Circuit Theory/Series Resistance

Series Resistance
Two or more resistor can be connected in series to increase the total resistance. The Total Resistance is equal to the sum of all the resistor's resistance. The total resistance In Series connected circuit Current and voltage will be reduced
 * [[Image:Resistors_in_Series.svg|200px]]
 * $$R_{tot}=R_1+R_2+R_3+...+R_n \,$$

Series Impedance


A branch is defined as any group of resistors, capacitors and inductors that can be circled with only two wires crossing the circle boundary.

A branch connects two, non-trivial nodes or junctions.

Within a branch the components are said to be "in series."

Consider a branch containing a Resistor, Capacitor and Inductor.

Say the driving function or source is
 * $$V_s = 10*cos(22400t+30^\circ)$$
 * $$V_s = 10*e^{-5000t}cos(22400t+30^\circ)$$

There is just one current, $$I$$.

Symbolic Derivation
The terminal equations are:
 * $$\mathbb{V}_r = \mathbb{I}*R$$
 * $$\mathbb{V}_L = \mathbb{I}*j\omega L$$ or $$\mathbb{V}_L = \mathbb{I}*sL$$
 * $$\mathbb{I} = \mathbb{V}_c *j\omega C$$ or $$\mathbb{I} = \mathbb{V}_c *sC$$

There are no junction equations and the loop equation is:
 * $$\mathbb{V}_r + \mathbb{V}_L + \mathbb{V}_c - \mathbb{V}_s = 0$$

Solving the terminal equations for voltage, substituting and then dividing by $$\mathbb{I}$$ yields:
 * $$\frac{\mathbb{V}_s}{\mathbb{I}} = R + j\omega L + \frac{1}{j\omega C}$$
 * $$\frac{\mathbb{V}_s}{\mathbb{I}} = R + sL + \frac{1}{sC}$$

In terms of impedance, if:
 * $$Z = R + j\omega L + \frac{1}{j\omega C}$$
 * $$Z = R + sL + \frac{1}{sC}$$

Then:
 * $$\frac{\mathbb{V}_s}{\mathbb{I}} = Z$$

In general, impedances add in series like resistors do in the time domain:
 * $$ Z = \sum R_i + \sum j\omega L_i + \sum\frac{1}{j\omega C_i}$$
 * $$ Z = \sum R_i + \sum sL_i + \sum\frac{1}{sC_i}$$



Numeric Example
In rectangular form:
 * $$Z = 100 + .15714j$$
 * $$Z = 80.508 + 2.2759j$$

In polar form (remember impedance is not a phasor, it is a concept in the phasor or complex frequency domain):
 * $$Z = 100.0001\angle 0.0016 (0.09^\circ)$$
 * $$Z = 80.5402\angle 0.0283 (1.6193^\circ)$$