Block 01 — Physical quantities and SI system
Learning objectives
- Use the SI base quantities, units, and symbols correctly; convert between units with prefixes.
- Distinguish base vs. derived quantities; express key EE units (e.g. $\rm V$, $\rm \Omega$) in SI base units.
- Apply quantity equations and perform unit (dimensional) checks; contrast with normalized (dimensionless) equations.
- Read and use common Latin/Greek letter symbols; distinguish uppercase/lowercase and instantaneous vs. constant quantities.
90-minute plan
- Warm-up (10 min):
- “What is the unit of conductivity? of energy?”
- Quick prefix quiz; everyday magnitude estimates ($\rm mA$, $\rm k\Omega$, $\rm \mu F$).
- Core concepts & derivations (60 min):
- SI base set → derived units; prefix rules;
- quantity vs. normalized equations;
- dimensional checks.
- Prefix ladder ($\rm E$…$\rm a$) and best-practice rounding/checks.
- Symbols & Greek letters in EEE1; time-varying vs constant symbols.
- Practice (15 min): Fast conversions and unit checks (individual → pair).
- Wrap-up (5 min): Summary table; common pitfalls checklist.
Conceptual overview
- Units are the grammar of engineering and physics.
- The SI defines seven base quantities and units; all other (derived) units are built from these without extra numerical factors. The SI defines seven base quantities and units.
- In EEE1 we work strictly in the SI system, combining numerical value × unit and tracking dimensions at every step (e.g., $I=2~\rm A$ means “two times one ampere”).
- Derived units (e.g., $\rm V$, $\rm \Omega$, $\rm S$) must reduce to base units without hidden factors.
- Prefixes scale units by powers of ten to keep numbers readable. Prefixes compress very large and very small numbers so we can compute and compare safely.
- Quantity equations keep units; normalized equations cancel units to yield dimensionless ratios (e.g., efficiency).
- In EE, symbol choices and letter case matter: $U$ vs. $u(t)$, $\rm M$ (mega) vs. $\rm m$ (milli). We adopt a consistent symbol set (Latin + Greek), and distinguish constants (capital letters) from time functions (lowercase, e.g., $u(t)$).
- Finally, we preview the three anchor quantities for the next blocks: charge (what moves), current (how fast charge moves), and voltage (energy per charge). Physics describes quantities with a numerical value × unit (e.g., $I=2~\rm{A}$).
Core content
SI base quantities and units
| Base quantity | Name | Unit | Definition |
|---|---|---|---|
| Time | Second | ${\rm s}$ | Oscillation of $Cs$-Atom |
| Length | Meter | ${\rm m}$ | by s und speed of light |
| el. Current | Ampere | ${\rm A}$ | by s and elementary charge |
| Mass | Kilogram | ${\rm kg}$ | still by kg prototype |
| Temperature | Kelvin | ${\rm K}$ | by triple point of water |
| amount of substance | Mol | ${\rm mol}$ | via number of $^{12}C$ nuclides |
| luminous intensity | Candela | ${\rm cd}$ | via given radiant intensity |
- For practical applications of physical laws of nature, physical quantities are put into mathematical relationships.
- There are basic quantities based on the SI system of units (French for Système International d'Unités), see below.
- In order to determine the basic quantities quantitatively (quantum = Latin for how big), physical units are defined, e.g. ${\rm metre}$ for length.
- In electrical engineering, the first three basic quantities (cf. Tabelle 1 ) are particularly important.
Mass is important for the representation of energy and power.
- Each physical quantity is indicated by a product of numerical value and unit:
e.g. $I = 2~{\rm A}$- This is the short form of $I = 2\cdot 1~{\rm A}$
- $I$ is the physical quantity, here: electric current strength
- $\{I\} = 2 $ is the numerical value
- $ [I] = 1~{\rm A}$ is the (measurement) unit, here: ${\rm Ampere}$
Common derived quantities
- Besides the basic quantities, there are also quantities derived from them, e.g. $[F] = [m]\cdot [a] \rightarrow 1~{\rm N} = 1 ~{\rm kg} \cdot {{1 ~{\rm m}}\over{1 ~{\rm s}^2}}$.
- SI units should be preferred for calculations. These can be derived from the basic quantities without a numerical factor.
example:- The pressure unit bar (${\rm bar}$) is an SI unit.
- BUT: The obsolete pressure unit „Standard atmosphere“ ($=1.013~{\rm bar}$) is not an SI unit.
- To prevent the numerical value from becoming too large or too small, it is possible to replace a decimal factor with a prefix.
We will see, that a lot of electrical quantities are derived quantities.
Prefixes
| prefix | prefix symbol | meaning |
|---|---|---|
| Deci | ${\rm d}$ | $10^{-1}$ |
| Centi | ${\rm c}$ | $10^{-2}$ |
| Milli | ${\rm m}$ | $10^{-3}$ |
| Micro | ${\rm u}$, $µ$ | $10^{-6}$ |
| Nano | ${\rm n}$ | $10^{-9}$ |
| Piko | ${\rm p}$ | $10^{-12}$ |
| Femto | ${\rm f}$ | $10^{-15}$ |
| Atto | ${\rm a}$ | $10^{-18}$ |
| Zeppto | ${\rm z}$ | $10^{-21}$ |
| Yocto | ${\rm y}$ | $10^{-24}$ |
- Instead of writing zeroes for like in $0.000000004 ~\rm C $ is is easier to write $4 \rm ~nC $.
- For calculation it is often easier to write $4 ~\rm nC = 4 \cdot 10^{-9} ~C$ or the notation
\rm 4e-9 C
Physical equations
- Physical equations allow a connection of physical quantities.
- There are two types of physical equations to distinguish:
- Quantity equations (in German: Größengleichungen )
- Normalized quantity equations (also called related quantity equations, in German normierte Größengleichungen)
Quantity Equations
The vast majority of physical equations result in a physical unit that does not equal $1$.
Example: Force $F = m \cdot a$ with $[{\rm F}] = 1~\rm kg \cdot {{{\rm m}}\over{{\rm s}^2}}$
- A unit check should always be performed for quantity equations
- Quantity equations should generally be preferred
normalized Quantity Equations
In normalized quantity equations, the measured value or calculated value of a quantity equation is divided by a reference value. This results in a dimensionless quantity relative to the reference value.
Example: The efficiency $\eta = {{P_{\rm O}}\over{P_{\rm I}}}$ is given as quotient between the outgoing power $P_{\rm O}$ and the incoming power $P_{\rm I}$.
As a reference the following values are often used:
- Nominal values (maximum permissible value in continuous operation) or
- Maximum values (maximum value achievable in the short term)
For normalized quantity equations, the units should always cancel out.
Example for a quantity equation
Let a body with the mass $m = 100~{\rm kg}$ be given. The body is lifted by the height $s=2~{\rm m}$.What is the value of the needed work?
physical equation:
Work = Force $\cdot$ displacement
$W = F \cdot s \quad\quad\quad\;$ where $F=m \cdot g$
$W = m \cdot g \cdot s \quad\quad$ where $m=100~{\rm kg}$, $s=2~m$ and $g=9.81~{{{\rm m}}\over{{\rm s}^2}}$
$W = 100~kg \cdot 9.81 ~{{{\rm m}}\over{{\rm s}^2}} \cdot 2~{\rm m} $
$W = 100 \cdot 9.81 \cdot 2 \;\; \cdot \;\; {\rm kg} \cdot {{{\rm m}}\over{{\rm s}^2}} \cdot {\rm m}$
$W = 1962 \quad\quad \cdot \quad\quad\; \left( {\rm kg} \cdot {{{\rm m}}\over{{\rm s}^2}} \right) \cdot {\rm m} $
$W = 1962~{\rm Nm} = 1962~{\rm J} $
Letters for physical quantities
| Uppercase letters | Lowercase letters | Name | Application |
|---|---|---|---|
| $A$ | $\alpha$ | Alpha | angles, linear temperature coefficient |
| $B$ | $\beta$ | Beta | angles, quadratic temperature coefficient, current gain |
| $\Gamma$ | $\gamma$ | Gamma | angles |
| $\Delta$ | $\delta$ | Delta | small deviation, length of a air gap |
| $E$ | $\epsilon$, $\varepsilon$ | Epsilon | electrical field constant, permittivity |
| $Z$ | $\zeta$ | Zeta | - (math function) |
| $H$ | $\eta$ | Eta | efficiency |
| $\Theta$ | $\theta$, $\vartheta$ | Theta | temperature in Kelvin |
| $I$ | $\iota$ | Iota | - |
| $K$ | $\kappa$ | Kappa | specific conductivity |
| $\Lambda$ | $\lambda$ | Lambda | - (wavelength) |
| $M$ | $\mu$ | Mu | magnetic field constant, permeability |
| $N$ | $\nu$ | Nu | - |
| $\Xi$ | $\xi$ | Xi | - |
| $O$ | $\omicron$ | Omicron | - |
| $\Pi$ | $\pi$ | Pi | math. product operator, math. constant |
| $R$ | $\rho$, $\varrho$ | Rho | specific resistivity |
| $\Sigma$ | $\sigma$ | Sigma | math. sum operator, alternatively for specific conductivity |
| $T$ | $\tau$ | Tau | time constant |
| $\Upsilon$ | $\upsilon$ | Upsilon | - |
| $\Phi$ | $\phi$, $\varphi$ | Phi | magnetic flux, angle, potential |
| $X$ | $\chi$ | Chi | - |
| $\Psi$ | $\psi$ | Psi | linked magnetic flux |
| $\Omega$ | $\omega$ | Omega | unit of resistance, angular frequency |
Latin/Greek letters are reused across physics.
In physics and electrical engineering, the letters for physical quantities are often close to the English term.
Thus explains $C$ for Capacity, $Q$ for Quantity and $\varepsilon_0$ for the Electical Field Constant. But, maybe you already know that $C$ is used for the thermal capacity as well as for the electrical capacity. The Latin alphabet does not have enough letters to avoid conflicts for the scope of physics. For this reason, Greek letters are used for various physical quantities (see Tabelle 4).
Especially in electrical engineering, upper/lower case letters are used to distinguish between
- a constant (time-independent) quantity,
e.g. the period $T$ - or a time-dependent quantity,
e.g. the instantaneous voltage $u(t)$ - EE relies on case and context (e.g., $U$ vs. $u(t)$). Time-varying quantities often use lowercase, constants uppercase.
Notation & units
The course consistently uses the following symbols, units, and typical values:
| Symbol | Quantity | SI unit | name of the unit | Typical values |
|---|---|---|---|---|
| $q$ | Electric charge | $\rm C$ | Coulomb | $10^{-19} ~\rm C$ (electron) to $\rm mC$ |
| $I$ | Electric current | $\rm A$ | Ampere | $\rm \mu A$ (sensors) to $\rm kA$ (lightning) |
| $U$ | Voltage (potential difference) | $\rm V$ | Volt | $\rm \mu V$ (noise) to $\rm MV$ (transmission lines) |
| $\varphi$ | Electric potential | $\rm V$ | Volt | — |
| $P$ | Power | $\rm W$ | Watt | $\rm mW$ (electronics) to $\rm MW$ (machines) |
| $W$ | Energy | $\rm J$ | Joule | $\rm µJ$ (capacitors) to $\rm MJ$ (batteries) |
| $R$ | Resistance | $\rm \Omega$ | Ohm | $\rm mΩ$ to $\rm MΩ$ |
| $G$ | Conductance | $\rm S$ | Siemens | $\rm µS$ to $\rm S$ |
| $\rho$ | Resistivity | $\rm \Omega \cdot m$ | — | $1.7 \cdot 10^{-8} ~\rm \Omega m$ (Cu) |
| $\sigma$ | Conductivity | $\rm S/m$ | — | $5.8 \cdot 10^{7} ~\rm S/m$ (Cu) |
| $C$ | Capacitance | $\rm F$ | Farad | $\rm pF$ (ceramic) to $\rm F$ (supercaps) |
| $L$ | Inductance | $\rm H$ | Henry | $\rm \mu H$ to $\rm H$ |
| $E$ | Electric field strength | $\rm V/m$ | — | $\rm 1 ~\rm V/m$ to $\rm MV/m$ (breakdown) |
| $D$ | Electric flux density | $\rm C/m²$ | — | — |
| $B$ | Magnetic flux density | $\rm T$ | Tesla | $\rm \mu T$ (Earth) to several $\rm T$ (MRI) |
| $H$ | Magnetic field strength | $\rm A/m$ | — | — |
| $\Phi$ | Magnetic flux | $\rm Wb$ | Weber | $\rm \mu Wb$ to $\rm mWb$ |
| $\theta$ | magnetic voltage (Magnetomotive force) | $\rm A \cdot turn$ | — | — |
| $R$ | Reluctance | $\rm A/Wb$ | — | — |
Common pitfalls & misconceptions
- Case matters: $\rm M$ (mega, $10^6$) vs. $\rm m$ (milli, $10^{-3}$);
- Micro symbol: use $\rm \mu$ (or
uonly when typing constraints exist); - usage of prefixes never stack prefixes (no “$\rm mµF$”).
- Mixed units: keep SI consistently; avoid mixing $\rm hours$/$\rm Wh$ inside SI derivations.
- Units vs. variables: don’t confuse $W$ (work) with $\rm W$ (Watt = unit of power $\rm P$ = work per second).
Don’t confuse $C$ (capacity = charge per voltage) with $\rm C$ (Coulomb = unit of charge $\rm Q$). - Units vs. prefixes: don’t confuse $\rm mN$ (Millinewton) with $\rm Nm$ (Newton meter).
- Normalized vs. quantity equations: dimensionless ratios should cancel units; if not, something’s wrong.
Exercises
Quick checks
Exercise E1.1 Unit check (quantity equation)
Show that $P=U\cdot I$ has unit watt. (Better to be calulcated after reading Block02)
- $[U]=\rm{V}=\rm{kg}\,\rm{m}^2\,\rm{s}^{-3}\,\rm{A}^{-1}$, $[I]=\rm{A}$.
- $[P]=[U][I]=\rm{kg}\,\rm{m}^2\,\rm{s}^{-3}=\rm{W}$.
Exercise E2.2 Work from lifting (quantity equation)
How much energy is needed to lift 100 kg for 2 meters?
- $W=mgs$ with $m=100~\rm{kg},\,g=9.81~\rm{m/s^2},\,s=2~\rm{m}$
- $W=100\cdot9.81\cdot2~\rm{Nm}=1962~\rm{J}$
Exercise E3.1 Conversion
Convert $47~\rm{k\Omega}$ to $\rm{M\Omega}$ and $\Omega$.
Exercise E4.2 Dimension
Is $\eta=\dfrac{P_\rm{O}}{P_\rm{I}}$ dimensionless?
Exercise E5.3 Conversion
Which is larger: $5~\rm{mA}$ or $4500~\rm{\mu A}$?
Exercise E6.4 Conversion
True/False: $1~\rm{V}=1~\rm{Nm/As}$.
Longer exercises
Exercise E7 Conversions: Battery
1. How many minutes could a consumer with $3~{\rm W}$ get supplied by an ideal battery with $10~{\rm kWh}$ ?
\begin{align*} W &= 10~{\rm kWh} &&= 10'000~{\rm Wh}\\ \\ t &= {{W }\over{P }} &&= {{10'000~{\rm Wh}}\over{3~{\rm W}}} \\ &&&= 3'333~{\rm h}= 200'000~{\rm min} \quad (= 139~{\rm days}) \end{align*}
2. Hard: Why is a real Lithium-ion battery with $10~{\rm kWh}$ unable to provide the given power for the calculated time?
- The battery has an internal resistance. Depending on the current the battery provides, this leads to internal losses.
- The internal resistance of the battery depends on the state of charge (SoC) of the battery.
- The wires also add additional losses to the system.
Exercise E8 Conversions: Speed, Energy, and Power
Convert the following values step by step:
1. A vehicle speed of $80.00~{\rm {km}\over{h}}$ in ${\rm {m}\over{s}}$
2. An energy of $60.0~{\rm J}$ in ${\rm kWh}$ ($1~{\rm J} = 1~{\rm Joule} = 1~{\rm Watt}\cdot {\rm second}$).
3. The number of electrolytically deposited single positively charged copper ions of $1.2 ~\rm Coulombs$ (a copper ion has the charge of about $1.6 \cdot 10^{-19}{\rm C}$)
4. Absorbed energy of a small IoT consumer, which consumes $1~{\rm µW}$ uniformly in $10 ~\rm days$
Exercise E9 Conversion: Vacuum Cleaner
Your $18~{\rm V}$ vacuum cleaner is equipped with a $4.0~{\rm Ah}$ battery, it runs $15~{\rm minutes}$.
How much electrical power is consumed by the motor during this time on average?
\begin{align*} W &= 18~{\rm V} \cdot 4.0~{\rm Ah} = 72~{\rm Wh} \\ \\ t &= 15~{\rm min} = 0.25~{\rm h} \\ \\ P &= {{W } \over {t }} \\ &= {{72~\rm Wh} \over {0.25~\rm h }} = 288~{\rm W} \end{align*}
Exercise E10 Conversions: Video on Prefixes
Exercise E11 Conversion: Energy Consumption
Convert the following values step by step:
How much energy does an average household consume per day when consuming an average power of $ 500~{\rm W} $?
How many chocolate bars ($2'000~{\rm kJ}$ each) does this correspond to?
Exercise E1 Conversion: Energy, Power and Area
A Tesla Model 3 has an average power consumption of $ {{16~{\rm kWh}}\over{100~{\rm km}}}$ and an usable battery capacity of $60~{\rm kWh}$. Solar panels produces per $1~\rm m^2$ in average in December $0.2~{{{\rm kWh}}\over{{\rm m^{2}}}}$. The car is driven $50~{\rm km}$ per day. The size of a distinct solar module with $460~{\rm W_p}$ (Watt peak) is $1.9~{\rm m} \times 1.1~{\rm m}$.
1. What is the average power consumption of the car per day?
2. How many square meters (=$\rm m^2$) of solar panels are needed on average in December?
3. How many panels are at least needed to cover this surface?
4. What is the combined $\rm kW_p$ of the panels you calculated in 3. ?
Embedded resources
A nice 10-minute intro into some of the main topics of this chapter
Short presentation of the SI units
Orders of magnitude and why prefixes matter.
Mini-assignment / homework (optional)
List 10 everyday EE-relevant quantities (e.g., USB current, phone battery energy, LED forward voltage). For each:
- write as value × unit with an appropriate prefix,
- convert to base SI units, and
- if a formula applies (e.g., $P=U\cdot I$), do a unit check.