Fundamentals

SI Units

• Many units will be introduced over the course of the semester.

• Some of the units used in this text are in Table 1.

• SI prefixes will be used and are considered part of your answer.

• See NIST [2010] for more information.

Table 1 Basic Units

Unit	Abbreviation	Variable
Coulomb	C	\(Q,q\)
Ampere	A	\(I,i\)
Volt	V	\(V,v\)
Ohm	\(\Omega\)	\(R,r,Z,z,X,x\)

Engineering Notation

• Engineering notation similar to scientific notation but uses only multiples of three in the exponent.

• Multiples of three follow our natural language for large numbers (Thousands, millions, billions, etc.) and small (thousandth, millionth, billionth, etc.)

Table 2 SI Prefixes

Factor	Name	Symbol
\(10^{15}\)	peta	P
\(10^{12}\)	tera	T
\(10^{9}\)	giga	G
\(10^{6}\)	mega	M
\(10^{3}\)	kilo	k
\(10^{0}\)	\(\cdot\)	\(\cdot\)
\(10^{-3}\)	milli	m
\(10^{-6}\)	micro	\(\mu\)
\(10^{-9}\)	nano	n
\(10^{-12}\)	pico	p
\(10^{-15}\)	femto	f

Constant vs. Time-variant Notation

• Constant values use capital variables.

• Time-variant values use lower-case variables.

Charge

• Fundamental focus of studying electronics

• Charge is measured in Coulombs

• Charge of a single electron

\[1\ \text{electron}=1.602\mathrm{e}{-19} C\]

• It takes a lot of electrons to get one Coulomb

• You’re likely to see smaller quantities of charge indicated by an SI prefix: \(pC\), \(nC\), \(\mu C\)

• Variable used is \(Q\) or \(q\)

Current

• In circuit analysis we are most concerned with the movement of charge a.k.a. current

• Current is defined as

\[i=\frac{dq}{dt} \ \text{or} \ I=\frac{\Delta Q}{\Delta t}\]

• Given a function, \(q(t)\), we can now find a function, \(i(t)\).

• Alternately state

\[Q=\int_{t_0}^ti(\tau)d\tau+Q_0\]

Conventional vs. Electron Current

• Electron flow - direction of negative charge (i.e. the charge on electrons)

• Conventional flow - direction of positive charge, opposite the movement of electrons. Positive charge in this case is the absence of an electron, a.k.a. a “hole”.

Example

What is the current through a conductor if \(6.24e18\) electrons pass through a cross section in 1 second?

Solution

\[6.24e18\ \text{electrons}\times\frac{1.602e{-19} C}{1\ \text{electron}} \times \frac{1}{s}=1 \frac{C}{s}=1 A\]

Current Sources in Schematics

We’ll use two types of current sources. The arrow in both points in the direction of conventional current flow.

Independent Current Source:

Has a constant value. An example is shown here.

Fig. 1 Independent Current Source.

Dependent Current Source:

Has a value dependent on either a current or voltage measured elsewhere in the circuit. An example is shown here.

Fig. 2 Dependent Current Source.

Voltage

• It takes work to move charge, we must talk about energy

• Voltage is a measure of potential energy in an electrical circuit

• How we measure potential energy in physics?

\[ E=mgh \]

• We must have two points between which to measure the height of the object, \(h\).

• Similarly, we need two points in the circuit \underline{across} which to measure voltage.

• Voltage tells us how much work is performed in order to move a quantity of charge

  \[v=\frac{dw}{dq}\]

Voltage

The amount of energy needed to move a quantity of charge.

Ground

If voltage is potential energy we often choose a point in the circuit to have 0 V. This is an equivalent concept to measuring potential energy with respect to the ‘ground’. Mechanical engineers might call this the datum line.

There are more than one type of ground. Here are some common symbols.

Fig. 3 Different types of ground.

Earth Ground

Earth ground is the potential of the earth (ie the dirt you’re standing on). Connection to this potential is typically made by driving a rod into the ground or connecting to buried, conductive piping.

This symbol should only be used when a similar connection to the earth is made. It should not be used as the general symbol for ground. It should definitely not be used on circuits that fly in helicopters or swim in submarines.

Signal Ground

This is the closest we have to a general symbol for ground. The only thing that this symbol designates is the point(s) in the circuit that we will consider to be at \(0 V\) potential.

Chassis Ground

At times the ground of a circuit will be connected to the chassis of the device. This potentially accomplishes two things. First, it can prevent the circuit from shocking anybody that makes contact with the chassis. Second, it can provide shielding from electromagnetic interference. Again, this symbol should only be used when a similar connection to the chassis is made.

Voltage Sources in Schematics

There are many types of voltage sources we’ll see in this class.

Independent Voltage Sources:

Has a constant value. Battery (shown on left) has a positive voltage on the side with the longer bar than the side with the short bar.

A generic voltage source is shown (on the right). The plus/minus indicate which side has the higher voltage.

Fig. 4 A battery and an generic independent voltage source.

Dependent Voltage Source:

Has a value dependent on either a current or voltage measured elsewhere in the circuit. An example is shown here. Again, the plus/minus indicate which side has the higher voltage.

Fig. 5 A dependent voltage source.

Labeled Voltages

Larger schematics will often use voltage labels to indicate a voltage at a point on the circuit. These voltages are always in reference to the circuit ground. Here are some examples:

Fig. 6 Different voltages labeled.

In all cases these supplies can be replaced by a voltage supply with the negative side connected to ground and the positive to the labeled point in the circuit. The value of that supply then matches the labeled value. For instance the circuit on the left can be redrawn as the equivalent circuit on the right:

Fig. 7 Different voltages labeled.

Resistance

The first of the passive elements we will study.

All materials have a value of resistance. Some are good conductors (low resistance), others are good insulators (high resistance), and others are somewhere in between. Resistance is measured in Ohms (\(\Omega\)), is denoted by the variable \(R\).

From Physical Properties

Resistance can be calculated with knowledge of physical properties as depicted in Fig. 8.

Fig. 8 Cross-section of a resistive element.

The properties include:

1. Length, \(l\), measure in meters (\(m\)).

2. Cross-sectional area, \(A\), measure in meters-squared (\(m^2\)).

3. Resistivity, \(\rho\), a property of the material measured in ohms times meters (\(\Omega\cdot\)m).

Using values for each the resistance, \(R\), is calculated as

\[R=\frac{\rho\cdot l}{A}\]

Table 3 Some material properties

Material	Resistivity	Temperature Coefficient per \(^\circ\)C,
Name	\(\rho\) (\(\Omega\cdot m\))	\(\alpha\)
Silver	1.59e-8	0.0061
Copper	1.68e-8	0.0068
Aluminum	2.65e-8	0.00429
Tungsten	5.6e-8	0.0045
Iron	9.71e-8	0.00651
Platinum	10.6e-8	0.003927
Manganin	48.2e-8	0.000002
Lead	22e-8	0.000002
Mercury	98e-8	0.0009
Nichrome	100e-8	0.0004
Constantan	49e-8	0.0004
Carbon/Graphite	3-60e-5	-0.0005
Germanium	1-500e-3	-0.05
Silicon	0.1-60e0	-0.07
Glass	1-10000e9	-0.07
Hard Rubber	1-100e13	-0.07
Quartz	7.5e17	-0.07

Conductance

Conductance is another means of characterizing an element’s ability to pass charge. It is the mathematical inverse of resistance: \( G=\frac{1}{R} \) It is measured in Siemens (S) and denoted by the variable \(G\).

Effect of Temperature

Thermal effects on electronics is a common issue. Resistors change value as temperature varies according to

\[ R=R\textss{ref}\left[1+\alpha(T-T\textss{ref})\right] \]

where,

• \(R\) - Resistance(\Omega) at temperature given by \(T\)

• \(R\textss{ref}\) - Resistance at a reference temperature given by \(T\textss{ref}\) (\(\Omega\))

• \(\alpha\) - Temperature coefficient (1/\(^\circ\)C or )

• T - Current temperature (\(^\circ\)C)

• \(T\textss{ref}\) - Reference temperature (\(^\circ\)C)

Example

A carbon resistor has a nominal value of 1.2 k\(\Omega\) at room temperature (21 \(\Deg\) C). Find the range of resistance as the temperature varies from -40 \(\Deg\) C to 90\(\Deg\) C

Solution

For \(T=-40 \Deg\) C,

\(R=1.2 k\Omega\left[1+(-0.0005)(-40 ^\circ C-21 ^\circ C)\right]=1.237 k\Omega\)

For \(T=90 \Deg\) C,

\(R=1.2 k\Omega\left[1+(-0.0005)(90 ^\circ C-21 ^\circ C)\right]=1.159 k\Omega\)

Alternate unit for temperature coefficient

It is common to find the temperature coefficient specified in parts per million per degree Celsius (ppm/\(\Deg\) C).
A similar formula is used as above though the coefficient has an additional factor of 1 million.

\[ R=R\textss{ref}\left[1+\frac{\alpha}{1e6}(T-T\textss{ref})\right] \]

Ohm’s Law

• Three laws form the basis of circuit analysis.

• The first we will learn is a law that relates the voltage across a resistor to the current flowing through that resistor.

Fig. 9 Explanation of Ohm’s law.

\[v=iR\]

Example

Find the magnitude of the current through the resistor and label the direction of the current.

Fig. 10 Example of Ohm’s law.

Solution

The voltage across the resistor can be found as \(8 \text{V}-(-2 \text{V})=10 \text{V}\). The current flows from the higher potential to the lower potential, or down as pictured below. The magnitude can be found by rearranging Ohm’s law

\[ I=\frac{V}{R}=\frac{10 \text{V}}{1 k\Omega}=10 \text{mA} \]

Fig. 11 Solution for current via Ohm’s law.

Example

Find the resistance of the element below:

Fig. 12 Find the resistance.

Solution

The answer can be found by rearranging Ohm’s law

\[ R=\frac{V}{I}=\frac{12 \text{V}}{8 \mu\text{A}}=1.5 M\Omega \]

Power

Recall from physics that power is work divided by time

\[p=\frac{dw}{dt}\]

Additionally, from above voltage is

\[v=\frac{dw}{dq}\]

and current is

\[i=\frac{dq}{dt}\]

Multiplying voltage and current results in

\[p=\frac{dw}{dt}=\frac{dw}{dq}\frac{dq}{dt}=vi\]

Passive Sign Convention

Note in the diagram

Fig. 13 Signs matter!

that the signs matter: in a passive element (like this resistor) current flows from the higher potential (marked with \(+\)) to the lower potential (marked with \(-\)) just like objects fall from a higher potential energy to a lower potential energy.

Power

Always answers the question: “How much power is begin dissipated?” Negative sign tells you whether it is supplied or absorbed.

Conservation of power

Example

Demonstrate that the circuit obeys conservation of power.

Fig. 14 Show conservation of power.

Solution

The trick here is to be completely sure about applying the passive sign convention so that it’s clear when an item is generating or dissipating power; this determines whether the power dissipated is positive or negative.

First, find the voltages at each node. The center node is at \(12V\). The node to the left is \(4V\) above that at \(16V\). Because the right resistor has a negative voltage, the right node is \(3V\) above the center node at \(15V\).

Table 4 Conservation of power calculations.

Circuit Element	Power Calculation	Power Dissipated
\(3A\) supply	\(-3 A \times 16V\)	\(-48W\)
\(2A\) Supply	\(-2A \times -1V\)	\(2W\)
\(-1.5A\) Supply	\(1.5A \times 15V\)	\(22.5W\)
\(4V\) Resistor	\(1A \times 4V\)	\(4W\)
\(12V\) Resistor	\(1.5A \times 12V\)	\(18W\)
\(-3V\) Resistor	\(-0.5A \times -3V\)	\(1.5W\)

Summing the Power Dissipated column shows that the sum is zero, so power is conserved.

Stupid Units of Energy

For SI units (NIST [2010]) energy is measured in joules (J) and power is measured in joules per second (J/s). One joule per second is equal to one watt.

Because of the usual scale of usage, the power company tends to report power in kilowatts (kW) or megawatts (MW). The power company uses a measure of energy that is easier to relate to residential or commercial power usage: multiplying the power in kW by the time of the usage in hours to get kilowatt-hours (kW-h).

Eversource (Eversource [2023]) currently charges 23.031 cents per kilowatt-hour.

If you’ve ever bought a power bank (Witman [2022]), then you’ll have noticed these are usually rated by the milliamp-hours (mA-h). As you can probably deduce, this is neither a measure of energy nor a measure of power. If we work it out, it’s really a measure of charge: milliamps are milli-coulombs per second, and hours are a measure of time, so we are left with a (scaled) measure of charge.

Similarly, car batteries are also usually rated in amp-hours (Ah). This is still a measure of charge, but because car batteries are usually assumed to output 12V1

12V 5AH battery How many watt hours? how long will it last if discharged at 1.3A? 50mA? How many amps can be drawn if it needs to last 40hrs?

Example

Suppose we have a 5Ah car battery. How many watt-hours of energy is that? How long will it last if discharged at 1.3A? How about 50 mA? How many amps can be drawn if it needs to last 40 hours?

Solution

Assume the 5Ah car battery outputs 12V. Over one hour, this battery can output 5A @ 12V for a total of 60 watts or 60 Wh.

If it needs to discharge 1.3A, then it can do this for

\[ \frac{ 5 {\rm A h}}{ 1.3 {\rm A}} \approx 3.85 {\rm h} \]

If it needs to discharge 50 mA, then it can do this for

\[ \frac{ 5 {\rm A h}}{ 0.05 {\rm A}} = 100 {\rm h} \]

Over 40 hours, the battery can discharge

\[ \frac{5 {\rm A h} }{ 40 {\rm h}} = 0.125 {\rm A} = 125 {\rm mA} \]

References

Eve23

Eversource. Electric Supply Rates | Connecticut | Eversource. https://www.eversource.com/content/residential/account-billing/manage-bill/about-your-bill/rates-tariffs/electric-supply-rates, 2023. [Online; accessed 16-January-2023].

NIS10(1,2)

NIST. SI Units. https://www.nist.gov/pml/owm/metric-si/si-units, 2010. [Online; accessed 16-January-2023].

Wit22

Sarah Witman. The Best Portable Chargers and Power Banks for Phones and Tablets. https://www.nytimes.com/wirecutter/reviews/best-usb-battery-packs/, 2022. [Online; accessed 16-January-2023].

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1

Despite the actual voltage being somewhere between 12.6V and 14.4V.