Inhaltsverzeichnis

Block 01 — Physical quantities and SI system

Learning objectives

After this 90-minute block, you can

90-minute plan

  1. Warm-up (10 min):
    1. “What is the unit of conductivity? of energy?”
  2. Quick prefix quiz; everyday magnitude estimates ($\rm mA$, $\rm k\Omega$, $\rm \mu F$).
  3. Core concepts & derivations (60 min):
    1. SI base set → derived units; prefix rules;
  4. quantity vs. normalized equations;
  5. dimensional checks.
    1. Prefix ladder ($\rm E$…$\rm a$) and best-practice rounding/checks.
    2. Symbols & Greek letters in EEE1; time-varying vs constant symbols.
  6. Practice (15 min): Fast conversions and unit checks (individual → pair).
  7. Wrap-up (5 min): Summary table; common pitfalls checklist.

Conceptual overview

  1. Units are the grammar of engineering and physics.
  2. 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.
  3. 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”).
  4. Derived units (e.g., $\rm V$, $\rm \Omega$, $\rm S$) must reduce to base units without hidden factors.
  5. 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.
  6. Quantity equations keep units; normalized equations cancel units to yield dimensionless ratios (e.g., efficiency).
  7. 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., $u(t)$).
  8. 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
Tab. 1: SI base quantities (SI)

Common derived quantities

We will see, that a lot of electrical quantities are derived quantities.

Prefixes

prefix prefix symbol meaning
Yotta ${\rm Y}$ $10^{24}$
Zetta ${\rm Z}$ $10^{21}$
Exa ${\rm E}$ $10^{18}$
Peta ${\rm P}$ $10^{15}$
Tera ${\rm T}$ $10^{12}$
Giga ${\rm G}$ $10^{9}$
Mega ${\rm M}$ $10^{6}$
Kilo ${\rm k}$ $10^{3}$
Hecto ${\rm h}$ $10^{2}$
Deka ${\rm de}$ $10^{1}$
Tab. 2: Prefixes I
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 ${\rm y}$ $10^{-24}$
Tab. 3: Prefixes II

Physical equations

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
Tab. 4: greek letters

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

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 $\rm A/Wb$
Tab. 5: Course-wide notation and units

Common pitfalls & misconceptions

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)

Result

  1. $[U]=\rm{V}=\rm{kg}\,\rm{m}^2\,\rm{s}^{-3}\,\rm{A}^{-1}$, $[I]=\rm{A}$.
  2. $[P]=[U][I]=\rm{kg}\,\rm{m}^2\,\rm{s}^{-3}=\rm{W}$.

Exercise E2.2 Work from lifting (quantity equation)

How much energy is needed to lift 100 kg for 2 meters?

Result

  1. $W=mgs$ with $m=100~\rm{kg},\,g=9.81~\rm{m/s^2},\,s=2~\rm{m}$
  2. $W=100\cdot9.81\cdot2~\rm{Nm}=1962~\rm{J}$

Exercise E3.1 Conversion

Convert $47~\rm{k\Omega}$ to $\rm{M\Omega}$ and $\Omega$.

Result

$47~\rm{k\Omega}=0.047~\rm{M\Omega}=47{,}000~\Omega$.

Exercise E4.2 Dimension

Is $\eta=\dfrac{P_\rm{O}}{P_\rm{I}}$ dimensionless?

Result

Yes. Units cancel ($\rm W/W$); normalized equation.

Exercise E5.3 Conversion

Which is larger: $5~\rm{mA}$ or $4500~\rm{\mu A}$?

Result

$5~\rm{mA}=5000~\rm{\mu A}$, so $5~\rm{mA}$ is larger.

Exercise E6.4 Conversion

True/False: $1~\rm{V}=1~\rm{Nm/As}$.

Result

True (from $W=U \cdot Q$).

Longer exercises

Exercise E1 Conversions: Battery

1. How many minutes could a consumer with $3~{\rm W}$ get supplied by an ideal battery with $10~{\rm kWh}$ ?

Solution

\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*}

Result

\begin{align*} t = 200'000~{\rm min} \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?

Result

There are additional losses:

  • 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 E2 Conversions: Speed, Energy, and Power

Convert the following values step by step:

1. A vehicle speed of $80.00~{\rm {km}\over{h}}$ in ${\rm {m}\over{s}}$

Fast Solution

\begin{align*} v = 80.00~{\rm {km}\over{h}} \qquad \rightarrow \qquad {{\rm {numerical~value}}\over{3.6}} \qquad \rightarrow \qquad v = 22.22~{\rm { m}\over{s}} \end{align*}

Solution

\begin{align*} v &= 80.00~{\rm {km}\over{h}} = 80~{\rm {1'000~m}\over{3'600~s}} = 80~{\rm {m}\over{3.6~s}} \\ &= 22.22~{\rm {m}\over{s}} \end{align*}

Result

\begin{align*} v = 22.22~{\rm {m}\over{s}} \end{align*}

2. An energy of $60.0~{\rm J}$ in ${\rm kWh}$ ($1~{\rm J} = 1~{\rm Joule} = 1~{\rm Watt}\cdot {\rm second}$).

Solution

\begin{align*} P = 60.0~{\rm J} = 60.0~{\rm Ws} = 0.06~{\rm kWs} = 0.0000167~{\rm kWh}= 1.67 \cdot 10^{-5}~{\rm kWh} \end{align*}

Result

\begin{align*} P = 1.67 \cdot 10^{-5}~{\rm kWh} \end{align*}

3. The number of electrolytically deposited single positively charged copper ions of $1.2 ~\rm Coulombs$ (a copper ion has the charge of about $1.6 \cdot 10^{-19}{\rm C}$)

Solution

\begin{align*} n_{\rm ions}= {{{\rm charge}}\over{{\rm charge\,per\,ion}}}={{1.2~{\rm C}}\over{1.6\cdot10^{-19}~{\rm C/ion}}}=7.5 \cdot 10^{18}~{\rm ions} \end{align*}

Result

\begin{align*} n_{\rm ions}= 7.5 \cdot 10^{18}~{\rm ions} \end{align*}

4. Absorbed energy of a small IoT consumer, which consumes $1~{\rm µW}$ uniformly in $10 ~\rm days$

Solution

\begin{align*} 10 ~{\rm days} &= 10 \cdot {{24 ~\rm h}\over{1 ~\rm day}} \cdot {{60 ~\rm min}\over{1 ~\rm h}} \cdot {{60 ~\rm s}\over{1 ~\rm min}} \\ &= 10 \cdot 24 \cdot 60 \cdot 60~{\rm s} \\ &= 864'000 ~{\rm s} \\ \\ W_{\rm absorbed} &= 1~{\rm µW} \cdot 864'000~{\rm s} \\ &= 864'000~{\rm µWs} \\ &= 864~{\rm mWs} \\ &= 0.864~{\rm Ws} \\ &= 0.864~{\rm J} \end{align*}

Result

\begin{align*} W_{\rm absorbed} = 0.864~{\rm Ws} = 0.864~{\rm J} \end{align*}

Exercise E3 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?

Solution

\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*}

Result

\begin{align*} 288~{\rm W} \end{align*}

Exercise E4 Conversions: Video on Prefixes

Exercise E5 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?

Solution

\begin{align*} W &= 500~{\rm W} \cdot 24~{\rm h} = 12000~{\rm Wh}= 43'200'000~{\rm Ws} = 43'200~{\rm kWs} &&= 43'200~{\rm kJ} \\ \text{Or: } W &= 0.5~{\rm kW} \cdot 24~{\rm h} = 12~{\rm kWh} = 43'200~{\rm kWs} &&= 43'200~{\rm kJ} \\ \\ n_{\rm bars} &={{43'200~{\rm kJ}}\over{2'000~{\rm kJ}}} = 21.6~{\rm chocolate~bars} \\ \end{align*}

Result

\begin{align*} 22~{\rm chocolate~bars} \end{align*}

Exercise E6 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?

Solution

\begin{align*} {{W}\over{l}} &= {{16~{\rm kWh}}\over{100~{\rm km}}} = 0.16 {{~{\rm kWh}}\over{~{\rm km}}} \\ W &= 50~{\rm km} \cdot 0.16 {{~{\rm kWh}}\over{~{\rm km}}} = 8~{\rm kWh} \end{align*}

Result

\begin{align*} W = 8~{\rm kWh}\end{align*}

2. How many square meters (=$\rm m^2$) of solar panels are needed on average in December?

Solution

\begin{align*} A = {{8~{\rm kWh}}\over{0.2 {{~{\rm kWh}}\over{~{\rm m^{2}}}}}} = 40~{\rm m^{2}} \end{align*}

Result

\begin{align*} A=40~{\rm m^{2}} \end{align*}

3. How many panels are at least needed to cover this surface?

Solution

\begin{align*} A_{\rm panel} &= 1.9~{\rm m} \cdot 1.1~{\rm m} = 2.1~{\rm m^{2}} \\ n_{\rm panels} &= {{A }\over{ A_{\rm panel }}} = {{40~{\rm m^{2}}}\over{2.1~{\rm m^{2}/panel}}} \\ &= 19.04~{\rm panels} \rightarrow 20~{\rm panels} \end{align*}

Result

\begin{align*} n_{\rm panels} = 20~{\rm panels} \end{align*}

4. What is the combined $\rm kW_p$ of the panels you calculated in 3. ?

Solution

\begin{align*} P &= 20~{\rm panels} \cdot 460~{\rm kW_p / panel} \\ &= 9'200~{\rm kW_p} \end{align*}

Result

\begin{align*} P = 9'200~{\rm kW_p} \end{align*}

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: