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| electrical_engineering_1:preparation_properties_proportions [2023/07/21 23:18] – mexleadmin | electrical_engineering_1:preparation_properties_proportions [2025/09/15 15:27] (current) – mexleadmin | ||
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| # | # | ||
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| - | ====== 1. Preparation, | + | ====== 1 Preparation, |
| ===== 1.1 Physical Proportions ===== | ===== 1.1 Physical Proportions ===== | ||
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| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== derived | + | ==== derived |
| * Besides the basic quantities, there are also quantities derived from them, e.g. $1~{{{\rm m}}\over{{\rm s}}}$. | * Besides the basic quantities, there are also quantities derived from them, e.g. $1~{{{\rm m}}\over{{\rm s}}}$. | ||
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| ~~PAGEBREAK~~ ~~CLEARFIX~~ | ~~PAGEBREAK~~ ~~CLEARFIX~~ | ||
| - | ==== Physical | + | ==== Physical |
| * Physical equations allow a connection of physical quantities. | * Physical equations allow a connection of physical quantities. | ||
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| <callout color=" | <callout color=" | ||
| - | === Quantity | + | === Quantity |
| The vast majority of physical equations result in a physical unit that does not equal $1$. | The vast majority of physical equations result in a physical unit that does not equal $1$. | ||
| \\ \\ | \\ \\ | ||
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| < | < | ||
| <callout color=" | <callout color=" | ||
| - | === normalized | + | === normalized |
| In normalized quantity equations, the measured value or calculated value of a quantity equation is divided by a reference value. | In normalized quantity equations, the measured value or calculated value of a quantity equation is divided by a reference value. | ||
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| </ | </ | ||
| - | ==== Letters for physical | + | ==== Letters for physical |
| In physics and electrical engineering, | In physics and electrical engineering, | ||
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| - | ==== Conductivity ==== | + | ==== Conductivity |
| <WRAP group>< | <WRAP group>< | ||
| <callout color=" | <callout color=" | ||
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| {{tagtopic> | {{tagtopic> | ||
| - | ===== 1.3 Effects of electric charges | + | ===== 1.3 Effects of Electric Charges |
| < | < | ||
| === Learning Objectives === | === Learning Objectives === | ||
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| * Qualitative investigation using a second experiment | * Qualitative investigation using a second experiment | ||
| * two charges ($Q_1$ and $Q_2$) at distance $r$ | * two charges ($Q_1$ and $Q_2$) at distance $r$ | ||
| - | * additional measurement of the force $F_C$ (e.g. via spring balance) | + | * additional measurement of the force $F_{\rm C}$ (e.g. via spring balance) |
| * Experiment results: | * Experiment results: | ||
| - | * Force increases linearly with larger charge $Q_1$ or $Q_2$. \\ $ F_C \sim Q_1$ and $ F_C \sim Q_2$ | + | * Force increases linearly with larger charge $Q_1$ or $Q_2$. \\ $ F_{\rm C} \sim Q_1$ and $ F_{\rm C} \sim Q_2$ |
| - | * Force falls quadratic with greater distance $r$ \\ $ F_C \sim {1 \over {r^2}}$ | + | * Force falls quadratic with greater distance $r$ \\ $ F_{\rm C} \sim {1 \over {r^2}}$ |
| - | * with a proportionality factor $a$: \\ $ F_C = a \cdot {{Q_1 \cdot Q_2} \over {r^2}}$ | + | * with a proportionality factor $a$: \\ $ F_{\rm C} = a \cdot {{Q_1 \cdot Q_2} \over {r^2}}$ |
| * Proportionality factor $a$ | * Proportionality factor $a$ | ||
| - | * The proportionality factor $a$ is defined | + | * The proportionality factor $a$ is defined |
| * $a$ thus becomes: | * $a$ thus becomes: | ||
| * $a = {{1} \over {4\pi\cdot\varepsilon}}$ | * $a = {{1} \over {4\pi\cdot\varepsilon}}$ | ||
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| <callout icon=" | <callout icon=" | ||
| - | The Coulomb force (in a vacuum) can be calculated via. \\ $\boxed{ | + | The Coulomb force (in a vacuum) can be calculated via. \\ $\boxed{ |
| where $\varepsilon_0 = 8.85 \cdot 10^{-12} \cdot ~{{\rm C}^2 \over {{\rm m}^2\cdot {\rm N}}} = 8.85 \cdot 10^{-12} \cdot ~{{{\rm As}} \over {{\rm Vm}}}$ | where $\varepsilon_0 = 8.85 \cdot 10^{-12} \cdot ~{{\rm C}^2 \over {{\rm m}^2\cdot {\rm N}}} = 8.85 \cdot 10^{-12} \cdot ~{{{\rm As}} \over {{\rm Vm}}}$ | ||
| </ | </ | ||
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| The current of $1~{\rm A}$ flows when an amount of charge of $1~{\rm C}$ is transported in $1~{\rm s}$ through the cross-section of the conductor. | The current of $1~{\rm A}$ flows when an amount of charge of $1~{\rm C}$ is transported in $1~{\rm s}$ through the cross-section of the conductor. | ||
| - | Before 2019: The current of $1~{\rm A}$ flows when two parallel conductors, each $1~{\rm m}$ long and $1~{\rm m}$ apart, exert a force of $F_C = 0.2\cdot 10^{-6}~{\rm N}$ on each other. | + | Before 2019: The current of $1~{\rm A}$ flows when two parallel conductors, each $1~{\rm m}$ long and $1~{\rm m}$ apart, exert a force of $F_{\rm L} = 0.2\cdot 10^{-6}~{\rm N}$ on each other. |
| </ | </ | ||
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| <callout icon=" | <callout icon=" | ||
| An electrode is a connection (or pin) of an electrical component. \\ | An electrode is a connection (or pin) of an electrical component. \\ | ||
| - | As a rule, the dimension of an electrode is characterized | + | Looking at a component, the electrode is characterized |
| The name of the electrode is given as follows: | The name of the electrode is given as follows: | ||
| * **A**node: Electrode at which the current enters the component. | * **A**node: Electrode at which the current enters the component. | ||
| * Cathode: Electrode at which the current exits the component. (in German // | * Cathode: Electrode at which the current exits the component. (in German // | ||
| - | As a mnemonic, you can remember the structure, shape, and electrodes | + | As a mnemonic, you can remember the diode' |
| </ | </ | ||
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| 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. | 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:// | + | For larger resistors with wires, the value is coded by four to six colored bands (see __ BROKEN-LINK: |
| < | < | ||
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| <WRAP group>< | <WRAP group>< | ||
| <callout color=" | <callout color=" | ||
| - | === Linear | + | === Linear |
| < | < | ||
| </ | </ | ||
| {{drawio> | {{drawio> | ||
| - | * For linear resistors, the resistance value is $R={{U_R}\over{I_R}}=const.$ and thus independent of $U_R$. | + | * For linear resistors, the resistance value is $R={{U_R}\over{I_R}}={\rm const.} $ and thus independent of $U_R$. |
| * **Ohm' | * **Ohm' | ||
| * In <imgref BildNr13> | * In <imgref BildNr13> | ||
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| </ | </ | ||
| <callout color=" | <callout color=" | ||
| - | === Non-linear | + | === Non-linear |
| < | < | ||
| </ | </ | ||
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| * The point in the $U$-$I$ diagram in which a system rests is called the operating point. In the <imgref BildNr14> | * 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. | * 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: \\ $\boxed{r={{dU_R}\over{dI_R}} \approx{{\Delta U_R}\over{\Delta I_R}} }$ with unit $[R]=1~\Omega$. | + | * 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 $g$. | * As with the resistor $R$, the reciprocal of the differential resistance $r$ is the differential conductance $g$. | ||
| - | * In <imgref BildNr14> | + | * In <imgref BildNr14> |
| </ | </ | ||
| </ | </ | ||
| - | ==== Resistance as a material | + | ==== Resistance as a Material |
| < | < | ||
| - | Clear explanation of resistivity | + | Good explanation of resistivity |
| {{youtube> | {{youtube> | ||
| </ | </ | ||
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| <WRAP > | <WRAP > | ||
| < | < | ||
| - | ^ Material | + | ^ Material |
| - | | Silver | + | | Silver |
| - | | Copper | + | | Copper |
| - | | Aluminium | + | | Gold |
| - | | Gold | $2.2\cdot 10^{-2}$ | + | | Aluminium |
| - | | Lead | + | | Lead | $2.1\cdot 10^{-1}$ |
| - | | Graphite | + | | Graphite |
| - | | Battery Acid (Lead-acid Battery) | $1.5\cdot 10^4$ | | + | | Battery Acid (Lead-acid Battery) |
| - | | Blood | $1.6\cdot 10^{6}$ | + | | Blood |
| - | | (Tap) Water | $2 \cdot 10^{7}$ | + | | (Tap) Water |
| - | | Paper | $1\cdot 10^{15} ... 1\cdot 10^{17}$ | + | | Paper |
| </ | </ | ||
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| $R(\vartheta) = R_0 + c\cdot (\vartheta - \vartheta_0)$ | $R(\vartheta) = R_0 + c\cdot (\vartheta - \vartheta_0)$ | ||
| - | * The constant is replaced by $c = R_0 \cdot \alpha$ | + | * 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}}} $ | + | * $\alpha$ 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. | + | * 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: $\alpha$, $\beta$, $\gamma$, ... | + | * These temperature coefficients are described with Greek letters: $\alpha$, $\beta$, $\gamma$, ... |
| + | * Sometimes in the datasheets the value $\alpha$ is named as TCR (" | ||
| <WRAP group>< | <WRAP group>< | ||
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| A series expansion can again be applied: $R(T) \sim {\rm e}^{{\rm A} + {{\rm B}\over{T}} + {{\rm C}\over{T^2}} + ...}$. | A series expansion can again be applied: $R(T) \sim {\rm e}^{{\rm A} + {{\rm B}\over{T}} + {{\rm C}\over{T^2}} + ...}$. | ||
| - | However, often only $B$ is given. \\ By taking the ratio of any temperature $T$ and $T_{25}=298.15~{\rm K}$ ($\hat{=} 25~°{\rm C}$) we get: | + | However, often only $B$ is given, for example {{electrical_engineering_1: |
| - | ${{R(T)}\over{R_{25}}} = {{exp \left({{\rm B}\over{T}}\right)} \over {{\rm exp} \left({{\rm B}\over{298.15 ~{\rm K}}}\right)}} $ with $R_{25}=R(T_{25})$ | + | ${{R(T)}\over{R_{25}}} = {{{\rm exp} \left({{\rm B}\over{T}}\right)} \over {{\rm exp} \left({{\rm B}\over{298.15 ~{\rm K}}}\right)}} $ with $R_{25}=R(T_{25})$ |
| This allows the final formula to be determined: | This allows the final formula to be determined: | ||
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| <panel type=" | <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 $0.20~{\rm µm}$? | + | 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>. | The resistivity is given by <tabref tab04>. | ||
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| <panel type=" | <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 $d_i=70~{\rm mm}$ and an outer diameter of $d_a = 120~{\rm mm}$. The number of turns is $n_W=1350$ turns, the wire diameter is $d=2.0~{\rm mm}$ and the specific conductivity of the wire is $\kappa_{Cu}=56 \cdot 10^6 ~{{{\rm S}}\over{{\rm m}}}$. | + | 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. | First, calculate the wound wire length and then the ohmic resistance of the entire coil. | ||
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| The power supply line to a consumer has to be replaced. Due to the application, | The power supply line to a consumer has to be replaced. Due to the application, | ||
| - | * The old aluminium supply cable had a specific conductivity $\kappa_{Al}=33\cdot 10^6 ~{\rm {S}\over{m}}$ and a cross-section $A_{Al}=115~{\rm mm}^2$. | + | * 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 $\kappa_{Cu}=56\cdot 10^6 ~{\rm {S}\over{m}}$ | + | * The new copper supply cable has a specific conductivity $\kappa_{\rm Cu}=56\cdot 10^6 ~{\rm {S}\over{m}}$ |
| - | Which wire cross-section $A_{Cu}$ must be selected? | + | Which wire cross-section $A_{\rm Cu}$ must be selected? |
| </ | </ | ||
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| < | < | ||
| - | === Goal === | + | === Learning Objectives |
| After this lesson you should be able to: | After this lesson you should be able to: | ||
| - Be able to calculate the electrical power and energy across a resistor. | - Be able to calculate the electrical power and energy across a resistor. | ||
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| What is the maximum current that can be measured? | 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: | The formulas $R = {{U} \over {I}}$ and $P = {U} \cdot {I}$ can be combined to get: | ||
| \begin{align*} | \begin{align*} | ||
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| \end{align*} | \end{align*} | ||
| - | # | + | # |
| - | # | + | # |
| \begin{align*} | \begin{align*} | ||
| I = 1.118... ~{\rm A} \rightarrow I = 1.12 ~{\rm A} | I = 1.118... ~{\rm A} \rightarrow I = 1.12 ~{\rm A} | ||
| \end{align*} | \end{align*} | ||
| - | # | + | # |
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| This process is also reversible: When cooled down, the conducting paths get re-connected. | This process is also reversible: When cooled down, the conducting paths get re-connected. | ||
| These components are also called **polymer positive temperature coefficient** components or PPTC. \\ | These components are also called **polymer positive temperature coefficient** components or PPTC. \\ | ||
| - | In the diagram below the internal structure and the resistance over the temperature are shown (more details about the structure and function can be found [[https:// | + | In the diagram below the internal structure and the resistance over the temperature are shown (more details about the structure and function can be found __ BROKEN-LINK: |
| {{drawio> | {{drawio> | ||
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| In the given circuit below, a fuse $F$ shall protect another component shown as $R_\rm L$, which could be a motor or motor driver for example. | In the given circuit below, a fuse $F$ shall protect another component shown as $R_\rm L$, which could be a motor or motor driver for example. | ||
| In general, the fuse $F$ can be seen as a (temperature variable) resistance. | In general, the fuse $F$ can be seen as a (temperature variable) resistance. | ||
| - | The source voltage $U_\rm S$ is $50~{\rm V}$ and $R_L=250~\Omega$. | + | The source voltage $U_\rm S$ is $50~{\rm V}$ and $R_{\rm L}=250~\Omega$. |
| {{drawio> | {{drawio> | ||
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| For this fuse, the component " | For this fuse, the component " | ||
| When this fuse trips, it has to carry nearly the full source voltage and dissipates a power of $0.8~{\rm W}$. | When this fuse trips, it has to carry nearly the full source voltage and dissipates a power of $0.8~{\rm W}$. | ||
| - | * First assume that the fuse is not blown. The resistance of the fuse at this is $1~\Omega$, which is negligible compared to $R_L$. What is the value of the current flowing through $R_L$? | + | * First assume that the fuse is not blown. The resistance of the fuse at this is $1~\Omega$, which is negligible compared to $R_{\rm L}$. What is the value of the current flowing through $R_{\rm L}$? |
| * Assuming for the next questions that the fuse has to carry the full source voltage and the given power is dissipated. | * Assuming for the next questions that the fuse has to carry the full source voltage and the given power is dissipated. | ||
| * Which value will the resistance of the fuse have? | * Which value will the resistance of the fuse have? | ||
| * What is the current flowing through the fuse, when it is tripped? | * What is the current flowing through the fuse, when it is tripped? | ||
| - | * Compare this resistance of the fuse with $R_L$. Is the assumption, that all of the voltage drops on the fuse feasible? | + | * Compare this resistance of the fuse with $R_{\rm L}$. Is the assumption, that all of the voltage drops on the fuse feasible? |
| </ | </ | ||