absorbing energy

𝑃𝑃 =𝑑𝑑𝑑𝑑𝑑𝑑𝑡𝑡

=𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑

∙𝑑𝑑𝑑𝑑𝑑𝑑𝑡𝑡

= 𝑉𝑉 ∙ 𝐼𝐼

𝑃𝑃 =𝑊𝑊𝑡𝑡

=𝑊𝑊𝑄𝑄∙𝑄𝑄𝑡𝑡

= 𝑉𝑉 ∙ 𝐼𝐼


Basic elements in electric circuits

5. Passive convention To avoid confusion with regard to the sign of power, the electrical engineering community uniformly adopts the passive sign convention, which simply states that the power dissipated by a load is a positive quantity (or, conversely, that the power generated by a source is a positive quantity). Another way of phrasing the same concept is to state that if current flows from a higher to a lower voltage (+ to −), the power is dissipated and will be a positive quantity.


Basic elements in electric circuits 6. SI System The International System of Units (SI) is the modern form of the metric system and is the world's most widely used system of measurement.


QUANTITY SYMBOL UNIT Energy = Work W Joule Power P Watt Efficiency η number (%) Eta Linear quantities Force F newton N Acceleration a m/s2 Velocity u m/s Energy W Joule Power P Watt Rotation quantities Torque T Angle θ rad, °degrees Angular Velocity Ω rad/s Energy W Joule Power P Watt


Circuit Elements

Electric circuits consist of interconnections of circuit elements.


Suppose that the excitation is the current i and the response is the voltage v. When the element is subjected to a current i_1, it provides a response v_1. Furthermore, when the element is subjected to a current i_2, it provides a response v_2. For a linear element, it is necessary that the excitation i_1+i_2 result in a response v_1+v_2. This is usually called the principle of superposition. Multiplying the input of a linear device by a constant must have the consequence of multiplying the output by the same constant. For example, doubling the size of the input causes the size of the output to double. This is called the property of homogeneity. An element is linear if, and only if, the properties of superposition and homogeneity are satisfied for all excitations and responses.

Circuit Elements

Linear System in Mathematics

A circuit element is linear if it satisfies both superposition and homogeneity properties


Superposition principle

The superposition principle says that, in a linear circuit containing several independent sources, the current or voltage of a circuit element equals the algebraic sum of the component voltages or currents produced by the independent sources acting alone. Put another way, the voltage or current contribution from each independent source can be found separately, and then all the contributions can be added to obtain the actual voltage or current with all independent sources in the circuit.

Circuit Elements

=


Circuit Elements

Linear System vs. Nonlinear System A device that does not satisfy either the superposition or the homogeneity principle is said to nonlinear

Ex: Consider the element represented by the relationship between current and voltage as:

𝑉𝑉 = 𝑅𝑅𝐼𝐼

The response to a current I_1 The response to a current I_2 The sum of these responses is Because the sum of the responses to i_1 and i_2 is equal to the response to i_1 + i_2, the principle of superposition is satisfied. we have for an excitation

𝑉𝑉1 = 𝑅𝑅𝐼𝐼1 𝑉𝑉2 = 𝑅𝑅𝐼𝐼2

𝑽𝑽𝟏𝟏 + 𝑽𝑽𝟔𝟔 = 𝑹𝑹𝑰𝑰𝟏𝟏 + 𝑹𝑹𝑰𝑰𝟔𝟔 = 𝐑𝐑(𝐈𝐈𝟏𝟏 + 𝐈𝐈𝟔𝟔)

𝐼𝐼2 = 𝑘𝑘𝐼𝐼1

𝑽𝑽𝟔𝟔 = 𝑹𝑹𝑰𝑰𝟔𝟔 = 𝑹𝑹𝑹𝑹𝑰𝑰𝟏𝟏 = 𝐤𝐤𝐤𝐤𝟏𝟏

Ex: Now let us consider an element represented by the relationship between current and voltage:

𝑉𝑉 = 𝐼𝐼2

The response to a current I_1 The response to a current I_2 The sum of these responses is Because the sum of the responses to i_1 and i_2 is not equal to the response to i_1 + i_2, the principle of superposition is not satisfied. we have for an excitation

𝑉𝑉1 = 𝐼𝐼12 𝑉𝑉2 = 𝐼𝐼22

𝑽𝑽𝟏𝟏 + 𝑽𝑽𝟔𝟔 = 𝑰𝑰𝟏𝟏𝟔𝟔 + 𝑰𝑰𝟔𝟔𝟔𝟔 ≠ 𝐈𝐈𝟏𝟏 + 𝐈𝐈𝟔𝟔 𝟔𝟔

𝐼𝐼2 = 𝑘𝑘𝐼𝐼1

𝑽𝑽𝟔𝟔 = 𝑰𝑰𝟔𝟔𝟔𝟔 = 𝑹𝑹𝟔𝟔𝑰𝑰𝟏𝟏𝟔𝟔 ≠ 𝐤𝐤𝐤𝐤𝟏𝟏


Circuit Elements Active and passive circuit elements A passive circuit element absorbs energy (including zero energy), e.g. resistor. Current flows into the +ve terminal and out of the -ve terminal An active element generates or supplies energy, e.g. battery, generator. Current flows into the –ve terminal and out of the +ve terminal.

Example for systems: Resistor satisfies v = Ri, so it is linear; A diode having a response v=i2 is nonlinear; A rectifier having a response v=iis nonlinear.


Circuit Elements

Resistors Resistance is the physical property of an element or device that impedes the flow of current; it is represented by the symbol R.


Circuit Elements Resistors Georg Simon Ohm was able to show that the current in a circuit composed of a battery and a conducting wire of uniform cross-section could be expressed as

𝑖𝑖 =𝐴𝐴𝐴𝐴𝜌𝜌𝜌𝜌

where A is the cross-sectional area, 𝜌𝜌 the resistivity, L the length, and v the voltage across the wire element.

𝑅𝑅 =𝜌𝜌𝜌𝜌𝐴𝐴

𝑉𝑉 = 𝐼𝐼𝑅𝑅

Resistance R, unit ohm Ω (capital omega), where 1 Ω = 1 V/A. The resistance of a 10m length of common TV cable is 2 mΩ.

Ohm’s Law


Circuit Elements Resistors • Resistance is proportional to length (wire of the same material and same cross-

section area) • Resistance is inversely proportional to A

Example: A coil is wound from a 10m length of copper wire having a cross-sectional area of 1.0 mm2. Calculate the resistance of the coil.

𝑅𝑅 =𝜌𝜌𝜌𝜌𝐴𝐴 =

1.59 × 10−8 × 101 × 10−6 = 0.159Ω


Temperature coefficient of resistance For most building temperature ranges of interest, say from 0 °C -100 °C, the change of resistance is usually proportional to the change of temperature. The ratio of the change of resistance per degree change of temperature to the resistance at some standard temperature is called the temperature coefficient of resistance, α.

𝑅𝑅𝑇𝑇 = 𝑅𝑅0(1 + 𝛼𝛼0 𝑇𝑇 − 𝑇𝑇0 ) where 𝛼𝛼0= the temperature coefficient of resistance at temperature T0 𝑅𝑅0 = resistance at temperature T0 𝑅𝑅𝑇𝑇 = resistance at temperature T

𝑅𝑅1𝑅𝑅2

=1 + 𝛼𝛼0(𝑇𝑇1 − 𝑇𝑇0)1 + 𝛼𝛼0(𝑇𝑇2 − 𝑇𝑇0)

Circuit Elements


Temperature coefficient of resistance

Circuit Elements

Example: A coil of copper wire has a resistance of 200 Ω when its mean temperature is 0 . Calculate the resistance of the coil when its mean temperature is 80 .

𝑅𝑅1 = 𝑅𝑅0 1 + 𝛼𝛼0𝜃𝜃1 = 200 1 + 0.00428 × 80 = 268.5Ω

• Most metals have a positive temperature coefficient of resistance α >0, which means their R↑ when T↑. (What is the effect of two lamps connected in series if their temperatures have risen to 2500 °C from room temperature?)

• Some alloys can be designed to have α = 0 for a certain temperature range. These are useful for measuring instruments.

• Some materials such as carbon have a negative temperature coefficient of resistance, i.e. their resistances R↓ when T↑. When semiconductors are connected to metals, this may give rise to thermal runaways.


Temperature coefficient of resistance

Circuit Elements

Example: When a voltage of 10V is applied to a coil of copper wire of mean temperature 20°C, a current of 1 A flows in the coil. After some time the current falls to 0.95 A yet the supply voltage remains unaltered. Determine the mean temperature of the coil if the temperature coefficient of resistance of copper is 4.28 x 10-3/°C referred to 0 °C. At 20°C

𝑅𝑅1 =𝑉𝑉1𝐼𝐼1

=101

= 10.0Ω

At T

𝑅𝑅2 =𝑉𝑉2𝐼𝐼2

=10

0.95= 10.53Ω

𝑅𝑅1𝑅𝑅2

=1 + 𝛼𝛼1𝑇𝑇11 + 𝛼𝛼2𝑇𝑇2

=10.0

10.53=

1 + 0.00428 × 201 + 0.00428 × 𝑇𝑇

Where 𝑇𝑇 = 33.4

We can use resistance to measure temperature. For instance, if the resistance of a coil, such as a field winding of an electrical machine, is measured at the beginning and at the end of a test, the mean temperature rise of the whole coil can be calculated.


Circuit Elements

Power delivered from a resistor

𝑃𝑃 = 𝑉𝑉𝐼𝐼 = 𝑉𝑉𝑉𝑉𝑅𝑅 =

𝑉𝑉2

𝑅𝑅 𝑃𝑃 = 𝑉𝑉𝐼𝐼 = 𝐼𝐼𝑅𝑅𝐼𝐼 = 𝐼𝐼2𝑅𝑅

Energy delivered from a resistor

𝑊𝑊 = 𝑃𝑃𝑡𝑡 = 𝑉𝑉𝐼𝐼𝑡𝑡 = 𝐼𝐼2𝑅𝑅𝑡𝑡

*Note for AC:

𝑊𝑊 = 𝑃𝑃𝑑𝑑𝑃𝑃𝑡𝑡

−∞= 𝐼𝐼2𝑅𝑅𝑑𝑑𝑃𝑃

𝑡𝑡

−∞


Circuit Elements

Independent sources A source is a voltage or current generator capable of supplying energy to a circuit. An independent source is a voltage or current generator not dependent on other circuit variables. Example: 9-volt battery, or a current generator i(t)=200 sin 30t mA.

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. (in contrast, a resistor is a circuit element in which the voltage of the element is proportional to the current in the same element.) Dependent sources are used to model electronic devices such as transistors and amplifiers. For example, the output voltage of an amplifier is proportional to the input voltage of that amplifier, so an amplifier can be modeled as a dependent source.


Circuit Elements

Transducer Transducers are devices that convert physical quantities to electrical quantities. This section describes two transducers: potentiometers and temperature sensors. Potentiometers convert position to resistance, and temperature sensors convert temperature to current.

Potentiometer Temperature sensors


Amplifier Amplifier is a electronic device can arise low-level electrical signals to higher electrical signal frequency.

Circuit Elements


Circuit Elements

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.

(a) single-pole, single-throw (SPST) switch (b) single-pole, double-throw (SPDT) switch This SPDT switch acts like two SPST switches, one between terminals c and a, another between terminals c and b. Before t = 0 s, the switch between c and a is closed and the switch between c and b is open. At t = 0 s both switches change state; that is, the switch between a and c opens, and the switch between c and b closes.


Circuit Elements Example: What is the value of the current I in figure at time t = 4s? What is the value of the current I at t=2s? Answer: i=0 when t=4s; i=2 mA when t=2s. What is the value of the voltage v in figure at time t=4s? ant 6s? Answer: v=6 volts at t=4s; and v=0 volts at t= 6s


Kirchhoff Laws

Nodes (Junctions) and Loop An electric circuit consists of circuit elements that are connected together. The places where the elements are connected to each other are called nodes (a.k.a. junctions). The same circuit can be drawn in several ways. One drawing of a circuit might look much different from another drawing of the same circuit.

Nodes


Kirchhoff Laws Same Circuit More formally, we say that circuit drawings A and B represent the same circuit when the following three conditions are met. 1. There is a one-to-one correspondence between the nodes of drawing A and the nodes of

drawing B. (A one-to-one correspondence is a matching. In this one-to-one correspondence, each node in drawing A is matched to exactly one node of drawing B and vice versa. The position of the nodes is not important.)

2. There is a one-to-one correspondence between the elements of drawing A and the elements of drawing B.

3. Corresponding elements are connected to corresponding nodes.


Kirchhoff Laws

Answer: (b) and (c) are the same circuit; (a) and (d) are not.

Same Circuit


Kirchhoff Laws

We must take reference directions into account as we add up the currents of elements connected to a particular node. Use the convention: use a + sign when the current is directed away from the node and a - sign when the current is directed toward the node.

Kirchhoff's current law (KCL):

The algebraic sum of the currents into a node at any instant is zero

𝑖𝑖𝑛𝑛

𝑁𝑁

𝑛𝑛=1

= 0


Kirchhoff Laws

Known 𝐼𝐼0 = −2 𝐴𝐴, 𝐼𝐼1 = −4 𝐴𝐴, 𝐼𝐼𝑆𝑆 = −8𝐴𝐴

Example: Find the unknown currents.


Kirchhoff's voltage law (KVL):

The algebraic sum of the voltages around any loop in a circuit is identically zero for all time

Kirchhoff Laws

We must take polarity into account as we add up the voltages of elements that comprise a loop. Use the convention: move around the loop in the clockwise direction and write the voltage with a plus sign when we encounter the + of the voltage polarity before the -. Conversely, write the voltage with a minus sign when we encounter the – of the voltage polarity before the +.


Kirchhoff Laws Example: Consider the circuit shown in figure. Determine the power supplied by element C and the power received by element D. Apply Kirchhoff’s voltage law (KVL) to the loop consisting of elements C, D, and B to get −𝐴𝐴 − −4 − 6 = 0 → 𝐴𝐴 = −2 𝑉𝑉 The value of the current in element C is 7 A. The voltage and current of element C given adhere to the passive convention, so 𝑃𝑃𝑐𝑐 = 𝐴𝐴 7 = −14𝑊𝑊 Next, apply Kirchhoff’s current law (KCL) at node b to get 7 + −10 + 𝑖𝑖 = 0 → 𝑖𝑖 = 3𝐴𝐴 The value of the voltage across element D is -4V. The voltage and current of element D given in figure do not adhere to the passive convention, so the power supplied by element D is given by 𝑃𝑃𝐷𝐷 = −4 𝑖𝑖 = −12𝑊𝑊


Kirchhoff Laws Example:


Kirchhoff Laws

Voltage division

Resistors connected in series will act as a voltage divider.


Voltage division

Previous circuit demonstrates the principle of voltage division. Resistors act as a voltage divider. In General, we will have

𝑉𝑉𝑛𝑛 =𝑅𝑅𝑛𝑛

𝑅𝑅1 + 𝑅𝑅2 + ⋯+ 𝑅𝑅𝑁𝑁𝑉𝑉𝑠𝑠

Kirchhoff Laws


Current division Resistors connector in parallel will divide the current.

Kirchhoff Laws

𝑖𝑖𝑠𝑠 = 𝑖𝑖1 + 𝑖𝑖2 𝑖𝑖1 = 𝐴𝐴/𝑅𝑅1 and 𝑖𝑖2 = 𝐴𝐴/𝑅𝑅2 𝑖𝑖𝑠𝑠 =

𝐴𝐴𝑅𝑅1

+𝐴𝐴𝑅𝑅2

= 𝐺𝐺1𝐴𝐴 + 𝐺𝐺2𝐴𝐴 = 𝐺𝐺1 + 𝐺𝐺2 𝐴𝐴

𝑖𝑖1 =𝐺𝐺1𝑖𝑖𝑠𝑠

𝐺𝐺1 + 𝐺𝐺2

𝑖𝑖2 =𝐺𝐺2𝑖𝑖𝑠𝑠

𝐺𝐺1 + 𝐺𝐺2

Where 𝐺𝐺 = 1𝑅𝑅


Kirchhoff Laws

Current division Resistors connector in parallel will divide the current. In general,

𝑖𝑖𝑠𝑠 = 𝑖𝑖1 + 𝑖𝑖2 + 𝑖𝑖3 + ⋯+ 𝑖𝑖𝑁𝑁 𝑖𝑖𝑛𝑛 = 𝐺𝐺𝑛𝑛𝐴𝐴

𝑖𝑖𝑠𝑠 = 𝐺𝐺1 + 𝐺𝐺2 + 𝐺𝐺3 + ⋯+ 𝐺𝐺𝑁𝑁 𝑖𝑖𝑠𝑠 = 𝐴𝐴 ∑ 𝐺𝐺𝑁𝑁𝑁𝑁

𝑛𝑛=1

𝑖𝑖𝑛𝑛 =𝐺𝐺𝑛𝑛𝑖𝑖𝑠𝑠

∑ 𝐺𝐺𝑛𝑛𝑁𝑁𝑛𝑛=1

=𝐺𝐺𝑛𝑛𝑖𝑖𝑠𝑠𝐺𝐺𝑝𝑝

𝐺𝐺𝑝𝑝 =1𝑅𝑅𝑝𝑝

= 1𝑅𝑅𝑛𝑛

𝑁𝑁

𝑛𝑛=1


Kirchhoff Laws Equivalent circuits Two circuits are equivalent if they exhibit identical characteristics at the same two terminals. • We can reduce a circuit to an equivalent circuit by replacing a network of resistors

with an equivalent resistance. • In general, we may find the equivalent resistance (or conductance) for a portion of

a circuit consisting only of resistors and then replace that portion of the circuit with the equivalent resistance.


Kirchhoff Laws Sample Equivalent Circuit for Series and Parallel elements

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