# Circuit Theory/Parallel Resistance

## Parallel Resistance

Resistors in parallel share the same voltage. They split the current up.
Giving the current multiple paths to follow means that the overall
resistance decreases.

!A diagram of several resistors, side by side, both leads of each
connected to the same
wires.

$$R_\mathrm{total} =\frac{1}{ \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n}} = \frac{1}{\sum\frac{1}{R_i}}$$

The above equation is often called the \"one over \-- one over\"
equation.

## Parallel Impedance

!used to show how impedance add\'s in
parallel

A group of parallel branches split up the current, but share the same
voltage. The parallel branches connect the same two nodes.

The impedance of parallel branches can be combined into one impedance.

Consider these parallel branches: one with a resistor, another with a
capacitor and a third with an inductor.

Now drive them with a voltage source:

$$V_s = 10*cos(22400t+30^\circ)$$

$$V_s = 10*e^{-5000t}cos(22400t+30^\circ)$$

There is just one current, $V$.

## symbolic derivation

!the impedance symbol is .. a
box !Parallel
Impedance calculation with matlab
..m-file
The terminal equations are:

$$\mathbb{V}_s = \mathbb{I}_r*R$$

$$\mathbb{V}_s = \mathbb{I}_L*j\omega L$$ or
$\mathbb{V}_s = \mathbb{I}_L*sL$

$$\mathbb{I}_c = \mathbb{V}_s *j\omega C$$ or
$\mathbb{I}_c = \mathbb{V}_s *sC$ There are no loop equations and the
junction equation is:

$$\mathbb{I}_r + \mathbb{I}_L + \mathbb{I}_c - \mathbb{I}_s = 0$$
Solving the terminal equation for currents, substituting and dividing
both sides by $\mathbb{V}_s$ yields:

$$\frac{\mathbb{I}_s}{\mathbb{V}_s} = \frac{1}{R} + \frac{1}{j\omega L} + j\omega C$$

$$\frac{\mathbb{I}_s}{\mathbb{V}_s} = \frac{1}{R} + \frac{1}{sL} + sC$$
In terms of impedance, if:

$$\frac{1}{Z} = \frac{1}{R} + \frac{1}{j\omega L} + j\omega C$$

$$\frac{1}{Z} = \frac{1}{R} + \frac{1}{sL} + sC$$ Then:

$$\frac{\mathbb{I}_s}{\mathbb{V}_s} = \frac{1}{Z}$$

In general, impedances add in series like resistors do in the time
domain:

$$\frac{1}{Z} = \sum\frac{1}{R_i} + \sum\frac{1}{j\omega L_i} + \sum j\omega C_i$$

$$\frac{1}{Z} = \sum\frac{1}{R_i} + \sum\frac{1}{sL_i} + \sum sC_i$$ or

$$Z = \frac{1}{\sum\frac{1}{R_i} + \sum\frac{1}{j\omega L_i} + \sum j\omega C_i}$$

$$Z = \frac{1}{\sum\frac{1}{R_i} + \sum\frac{1}{sL_i} + \sum sC_i}$$

## numeric example

$$Z = 99.9938 - 0.7857i = 99.9969\angle -0.0079 (-0.4502^\circ)$$

$$Z = 186.85 -837.07i = 857.6701\angle -1.3512 (-77.4170^\circ)$$

So the exact same components hooked in series are dominated by the
inductor (large reactance of the inductor overshadows the smaller
reactance of the capacitor), but when the same components are hooked in
parallel, the small er capacitive reactance dominates.

## notes

What is the point of all this? The goal is to start with simple
circuits, explore them with words and symbols, find the patterns and
generalize, establish and name concepts and then wait for their
application. What are we waiting for? Selecting channels on a
TV/Radio/Wifi, ZigBee, RFID, Cellphones, Blue
Tooth and sending data over these channels as well as to hard drives and
SD cards, monitors and hubs, designing antennas, \....

