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electrical_engineering_1:simple_circuits [2023/03/19 19:04] – mexleadminelectrical_engineering_1:simple_circuits [2024/10/24 08:13] (aktuell) – mexleadmin
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-====== 2. Simple DC circuits ======+====== 2 Simple DC circuits ======
  
 So far, only simple circuits consisting of a source and a load connected by wires have been considered. \\  So far, only simple circuits consisting of a source and a load connected by wires have been considered. \\ 
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 {{youtube>VojwBoSHc8U}} {{youtube>VojwBoSHc8U}}
 </WRAP> </WRAP>
 +\\ \\
 +The current divider rule shows in which way an incoming current on a node will be divided into two outgoing branches.
 +The rule states that the currents $I_1, ... I_n$ on parallel resistors $R_1, ... R_n$ behave just like their conductances $G_1, ... G_n$ through which the current flows. \\
  
-The current divider rule can also be derived from Kirchhoff's current law. \\ +$\large{{I_1}\over{I_{\rm res}} = {{G_1}\over{G_{\rm res}}}$ 
-This states that, for resistors $R_1, ... R_n$ their currents $I_1, ... I_n$ behave just like the conductances $G_1, ... G_n$ through which they flow. \\ +
- +
-$\large{{I_1}\over{I_g}} = {{G_1}\over{G_g}}$ +
  
 $\large{{I_1}\over{I_2}} = {{G_1}\over{G_2}}$ $\large{{I_1}\over{I_2}} = {{G_1}\over{G_2}}$
  
-This can also be derived by Kirchhoff's current law: The voltage drop $U$ on parallel resistors $R_1, ... R_n$ is the same. When $U_1 = U_2 = ... = U$, then also $R_1 \cdot I_1 = R_2 \cdot I_2 = ... = R_{\rm eq} \cdot I_{\rm res}$. \\ +The rule also be derived from Kirchhoff's current law: \\ 
-Therefore, we get with the conductance: ${{I_1} \over {G_1}} = {{I_2} \over {G_2}}= ... = {{I_{\rm eq}} \over {G_{\rm res}}}$+  - The voltage drop $U$ on parallel resistors $R_1, ... R_n$ is the same.  
 +  - When $U_1 = U_2 = ... = U$, then the following equation is also true: $R_1 \cdot I_1 = R_2 \cdot I_2 = ... = R_{\rm eq} \cdot I_{\rm res}$. \\ 
 +  - Therefore, we get with the conductance: ${{I_1} \over {G_1}} = {{I_2} \over {G_2}}= ... = {{I_{\rm eq}} \over {G_{\rm res}}}$
  
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
  
  
 +<wrap anchor #exercise_2_4_1 />
 <panel type="info" title="Exercise 2.4.1 Current divider"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> <panel type="info" title="Exercise 2.4.1 Current divider"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%>
  
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 __In general__: The equivalent resistance of a series circuit is always greater than the greatest resistance. __In general__: The equivalent resistance of a series circuit is always greater than the greatest resistance.
 +
 +==== Application ====
 +
 +=== Kelvin-Sensing ===
 +
 +Often resistors are used to measure a current $I$ via the voltage drop on the resistor $U = R \cdot I$. Applications include the measurement of motor currents in the range of $0.1 ... 500 ~\rm A$. \\
 +Those resistors are called //shunt resistors// and are commonly in the range of some $\rm m\Omega$.
 +This measurement can be interfered by the resistor of the supply lines.
 +
 +To get an accurate measurement often Kelvin sensing, also known as {{wp>Four-terminal sensing}} or 4 wire sensing, is used.
 +This is a method of measuring electrical resistance avoiding errors caused by wire resistances. \\
 +The simulation in <imgref BildNr005> shows such a setup.
 +
 +Four-terminal sensing involves using:
 +  * a pair of //current leads// or //force leads// (with the resistances $R_{\rm cl1}$ and $R_{\rm cl2}$) to supply current to the circuit and 
 +  * a pair of //voltage leads// or //sense leads// (with the resistances $R_{\rm vl1}$ and $R_{\rm vl2}$) to measure the voltage drop across the impedance to be measured. 
 +The sense connections via the voltage leads are made immediately adjacent to the target impedance $R_{\rm s}$ at the device under test $\rm DUT$. 
 +By this, they do not include the voltage drop in the force leads or contacts. \\
 +Since almost no current flows to the measuring instrument, the voltage drop in the sense leads is negligible. 
 +This method can be a practical tool for finding poor connections or unexpected resistance in an electrical circuit.
 +
 +<WRAP>
 +<imgcaption BildNr005 | Example of a circuit>
 +</imgcaption> \\
 +{{url>https://www.falstad.com/circuit/circuitjs.html?running=false&ctz=CQAgjCAMB0l3BWKsAsYBsAOdAmSYBOdAdkwUkxRB2JCRQGY6BTAWjDACgAnEIkFJnBgcAoZGRdeOBOmGiZcsCioSp86hT5y84yZwDuGwdVljqh00pVXN4y4rsbdFo444KUE-pAdnlVDhe4OgSvgBuIOzEOlrRciZhAmqiSTAIlqyYEi78Lr4ADlHZ5lk5WkxhmWAxTmVOONW1JvUmjUXecmqhIJVQPNqDHoNh0FxG-D4jA-wYaiIhaWOW3Z2L-UatQvyJDsFzGge+bsGzCz6cAB5RHIEoogy0OJi0VGBCAGoA9gA2AC4AW2Yf2Y3AAOgBnACWADtIWCwTCcKwDFDuMxIRDmDDoTCAOaQgCGMIAJgiYSgUWiMRCsTjYXirnwIEEFBxqOgEuAhKj0RDIai-gALeGI3k0vlQiF-YkAY2YTJwDCYrI5BGoKHo3JAABEAKoAFU4jxyZja-gWECqeJutRc8WoQSg-WuCCQDCQJF6QjeQgASkz0OqGOrKAJRG9RLKflxrkHNBIw3gmJGQNHGq6kIQ5F7COrfSAA3HQ7QwwQJKnwjGmW6QJgmF7sAJtUW6FmTLmEBAC6342AEKIw-2tQsqxmOUJ+yqSv2ue9C4rMKJ+09MJOeqmIZwgA noborder}}
 +</WRAP>
  
 ~~PAGEBREAK~~ ~~CLEARFIX~~ ~~PAGEBREAK~~ ~~CLEARFIX~~
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 \begin{align*} \begin{align*}
-R_g = R || R + R || R || R || R + R || R || R || R + R || R = {{1}\over{2}}\cdot R + {{1}\over{4}}\cdot R + {{1}\over{4}}\cdot R + {{1}\over{2}}\cdot R = 1.5\cdot R+R_{\rm eq} = R || R + R || R || R || R + R || R || R || R + R || R = {{1}\over{2}}\cdot R + {{1}\over{4}}\cdot R + {{1}\over{4}}\cdot R + {{1}\over{2}}\cdot R = 1.5\cdot R
   \end{align*}   \end{align*}
  
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-<panel type="info" title="other Exercises"> <WRAP group><WRAP column 2%>{{fa>pencil?32}}</WRAP><WRAP column 92%> 
  
-More German exercises can be found online on the pages of [[https://www.eit.hs-karlsruhe.de/hertz/teil-b-gleichstromtechnik/zusammenschaltung-von-widerstaenden-und-idealen-quellen/uebungsaufgaben-zusammenschaltung-von-widerstaenden/berechnung-von-ersatzwiderstaenden.html|HErTZ]] (selection on the left in the menu). 
-</WRAP></WRAP></panel>