Mathwizurd.com is created by David Witten, a mathematics and computer science student at Vanderbilt University. For more information, see the "About" page.

## Jan 12 Circuit RIVIP Process

Sample circuit I will be discussing

When calculating things about a circuit, a useful and efficient process to do (for simple circuits) is called the RIVIP process. It stands for:

• R: Calculate the total Equivalent Resistance
• I: Calculate the total current
• VI: Calculate the voltages to different branches
• P: Calculate power

So, the first step is to calculate the equivalent resistance. There are two formulas for equivalent resistance: parallel and series.

For series circuits, or the most simple kind of circuits, the equivalent resistance is simply the sum of the resistances.

For parallel circuits (right part of the circuit above), the resistance is the reciprocal of the sum of the reciprocals of the resistances. (shown below)

MathJax TeX Test Page Parallel $$\dfrac{1}{r_{eq}} = \frac1{r_1} + \frac1{r_1} + ... + \frac1{r_n}$$ $$V = V_1 = V_2 = ... = V_n$$ Voltage is the same in every parallel branch.

Series $$r_{eq} = r_1 + r_2 + ... r_n$$ $$V_{eq} = V_1 + V_2 + ... V_n$$ So, now we must plug that formula in. There are two sections: the series part and the parallel part. A good way to think of it is to imagine replacing the parallel part with one resistor, whose resistance would be $$\dfrac{1}{\dfrac{1}{2} + 1} = \dfrac{2}{3}$$ The series part equals $11 + 4$. The total resistance equals $$11 + 4 + \dfrac{2}{3} = \boxed{\dfrac{47}{3}}$$ Ohm's Law is $V = IR$. Voltage = $17$ V, resistance = $\dfrac{47}{3}$. Therefore, the current equals $\dfrac{51}{47}$ Amperes.

Now we go through and calculate the voltage for each part: $$\text{First resistor: } V = \left(\dfrac{51}{47}\right)4 = \boxed{4.34}$$ $$\text{Combined parallel resistor: } V = \left(\dfrac{51}{47}\right)\dfrac{2}{3} = \boxed{0.72}$$ $$\text{Last resistor: } V = \left(\dfrac{51}{47}\right)11 = \boxed{11.94}$$ Total = $4.34 + 0.72 + 11.94 = 17$, which is our original voltage.

Finally, we calculate power. Power = current * voltage, which equals $$\left(\dfrac{51}{47}\right)17 = \boxed{18.447 W}$$
David Witten