# 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

:   ![](Resistors_in_Series.svg "fig:Resistors_in_Series.svg"){width="200"}
:   $R_{tot}=R_1+R_2+R_3+...+R_n \,$

## Series Impedance

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

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 impedance symbol is .. a
box 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:

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

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

!matlab screen shot of numeric calculation ..
/m-file/

## 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):

$$Z = 100.0001\angle 0.0016 (0.09^\circ)$$

$$Z = 80.5402\angle  0.0283 (1.6193^\circ)$$

## Notes

-   Calculations like this are only necessary when trying to find out
    what is going on in one little part of the circuit \... and
    Kirchhoff would take too long. This was true in the past, but given
    the modern capabilities of matLab and mupad, the importance of the
    series concept may diminish.

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-   It appears that the exponential term decreases the overall magnitude
    and increases the reactance \... increases in the inductive
    direction with the exponential term.

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-   The V/I ratio sets up/prepares for the transfer function discussion
    later.

