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    mHN=j9AH5CM)45+AqO3JE$8dgb3MFoigQ1Sw@|8Iz;fV@1Dm-kgq WZ+}ladnU^DXaLD1EUc!h&h&p|P#?hn diff --git a/Chapter1.tex b/Chapter1.tex index a5e0a7e..fa1634a 100644 --- a/Chapter1.tex +++ b/Chapter1.tex @@ -25,7 +25,7 @@ to[R=$R_2$,v=$V_{2}$] (2,0) to (0,0) \end{circuit} - By LVK and Ohm's law \[ V = V_{1} + V_{2} = R_{1}\cdot I + R_{2} \cdot I = (R_{1}+R_{2}) \cdot I = R \cdot I \] + By KVL and Ohm's law \[ V = V_{1} + V_{2} = R_{1}\cdot I + R_{2} \cdot I = (R_{1}+R_{2}) \cdot I = R \cdot I \] where \[\mans{R = R_{1} + R_{2}}\] is the resistance of $R_{1}$ and $R_{2}$ in series. Now, consider a simple parallel resistor circuit. \begin{circuit}{fig:1.3.2}{A basic parallel circuit.} @@ -36,7 +36,7 @@ to[R=$R_2$,i>^=$I_{2}$] (4,0) to (0,0) \end{circuit} - By LCK and Ohm's law \[ I = I_{1} + I_{2} = \frac{V}{R_{1}} + \frac{V}{R_{2}} = \left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right)\cdot V \] + By KCL and Ohm's law \[ I = I_{1} + I_{2} = \frac{V}{R_{1}} + \frac{V}{R_{2}} = \left(\frac{1}{R_{1}}+\frac{1}{R_{2}}\right)\cdot V \] solving for V as a function of I we get \[V = \dfrac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}}\cdot I = \frac{R_{1}R_{2}}{R_{1}+R_{2}}\cdot I = R\cdot I \] where \[\mans{R = \dfrac{1}{\frac{1}{R_{1}}+\frac{1}{R_{2}}} = \frac{R_{1}R_{2}}{R_{1}+R_{2}}}\] is the resistance of $R_{1}$ and $R_{2}$ in parallel.