Unterschiede
Hier werden die Unterschiede zwischen zwei Versionen angezeigt.
| Beide Seiten der vorigen Revision Vorhergehende Überarbeitung Nächste Überarbeitung | Vorhergehende Überarbeitung | ||
| electrical_engineering_and_electronics_1:block03 [2025/09/28 15:34] – mexleadmin | electrical_engineering_and_electronics_1:block03 [2025/09/29 01:01] (aktuell) – mexleadmin | ||
|---|---|---|---|
| Zeile 1: | Zeile 1: | ||
| - | ==== Ideal current source | + | ====== Block 03 — Electric resistance and power ====== |
| - | From circuit theory, we abstract | + | |
| - | * Delivers | + | ===== Learning objectives ===== |
| - | * Symbol: circle | + | < |
| - | * U–I characteristic: vertical | + | After this 90-minute block, you can |
| + | * Explain and use the passive/ | ||
| + | | ||
| + | * Apply **Ohm’s law** and the $U$–$I$ characteristic; | ||
| + | * Derive resistance from **geometry** and **material**: | ||
| + | * Quantify **temperature dependence** with a linear coefficient $\alpha$ and compute $R(T)$. | ||
| + | * Read **practical resistor values** (E-series, RKM code, color code); relate nominal value, tolerance, and power rating. | ||
| + | </ | ||
| + | |||
| + | ===== 90-minute plan ===== | ||
| + | - Warm-up (8 min): | ||
| + | - recall charge, | ||
| + | - “Where does the minus sign go?” Mini-quiz on passive/ | ||
| + | - quick **unit check** for $P=U\cdot I$. | ||
| + | - quick quiz about units of $\rho$, $\kappa$, and $P$. | ||
| + | - Core concepts & derivations (62 min): | ||
| + | - Power & energy in DC; sign convention; forms of $P$ for a resistor ($R I^2$, $U^2/R$). | ||
| + | - Ohm’s law; linear vs. non-linear $U$–$I$ curves; operating point and differential resistance. | ||
| + | - Material law: $R=\rho\, | ||
| + | - Temperature dependence $R(T)=R_{20}\, | ||
| + | - Practical resistors: E-series, markings, power ratings, derating idea. | ||
| + | - Practice (15 min): Two quick examples + 3 short checks (see below). | ||
| + | - Wrap-up (5 min): Summary box; common pitfalls checklist. | ||
| + | |||
| + | ===== Conceptual overview ===== | ||
| + | <callout icon=" | ||
| + | - **Power** is the rate of energy conversion: $P=\dfrac{\rm dW}{\rm dt}$. In DC, $W=U\cdot I\cdot t$ and $P=U\cdot I$. | ||
| + | - **Sign convention matters.** With the **passive** convention (consumer/ | ||
| + | - **Ohm’s law** models many conductors near room temperature and moderate fields: $U=R\,I$. The slope in the $U$–$I$ plot is the **conductance** $G=1/R$. | ||
| + | - **Geometry & material** set the resistance: long/thin increases $R$; short/thick decreases $R$. The material constant is **resistivity** $\rho$ (reciprocal to **conductivity** $\kappa$). | ||
| + | | ||
| + | - **Real components** come in E-series with tolerances and power ratings; choose values that balance accuracy, heating, and availability. | ||
| + | </ | ||
| + | |||
| + | ===== Core content ===== | ||
| + | |||
| + | <WRAP right> | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| + | |||
| + | Every electrical circuit consists of three elements: | ||
| + | - **Consumers** also called **Loads** (in German: Verbraucher or Last): consumers convert electrical energy into energy that is not purely electrical. \\ e.g. | ||
| + | - into electrostatic energy (capacitor) | ||
| + | - into magnetostatic energy (magnet) | ||
| + | - into electromagnetic energy (LED, light bulb) | ||
| + | - into mechanical energy (loudspeaker, | ||
| + | - into chemical energy (charging an accumulator) | ||
| + | - **Sources** also called **Generator** (in German: Quellen): sources convert energy from another form of energy into electrical energy. (e.g. generator, battery, photovoltaic). | ||
| + | - **Wires** also called **Interconnections** (in German Leitungen or Verbindungen): | ||
| + | |||
| + | These elements will be considered in more detail below. | ||
| + | |||
| + | ==== Consumer ==== | ||
| + | |||
| + | <WRAP right> | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| + | |||
| + | * The colloquial term ' | ||
| + | * A resistor is often also referred to as a consumer. In addition to pure ohmic consumers, however, there are also ohmic-inductive consumers (e.g. coils in a motor) or ohmic-capacitive consumers (e.g. various power supplies using capacitors at the output). Correspondingly the equation " | ||
| + | * Current-voltage characteristics (see <imgref BildNr4> | ||
| + | |||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ==== Ohmic resistance ==== | ||
| + | |||
| + | <WRAP right> | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| + | |||
| + | Given is an electrical conductor (" | ||
| + | * $\rightarrow$ Current flows | ||
| + | * Similar to the transport of a mass in the gravitational field, energy is needed to transport the charge in the electric field | ||
| + | * A point charge $q$ is moved from electrode ① to electrode ②. \\ The charge resembles a moving point of mass in the gravitational field. | ||
| + | * $\rightarrow$ there is a turnover of energy. | ||
| + | |||
| + | Current flow generally requires an energy input first. This energy is at some point extracted from the electric circuit and is usually converted into heat. | ||
| + | The reason for this conversion is the resistance e.g. of the conductor or other loads. | ||
| + | |||
| + | A resistor is an electrical component with two connections (or terminals). Components with two terminals are called two-terminal networks or one-port networks (<imgref BildNr11> | ||
| + | |||
| + | <WRAP right> | ||
| + | {{url> | ||
| + | </ | ||
| + | |||
| + | In general, the cause-and-effect relationship is such that an applied | ||
| + | In electrical engineering, | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | The values of the resistors are standardized in such a way, that there is a fixed number of different values between $1~\Omega$ and $10~\Omega$ or between $10~\rm k\Omega$ and $100~\rm k\Omega$. These ranges, which cover values up to the tenfold number, are called decades. In general, the resistors are ordered in the so-called {{wp>E series of preferred numbers}}, like 6 values in a decade, which is named E6 (here: $1.0~\rm k\Omega$, $1.5~\rm k\Omega$, $2.2~\rm k\Omega$, $3.3~\rm k\Omega$, $4.7~\rm k\Omega$, $6.8~\rm k\Omega$). As higher the number (e.g. E24) more different values are available in a decade, and as more precise the given value is. | ||
| + | |||
| + | For larger resistors with wires, the value is coded by four to six colored bands (see [[https:// | ||
| + | |||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ==== Linearity of Resistors ==== | ||
| + | <WRAP group>< | ||
| + | <callout color=" | ||
| + | === Linear Resistors == | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | |||
| + | * For linear resistors, the resistance value is $R={{U_R}\over{I_R}}={\rm const.} $ and thus independent of $U_R$. | ||
| + | * **Ohm' | ||
| + | * In <imgref BildNr13> | ||
| + | * The reciprocal value (inverse) of the resistance is called the conductance: | ||
| + | |||
| + | </ | ||
| + | |||
| + | </ | ||
| + | <callout color=" | ||
| + | === Non-linear Resistors | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | |||
| + | * The point in the $U$-$I$ diagram in which a system rests is called the operating point. In the <imgref BildNr14> | ||
| + | * For nonlinear resistors, the resistance value is $R={{U_R}\over{I_R(U_R)}}=f(U_R)$. This resistance value depends on the operating point. | ||
| + | * Often small changes around the operating point are of interest (e.g. for small disturbances of load machines). For this purpose, the differential resistance $r$ (also dynamic resistance) is determined: \\ $\boxed{r={{{\rm d}U_R}\over{{\rm d}I_R}} \approx{{\Delta U_R}\over{\Delta I_R}} }$ with unit $[R]=1~\Omega$. | ||
| + | * As with the resistor $R$, the reciprocal of the differential resistance $r$ is the differential conductance $g$. | ||
| + | * In <imgref BildNr14> | ||
| + | </ | ||
| + | |||
| + | </ | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ==== Ideal sources ==== | ||
| + | * Sources act as generators of electrical energy | ||
| + | * A distinction is made between ideal and real sources. \\ The real sources are described in the following chapter " | ||
| + | |||
| + | The **ideal voltage source** generates a defined constant output voltage $U_\rm s$ (in German often $U_\rm q$ for Quellenspannung). | ||
| + | In order to maintain this voltage, it can supply any current. | ||
| + | The current-voltage characteristic also represents this (see <imgref BildNr6> | ||
| + | The circuit symbol shows a circle with two terminals. In the circuit, the two terminals are short-circuited. \\ | ||
| + | Another circuit symbol shows the negative terminal of the voltage source as a "thick minus symbol", | ||
| + | |||
| + | The **ideal current source** produces a defined constant output current $I_\rm s$ (in German often $I_\rm q$ for Quellenstrom). | ||
| + | For this current to flow, any voltage can be applied to its terminals. | ||
| + | The current-voltage characteristic also represents this (see <imgref BildNr7> | ||
| + | The circuit symbol shows again a circle with two connections. This time the two connections are left open in the circle and a line is drawn perpendicular to them. | ||
| + | |||
| + | < | ||
| <WRAP group>< | <WRAP group>< | ||
| - | < | + | < |
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| + | <WRAP column 45%> | ||
| + | |||
| + | < | ||
| </ | </ | ||
| {{drawio> | {{drawio> | ||
| - | </ | + | |
| - | {{youtube> | + | </ |
| - | </ | + | |
| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ====Resistance as a Material Property ($\rho$, $\kappa$) ==== | ||
| + | |||
| + | The value of the resistance can also be derived from the geometry of the resistor. For this purpose, an experiment can be carried out with resistors of different shapes. Thereby it can be stated: | ||
| + | * the resistance $R$ increases proportionally with the distance $l$ the current has to travel: $R \sim l$ | ||
| + | * the resistance $R$ decreases inversely proportional with the cross-sectional area $A$ through which the current passes: $R \sim {{1}\over{A}}$ | ||
| + | * the resistance $R$ depends on the material (<tabref tab04>) | ||
| + | * thus one obtains: \\ $R \sim {{l}\over{A}}$ | ||
| + | |||
| + | <callout icon=" | ||
| + | The resistance can be calculated by \\ $\boxed{R = \rho \cdot {{l}\over{A}} } $ | ||
| + | * $\rho$ is the material dependent resistivity with the unit: $[\rho]={{[R]\cdot[A]}\over{l}}=1~{{\Omega\cdot {\rm m}^{\not{2}}}\over{\not{{\rm m}}}}=1~\Omega\cdot {\rm m}$ | ||
| + | * Often, instead of $1~\Omega\cdot {\rm m}$, the unit $1~{{\Omega\cdot {{\rm mm}^2}}\over{{\rm m}}}$ is used. It holds that $1~{{\Omega\cdot {{\rm mm}^2}}\over{{\rm m}}}= 10^{-6}~\Omega {\rm m}$ | ||
| + | </ | ||
| + | |||
| + | <callout icon=" | ||
| + | $[\rho]={\rm \Omega\, | ||
| + | </ | ||
| + | |||
| + | <WRAP > | ||
| + | < | ||
| + | ^ Material | ||
| + | | Silver | ||
| + | | Copper | ||
| + | | Gold | $2.2\cdot 10^{-2}$ | ||
| + | | Aluminium | ||
| + | | Lead | $2.1\cdot 10^{-1}$ | ||
| + | | Graphite | ||
| + | | Battery Acid (Lead-acid Battery) | ||
| + | | Blood | ||
| + | | (Tap) Water | ||
| + | | Paper | ||
| + | </ | ||
| + | </ | ||
| + | * There exists also a specific conductance $\kappa$, given by the conductance $G$ : \\ $G= \kappa \cdot {{A}\over{l}}$ | ||
| + | * The specific conductance $\kappa$ is the reciprocal of the specific resistance $\rho$: \\ $\kappa={{1}\over{\rho}}$ | ||
| ==== Conductivity of Matter ==== | ==== Conductivity of Matter ==== | ||
| Zeile 51: | Zeile 242: | ||
| </ | </ | ||
| </ | </ | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ==== Temperature dependence of resistance ==== | ||
| + | |||
| + | The resistance value is - apart from the influences of geometry and material mentioned so far - also influenced by other effects. These are e.g.: | ||
| + | * Heat (thermoresistive effect, e.g. in the resistance thermometer) | ||
| + | * Light (photosensitive effect, e.g. in the component photoresistor) | ||
| + | * Magnetic field (magnetoresistive effect, e.g. in hard disks) | ||
| + | * Pressure (piezoresistive effect e.g. tire pressure sensor) | ||
| + | * Chemical environment (chemoresistive effect e.g. chemical analysis of breathing air) | ||
| + | |||
| + | To summarize these influences in a formula, the mathematical tool of {{wp> | ||
| + | This will be shown here practically for the thermoresistive effect. | ||
| + | The thermoresistive effect, or the temperature dependence of the resistivity, | ||
| + | |||
| + | The starting point for this is again an experiment. The ohmic resistance is to be determined as a function of temperature. | ||
| + | For this purpose, the resistance is measured using a voltage source, a voltmeter (voltage measuring device), and an ammeter (current measuring device), and the temperature is changed (<imgref BildNr15> | ||
| + | |||
| + | <WRAP group>< | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| + | |||
| + | The result is a curve of the resistance $R$ versus the temperature $\vartheta$ as shown in <imgref BildNr22> | ||
| + | As a first approximation is a linear progression around an operating point. | ||
| + | This results in: | ||
| + | |||
| + | $R(\vartheta) = R_0 + c\cdot (\vartheta - \vartheta_0)$ | ||
| + | |||
| + | * The constant is replaced by $c = R_0 \cdot \alpha$ | ||
| + | * $\alpha$ here is the linear temperature coefficient with unit: $ [\alpha] = {{1}\over{[\vartheta]}} = {{1}\over{{\rm K}}} $ | ||
| + | * Besides the linear term, it is also possible to increase the accuracy of the calculation of $R(\vartheta)$ with higher exponents of the temperature influence. This approach will be discussed in more detail in the mathematics section below. | ||
| + | * These temperature coefficients are described with Greek letters: $\alpha$, $\beta$, $\gamma$, ... | ||
| + | * Sometimes in the datasheets the value $\alpha$ is named as TCR (" | ||
| + | |||
| + | <WRAP > | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | |||
| + | <callout icon=" | ||
| + | |||
| + | The temperature dependence of the resistance is described by the following equation: | ||
| + | $\boxed{ R(\vartheta) = R_0 (1 + \alpha \cdot (\vartheta - \vartheta_0) + \beta \cdot (\vartheta - \vartheta_0)^2 + ...)}$ | ||
| + | |||
| + | Where: | ||
| + | * $\alpha$ the (linear) temperature coefficient with unit: $ [\alpha] = {{1}\over{{\rm K}}} $ | ||
| + | * $\beta$ the (quadratic) temperature coefficient with unit: $ [\beta] = {{1}\over{{\rm K}^2}} $ | ||
| + | * $\vartheta_0$ is the given reference temperature, | ||
| + | |||
| + | </ | ||
| + | |||
| + | <callout icon=" | ||
| + | Some source pages use $\sigma$ for conductivity. In EEE1 we use **$\kappa$** in line with the course standard ($\kappa=1/ | ||
| + | </ | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ==== Power and energy ==== | ||
| + | |||
| + | We know that a movement of a charge across a potential difference corresponds to a change in energy. | ||
| + | Charge transport therefore automatically means energy expenditure. Often, however, the energy expenditure per unit of time is of interest. | ||
| + | |||
| + | < | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | |||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| + | |||
| + | The energy expenditure per time unit represents the **power**: \\ | ||
| + | $\boxed{P={{\Delta W}\over{\Delta t}}}$ with the unit $[P]={{[W]}\over{[t]}}=1~{\rm {J}\over{s}} = 1~{\rm {Nm}\over{s}} = 1 ~{\rm V\cdot A} = 1~{\rm W}$ | ||
| + | |||
| + | For a constant power $P$ and an initial energy $W(t=0~{\rm s})=0$ holds: \\ | ||
| + | $\boxed{W=P \cdot t}$ \\ | ||
| + | If the above restrictions do not apply, the generated/ | ||
| + | |||
| + | Besides the current flow from the source to the consumer (and back), also power flows from the source to the consumer. | ||
| + | In the following circuit, the color code shows the incoming and outgoing power. | ||
| + | |||
| + | < | ||
| + | {{url> | ||
| + | </ | ||
| + | |||
| + | If we only consider a __DC circuit__, the following energy is converted between the terminals (see also <imgref BildNr19> | ||
| + | $W=U_{12}\cdot Q = U_{12} \cdot I \cdot t$ | ||
| + | |||
| + | This gives the power (i.e. energy converted per unit time): \\ | ||
| + | $\boxed{P=U_{12} \cdot I}$ with the unit $[P]= 1 ~{\rm V\cdot A} = 1~{\rm W} \quad$ ... ${\rm W}$ here stands for the physical unit watts. | ||
| + | |||
| + | For ohmic resistors: | ||
| + | |||
| + | $\boxed{P=R\cdot I^2 = {{U_{12}^2}\over{R}}}$ | ||
| + | |||
| + | <callout icon=" | ||
| + | $[U]= {\rm kg\, | ||
| + | </ | ||
| + | |||
| + | ==== Passive vs. active sign convention ==== | ||
| + | |||
| + | For **loads** (passive convention): | ||
| + | For **sources** (active convention): | ||
| + | <callout icon=" | ||
| + | |||
| + | <WRAP group>< | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| + | < | ||
| + | </ | ||
| + | {{drawio> | ||
| + | </ | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ===== Common pitfalls ===== | ||
| + | * Don’t mix conventions: | ||
| + | * Always attach **units** (e.g., $I=2~{\rm A}$). Check $P$ with **two forms** ($R I^2$ vs. $U^2/R$) to catch arithmetic slips. | ||
| + | * Geometry units trap: in $R=\rho\, | ||
| + | * Temperature law is **approximate** near the reference (often $20^\circ\rm C$); large $\Delta T$ or semiconductors need better models. | ||
| + | |||
| + | ===== Exercises ===== | ||
| + | ==== Worked examples ==== | ||
| + | |||
| + | **Example 1 — Copper wire resistance.** | ||
| + | A copper lead ($\rho=0.0178~{\rm \Omega\, | ||
| + | \begin{align*} | ||
| + | R &= \rho\, | ||
| + | \end{align*} | ||
| + | |||
| + | **Example 2 — Power forms cross-check.** | ||
| + | A load takes $I=250~{\rm mA}$ at $U=12.0~{\rm V}$. | ||
| + | \begin{align*} | ||
| + | P&=U I=12.0~{\rm V}\cdot0.250~{\rm A}=3.00~{\rm W}. \\ | ||
| + | R& | ||
| + | P=R I^2=48.0~\Omega\cdot(0.250~{\rm A})^2=3.00~{\rm W}. | ||
| + | \end{align*} | ||
| + | |||
| + | **Example 3 — Temperature coefficient.** | ||
| + | A metal resistor has $R_{20}=1.00~{\rm k\Omega}$ and $\alpha=3.9\cdot10^{-3}~{\rm K^{-1}}$. Find $R$ at $T=60^\circ{\rm C}$. | ||
| + | \begin{align*} | ||
| + | R(60^\circ{\rm C})& | ||
| + | & | ||
| + | \end{align*} | ||
| + | |||
| + | **Example 4 — Choosing an E-series value.** | ||
| + | Need $\approx 3.3~{\rm k\Omega}$: nearest E12 is $3.3~{\rm k\Omega}$ (good match). For $\approx 3.0~{\rm k\Omega}$: E12 has no 3.0 → choose $3.0~{\rm k\Omega}$ from E24 or combine series/ | ||
| + | |||
| + | <panel type=" | ||
| + | A $R=220~\Omega$ resistor is across $U=9.0~{\rm V}$. | ||
| + | Compute $I$ and $P$. Is a $0.25~{\rm W}$ part safe at $25^\circ{\rm C}$? | ||
| + | |||
| + | <button size=" | ||
| + | Use $I=U/R$, $P=U^2/R$, then compare with the nominal rating. | ||
| + | </ | ||
| + | |||
| + | <button size=" | ||
| + | \begin{align*} | ||
| + | I& | ||
| + | P=\frac{(9.0~{\rm V})^2}{220~\Omega}=0.368~{\rm W}. | ||
| + | \end{align*} | ||
| + | A $0.25~{\rm W}$ part is **not** safe; choose $\ge 0.5~{\rm W}$ (and consider derating). | ||
| + | </ | ||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | {{youtube> | ||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | Assume that a soft pencil lead is $100 ~\%$ graphite. What is the resistance of a $5.0~{\rm cm}$ long and $0.20~{\rm mm}$ wide line if it has a height of $0.20~{\rm µm}$? | ||
| + | |||
| + | The resistivity is given by <tabref tab04>. | ||
| + | |||
| + | {{drawio> | ||
| + | |||
| + | <button size=" | ||
| + | \begin{align*} | ||
| + | R = 10~{\rm k} \Omega | ||
| + | \end{align*}</ | ||
| + | |||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | Let a cylindrical coil in the form of a multi-layer winding be given - this could for example occur in windings of a motor. | ||
| + | The cylindrical coil has an inner diameter of $d_{\rm i}=70~{\rm mm}$ and an outer diameter of $d_{\rm a} = 120~{\rm mm}$. | ||
| + | The number of turns is $n_{\rm W}=1350$ turns, the wire diameter is $d=2.0~{\rm mm}$ and the specific conductivity of the wire is $\kappa_{\rm Cu}=56 \cdot 10^6 ~{{{\rm S}}\over{{\rm m}}}$. | ||
| + | |||
| + | First, calculate the wound wire length and then the ohmic resistance of the entire coil. | ||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | The power supply line to a consumer has to be replaced. Due to the application, | ||
| + | * The old aluminium supply cable had a specific conductivity $\kappa_{\rm Al}=33\cdot 10^6 ~{\rm {S}\over{m}}$ and a cross-section $A_{\rm Al}=115~{\rm mm}^2$. | ||
| + | * The new copper supply cable has a specific conductivity $\kappa_{\rm Cu}=56\cdot 10^6 ~{\rm {S}\over{m}}$ | ||
| + | |||
| + | Which wire cross-section $A_{\rm Cu}$ must be selected? | ||
| + | </ | ||
| + | |||
| + | {{page> | ||
| + | |||
| + | <panel type=" | ||
| + | You need a lead with $R\le 0.10~\Omega$ (round-trip) over $l=3.0~{\rm m}$ each way (total conductor length $6.0~{\rm m}$). | ||
| + | Pick $A$ for copper ($\rho=0.0178~{\rm \Omega\, | ||
| + | |||
| + | <button size=" | ||
| + | \begin{align*} | ||
| + | R& | ||
| + | =0.0178~{\rm \Omega\, | ||
| + | \end{align*} | ||
| + | Use the next standard cross-section (e.g., $1.5~{\rm mm^2}$). | ||
| + | </ | ||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | A resistor has $R_{20}=4.70~{\rm k\Omega}$, $\alpha=3.7\cdot10^{-3}~{\rm K^{-1}}$. Find $R$ at $0^\circ{\rm C}$ and at $70^\circ{\rm C}$. | ||
| + | |||
| + | <button size=" | ||
| + | \begin{align*} | ||
| + | R(0^\circ{\rm C})& | ||
| + | R(70^\circ{\rm C})& | ||
| + | \end{align*} | ||
| + | </ | ||
| + | </ | ||
| + | |||
| + | <panel type=" | ||
| + | |||
| + | The first 5:20 minutes is a recap of the fundamentals of calculating the electric power | ||
| + | {{youtube> | ||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | </ | ||
| + | |||
| + | # | ||
| + | # | ||
| + | |||
| + | An SMD resistor is used on a circuit board for current measurement. The resistance value should be $R=0.20~\Omega$, | ||
| + | What is the maximum current that can be measured? | ||
| + | |||
| + | # | ||
| + | The formulas $R = {{U} \over {I}}$ and $P = {U} \cdot {I}$ can be combined to get: | ||
| + | \begin{align*} | ||
| + | P = R \cdot I^2 | ||
| + | \end{align*} | ||
| + | |||
| + | This can be rearranged into | ||
| + | |||
| + | \begin{align*} | ||
| + | I = + \sqrt{ {{P} \over{R} } } | ||
| + | \end{align*} | ||
| + | |||
| + | # | ||
| + | |||
| + | # | ||
| + | \begin{align*} | ||
| + | I = 1.118... ~{\rm A} \rightarrow I = 1.12 ~{\rm A} | ||
| + | \end{align*} | ||
| + | |||
| + | # | ||
| + | # | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | ===== Embedded resources ===== | ||
| + | <WRAP column half> | ||
| + | Explanation (video): {{youtube> | ||
| + | </ | ||
| + | |||
| + | <WRAP column half> | ||
| + | Good explanation of resistivity | ||
| + | {{youtube> | ||
| + | </ | ||
| + | |||
| + | <WRAP column half> | ||
| + | Video explainer on $R(T)$. | ||
| + | {{youtube> | ||
| + | </ | ||
| + | |||
| + | <WRAP column half> | ||
| + | {{youtube> | ||
| + | </ | ||
| + | |||
| + | <WRAP column half> | ||
| + | Voltage vs Power vs Energy | ||
| + | {{youtube> | ||
| + | </ | ||
| + | |||
| + | <WRAP column half> | ||
| + | Resistor packages | ||
| + | {{youtube> | ||
| + | </ | ||
| + | |||
| + | <WRAP column half> | ||
| + | A nice 10-minute intro into power and efficiency (a cutout from 2:40 to 12:15 from a full video of EEVblog) | ||
| + | {{youtube> | ||
| + | </ | ||
| + | |||
| + | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| + | |||
| + | |||