184_notes:examples:week6_node_rule

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184_notes:examples:week6_node_rule [2018/02/03 22:29] – tallpaul184_notes:examples:week6_node_rule [2021/06/08 00:51] (current) – schram45
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 +[[184_notes:current|Return to current in wires]]
 +
 =====Example: Application of Node Rule===== =====Example: Application of Node Rule=====
 Suppose you have the circuit below. You are given a few values: $I_1=8 \text{ A}$, $I_2=3 \text{ A}$, and $I_3=4 \text{ A}$. Determine all other currents in the circuit, using the [[184_notes:current#Current_in_Different_Parts_of_the_Wire|Current Node Rule]]. Draw the direction of the current as well. Suppose you have the circuit below. You are given a few values: $I_1=8 \text{ A}$, $I_2=3 \text{ A}$, and $I_3=4 \text{ A}$. Determine all other currents in the circuit, using the [[184_notes:current#Current_in_Different_Parts_of_the_Wire|Current Node Rule]]. Draw the direction of the current as well.
-{{ 184_notes:6_nodeless.png?300 |Circuit}}+[{{ 184_notes:6_nodeless.png?300 |Circuit}}]
  
 ===Facts=== ===Facts===
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 ===Representations=== ===Representations===
-  * For simplicity of discussion, we label the nodes in an updated representation: +For simplicity of discussion, we label the nodes in an updated representation: 
-{{ 184_notes:6_nodes.png?300 |Circuit with Nodes}}+[{{ 184_notes:6_nodes.png?300 |Circuit with Nodes}}] 
 + 
 +<WRAP TIP> 
 +===Assumption=== 
 +We will assume we have a perfect battery to supply a steady current to the circuit and will not die over time. 
 +</WRAP>
  
 ====Solution==== ====Solution====
 +Okay, there is a lot going on with all these nodes. Let's make a plan to organize our approach.
 +<WRAP TIP>
 +=== Plan ===
 +Take the nodes one at a time. Here's the plan in steps:
 +  * Look at all the known currents attached to a node.
 +  * Assign variables to the unknown currents attached to a node.
 +  * Set up an equation using the Node Rule. If not sure about whether a current is going in or coming out of the node, guess.
 +  * Solve for the unknown currents.
 +  * If any of them are negative, then we guessed wrong two steps ago. We can just flip the sign now.
 +  * Repeat the above steps for all the nodes.
 +</WRAP>
 +
 Let's start with node $A$. Incoming current is $I_1$, and outgoing current is $I_2$. How do we decide if $I_{A\rightarrow B}$ is incoming or outgoing? We need to bring it back to the Node Rule: $I_{in}=I_{out}$. Since $I_1=8 \text{ A}$ and $I_2=3 \text{ A}$, we need $I_{A\rightarrow B}$ to be outgoing to balance. To satisfy the Node Rule, we set Let's start with node $A$. Incoming current is $I_1$, and outgoing current is $I_2$. How do we decide if $I_{A\rightarrow B}$ is incoming or outgoing? We need to bring it back to the Node Rule: $I_{in}=I_{out}$. Since $I_1=8 \text{ A}$ and $I_2=3 \text{ A}$, we need $I_{A\rightarrow B}$ to be outgoing to balance. To satisfy the Node Rule, we set
 $$I_{A\rightarrow B} = I_{out}-I_2 = I_{in}-I_2 = I_1-I_2 = 5 \text{ A}$$ $$I_{A\rightarrow B} = I_{out}-I_2 = I_{in}-I_2 = I_1-I_2 = 5 \text{ A}$$
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 $$I_{D\rightarrow battery} = I_{out} = I_{in} = I_{B\rightarrow D}+I_{B\rightarrow D} = 8 \text{ A}$$ $$I_{D\rightarrow battery} = I_{out} = I_{in} = I_{B\rightarrow D}+I_{B\rightarrow D} = 8 \text{ A}$$
  
-Notice that $I_{D\rightarrow battery}=I_1$. This will always be the case for currents going in and out of the battery (approximating a few things that are usually safe to approximate, such as a steady current). In fact, we could have treated the battery as another node in this example. Notice also that if you incorrectly reason about the direction of a current (incoming or outgoing), the calculation will give a negative number for the current. The Node Rule is self-correcting. A final diagram with directions is shown below.+Notice that $I_{D\rightarrow battery}=I_1$. This will always be the case for currents going in and out of the battery (approximating a few things that are usually safe to approximate, such as a steady current). In fact, we could have treated the battery as another node in this example. Notice also that if you incorrectly reason about the direction of a current (incoming or outgoing), the calculation will give a negative number for the current. The Node Rule is self-correcting. A final representation with directions is shown below.
  
-{{ 184_notes:6_nodes_with_arrows.png?300 |Circuit with Nodes}}+[{{ 184_notes:6_nodes_with_arrows.png?300 |Circuit with Nodes}}]
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