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Chapter 1: Circuit Variables
Objectives
o Understand the use of circuit schematics in circuit modeling
BJT Circuit
o Understand basic concepts of voltage and current
o Understand sign conventions in voltage and current
o Be able to do dc power calculations and correctly interpret signs
Homework/Quiz/Exam Prep
o Units and labeling; homework format
o Math requirements
Trig functions: sin, cos
Equations of a straight line
Presentation
o Present BJT schematic and introduce
Modeling concept
Labeling of voltage and current
Intro to reference polarities
Activity: students label their own diagrams and exchange with neighbors to check
their work
o Definitions of voltage and current
A simple circuit
Water analogy
o Ideal Circuit Elements: the “box”
Reference vs. actual polarities
Direction of charge flow
Voltage drop and voltage rise
o Power and Energy
Delivered, absorbed
Relationship to charge flow
o Active and Passive Sign Conventions
Activity: simple source/resistor circuit with given current; find power delivered or
absorbed by source
Chapter 1: Circuit Variables
1.1 Electrical Engineering: An Overview
Electrical Engineers are concerned with the design, analysis, and operation of systems involving electrical signals. Examples:
Communications/signal engineering Computer systems Control systems/robotics Power systems Microelectronics
Theoretical Basis for Electrical Engineering: Electromagnetics and Maxwell’s Equations. We will not deal with this topic here. Instead we will talk about a specialization of electromagnetics…
Circuit Theory is an important special case of electromagnetics. It is appropriate where the spatial dimensions of the electrical system are small compared with the wavelength of an electromagnetic signal, i.e., where the shape of the circuit does not matter.
1.2 Units
We will use the International System of units:
Length: meters [m] Mass: kilogram [kg] Time: second [s] Current: Ampere [A] Temperature: Kelvin [K]
Also: 1 Ampere is the current that, maintained in two straight, parallel, infinite conductors of negligible circular cross section placed 1 m apart in vacuum, would produce a force between the conductors of 2 x 10-7^ [N/m].
The Coulomb is a unit of charge derived from the Ampere; more below.
1.3 Circuit Analysis: An Overview
Idea: We want to be able to make quantitative predictions about electrical circuit behavior. We do this using idealized connections among circuit elements called circuit models.
Formal Definitions
Potential and Voltage : To move a hypothetical positive test charge from point A to point B
in a region where electric forces are present requires work (energy). The work required is the difference in electric potential energy between point A and point B. Electric potential is the work per unit charge required to move the test charge. It is defined in the limit of a vanishingly small test charge (i.e., as q becomes 0):
BA B A AB^0
W
v v v q
q
Here, v B - v A is the difference in potential , or voltage v BA, between A and B; WAB is the work done in moving the charge from A to B; q is the charge in Coulombs. In differential form,
. dq
dw v
Units: 1 Volt = 1 Joule/Coulomb
1 Joule = 1 [kg m^2 /s^2 ] 1 Coulomb = 1 [Ampere]. 1 [s].
Important : voltage , like potential energy, is defined as a difference in potential. It is not a force, and it is not energy.
Simple Example Application
Imagine a 9 [V] battery: If I move a positive test charge +q from terminal A to terminal B, I need to do positive work on the test charge. (Basic electrostatics tells me that the test charge doesn’t want to be at the positive battery terminal because like charges repel, so I will have to push it there, which means I am doing work on it.) So in this case, the work done in moving from A to B, i.e., WAB , is positive.
If WAB is positive, then by the equation above, vBA is positive. This is as it should be: vBA = + 9 [V] for our battery.
+q
Ever-
Hardy 9
Volt
B A
Current: When charge flows in a conducting material, a current exists. Current is the defined
as the rate at which charge moves past an imaginary plane in a device, or in a wire. Formally,
dt
dq i
In this equation, i is the current; q is the charge; t is time.
Units:
1 Ampere = 1 Coulomb/second
We will not be concerned with the details of how current flows. We usually think of it as a flow of electrons in a wire, and that will be good enough for us here.
A simple circuit
The battery on the right provides a voltage of 9 [V], with “positive” and “negative” terminals as indicated. Chemical forces in the battery maintain an electric potential difference between the terminals. We can think of the terminals as being analogous to “up” and “down” in the water tower.
The battery doesn’t “do” anything until we include it in a complete circuit. Example:
What’s going on here? When wires and a light bulb are connected to the battery, the battery voltage causes a current to flow. The current is causing the light to glow. A resistor has been inserted since we may need to limit the current so the bulb doesn’t burn up!
Note that we need a connection to and from the bulb for the circuit to work; that is, we need a complete path for current to flow.
How does this work? We can go back to our Physics text to find out that the battery exerts a force on the charges in the wire and causes them to move, provided the circuit is complete. We do not need to know about these forces to do circuit analysis. Instead of forces, we talk about voltage.
Ever- Hardy 9 Volt
Ever- Hardy 9 Volt
Resistor
1.5 The Ideal Basic Circuit Elements
We will use a box like the one below to represent several things. It can be just one or perhaps many electrical components, or even an entire circuit or electrical system. For now, we assume it represents an ideal basic circuit element. Ideal basic circuit elements are the building blocks of circuit models.
The ideal basic circuit elements are:
Voltage source Resistor Inductor Capacitor Inductor
We can model any circuit, no matter how complex, with a combination of these basic circuit elements.
Ideal We are calling these things “ideal” circuit elements. This means that we should not expect real voltage sources, a battery for example, to act like ideal voltage sources. The same is true for the other elements. But as we will see, we can model non-ideal behavior using these basic circuit elements in combination.
Properties of the ideal basic circuit element:
It has two terminals (labeled “1” and “2”). It can be described mathematically in terms of a voltage v and/or a current i. It cannot be subdivided into other circuit elements, hence it is basic. It is ideal in the sense that it has idealized properties that do not necessarily hold for real circuit elements.
vE E
1^ iE
2
The box contains a basic circuit element ‘E’. It is connected to something at terminals 1 and 2. What it’s connected to is not shown and for now doesn’t matter.
Labeling Voltage: Reference and Actual Polarities
Because voltage v E across a circuit element is a potential difference , one side of the circuit element has a higher potential than the other. We need a way to indicate this. But : we won’t always know which side is the higher potential until we calculate or measure the voltage. So we also need a way to handle that issue.
The ‘+’ and ‘-‘ associated with v E in the boxes below indicate the reference polarity for the voltage. These are analogous to labels on an x-y graph, where arrows indicate “positive x” and “positive y”. The first box below is labeled as if the higher potential is at terminal 1. We can assign that polarity without knowing whether terminal 1 is in fact the higher potential; in other words, without knowing the actual polarity. The voltage across the first box may in fact be more positive at terminal 2. To handle this, we need to know both a magnitude and a sign (positive or negative) for v E. Both of these are needed if we want to know the actual potential difference between terminals 1 and 2.
Consider the circuit element to the right. My reference polarity, v E, shows the higher potential at terminal 1, but I don’t yet know the actual polarity.
a) If I measure v E and find that it is +5 [V], then terminal 1 is 5 [V] higher in potential than terminal 2. The actual polarity is the same as the reference polarity. b) If I measure v E and find that it is -5 [V], then terminal 2 is 5 [V] higher in potential than terminal 1. The actual polarity is opposite to the reference polarity.
If I like, I can change the reference polarity….
In this case, the measurements I made above will give the opposite sign. If terminal 1 is 5 [V] higher in potential than terminal 2, v E will be negative. If terminal 2 is higher in potential than terminal 1, v E will be positive.
Bottom line: The ‘+’ and ‘-‘ signs tell us the reference polarity. We don’t know the actual polarity until we are measure (or calculate, or we are given) the sign of v E. Often we will have to label a voltage before we know what the actual polarity is.
But…how do I “measure” v E?? To measure v E, I put the red lead of my voltmeter at the positive terminal of v E, and the black lead at the negative terminal.
Notation: Use a lower case ‘v’ with a subscript ( v E) to indicate a voltage. Always label the
polarity with + and -.
vE
-
+
1
2
1
2
vE
+
-
Labeling Current: Reference and Actual Polarities
We can think about the same kind of thing for current. Current flowing through a circuit element, like the box at the right, may be flowing from top to bottom (terminal 1 to terminal 2), or the other way around.
The arrow shown on the box to the right indicates the reference current polarity ( or reference current direction). It assumes that current is entering terminal 1 and leaving terminal 2.
a) If I measure i E and find that it is +30 [mA], then 30 [mA] of current is entering terminal 1 and leaving terminal 2. The actual current direction and the reference current direction are the same. b) If I measure i E and find that it is -30 [mA], then 30 [mA] of current is entering terminal 2 and leaving terminal 1. The actual and reference current directions are now opposite to each other.
If I change the reference direction, then the measurements above will give the opposite signs.
If current is entering terminal 1, i E will be negative; if it is leaving terminal 1, i E will be positive.
Bottom Line: The arrow indicates a reference current direction. We don’t know the actual current direction until we know the sign of i E, which we get by measurement or calculation. We will have to label currents even though we don’t know which way they are going. These labels will need to include reference directions.
But…how do I “measure” i E?? To measure i E, I put the red lead at the tail of the arrow, and the black lead at the head of the arrow.
Notation: Use a lower case ‘i’ with a subscript ( i E) to indicate a current. Always label the polarity (direction) with an arrow.
1 iE
2
1 iE
2
1.6 Power and Energy
Usually, we will be talking about electrical energy, as opposed to sound or light energy. Some important ideas:
Electrical energy can be either delivered or absorbed. For example: o A glowing flashlight bulb is absorbing electrical energy. (It is giving off (delivering) light and heat energy.) o A flashlight battery is delivering energy to the bulb (assuming the flashlight is on). o Rechargeable batteries can be re-charged, during which time the battery is absorbing electrical energy.
Power is the rate at which energy is delivered or absorbed. That is,
dt
dw p
Units: w is the energy in Joules, t is the time in seconds, and p is the power in Watts.
Since electrical energy can be delivered or absorbed, electrical power can be delivered or absorbed. It is very important that we keep track of whether a circuit element is delivering or absorbing power and energy.
We can relate electrical power to voltage and current.
vi dt
dq
dq
dw dt
dw p .
Power delivered vs. power absorbed
Electrical power is obtained by multiplying voltage and current. But how can we tell whether power is being delivered or absorbed? And what if the voltage is negative or the current is negative? To get all of that right, we need a rule for signs.
To illustrate, we will calculate the power absorbed by our circuit element. Here is the rule: In the diagram on the left, current is entering the positive terminal and leaving the negative terminal. So for the thing inside the box , the current is in the direction of the voltage drop. In that case, we write pabs,E = vE.iE. Here “pabs,E” means “the power being absorbed by element E”.
For the diagram on the right, where the current is in the direction of the voltage rise (for the thing
inside the box), we write pabs,E = - vE.iE. We need to use the appropriate sign, which we get by
looking at the diagram, and at which way the current is going relative to the voltage drop.
Some examples:
For the case on the left above:
If v E = 3 [V] and i E = 25 [mA], then p abs.E = v E. i E = (3) (0.025) = 0.075 [W] = 75 [mW]. If v E = -2 [V] and i E = 10 [mA], then pabs.E = v E. i E = (-2) (0.010) = - 20 [mW].
For the case on the right:
If v E = 2 [V] and i E = - 250 [mA], pabs,E = - v E. i E = - (2) (-0.250) = 0.5 [W]. If v E = - 10 [V] and i E = - 35 [mA], pabs = - v E. i E = - (-10) (-0.035) = - 350 [mW].
But wait: what does it mean that the absorbed power is negative? We interpret a negative absorbed power to mean that power is in fact being delivered , not absorbed. We could also have calculated delivered power as p del = - p abs.
vE E
-
+
iE
E
iE
vE
pabs , E vEiE abs E E E
p , v i
Summary
To calculate power…
Choose whether to calculate absorbed power or delivered power; this is your choice, even if you have no idea which it “actually” is. Then, use the appropriate formula, which is based on a diagram showing reference current direction and reference voltage polarity.
Plug voltage and current into the appropriate formula, including the signs of v and i.
Decide whether power is actually being absorbed or delivered based on the sign of the result.
Example
Given: v E = - 12 [V]; i E = - 0.2 [A].
Problem: Calculate the power absorbed by the circuit element E.
Solution: At the box, the current is in the direction of the voltage rise, so we write
p abs,E = - v E. i E = - (-12) (- 0.2) = - 2.4 [W].
So power is in fact being delivered. Thus, the delivered power is 2.4 [W]; the absorbed power is -2.4 [W].
vE E
-
+
iE
Sign Relationships: Active and Passive Sign Convention
We finish with a definition. Whether or not the reference direction for the current is in the direction of the voltage drop is an issue that will come up in future discussions, so we need a way to refer to it.
Passive Sign Convention: When the reference current direction is in the direction of the reference voltage drop, as it is in the figure to the right, we say that we are using the passive sign convention.
Active Sign Convention: The alternative is a situation in which the reference current direction is in the direction of the reference voltage rise, as it is n the figure to the left. This is the active sign convention.
The passive and active sign conventions provide names for dealing with these two cases.
vE
-
+
iE
i E
v E +
-
