# Foundations

## Voldate

Voldate represents the work per unit charge (J/C) required to move the charges from one point to the other.

## Resistance

Resistance is determined by the physical dimensions and the resistivity ρ of the material of which the resistor is composed. For a bar of resistive material of length l and cross-sectional area A the resistance is given by $$R=\frac{\rho l}{A} =\frac{l}{\sigma A}$$ ρ [Ω·m] is the resistivity of the material, and σ [S/m] is the conductivity of the material.

# Electric circuits

## Collection

In the electric circuits,there are three parts of electric devices.Source provide the power,conductor like wire will transfer electric to the load,and the load will transform electric to other forms.

## Circuit equations

An electric circuit may be described mathematically by ordinary differential equations.

All the linear equations are used to construct a line. A non-linear equation is such which does not form a straight line. It looks like a curve in a graph and has a variable slope value.

# Equivalent circuit

## Source

Two elements represents the transformation of electric energy from nonelectric energy. EMF source and current source.

## Resistance

Resistance is an equivalent circuit element which represents the transformation of electric energy into nonelectric energy.
Resistance(R)$$R=\frac{du}{di}$$ Conductance(g)$$g=\frac{di}{du}$$

## Inductor

An ideal inductor is an energy-storage circuit element (with no loss associated with it) representing the magnetic-field effect. $$L=\frac{d\psi }{di}$$ $$i(t)=\frac{1}{L}\int u(t)dt$$

## Capacitor

An ideal capacitor is an energy-storage circuit element (with no loss associated with it) representing the electric-field effect. $$C=\frac{dq}{du}$$ $$u(t)=\frac{1}{C}\int i(t)dt$$

When elements have their own signs,if the current flow is opposite to it the whole power is on the contrary.

# Circuit topology

## Series connection

$$R_{eq}=R_{1}+R_{2}+R_{3}+R_{4}+R_{5}$$ $$L_{eq}=L_{1}+L_{2}+L_{3}$$ $$E_{eq}=E_{1}+E_{2}+E_{3}-E_{4}+E_{5}$$ $$g_{eq1}=g_{2}+g_{3}\qquad g_{eq2}=g_{eq1}+g_{1}$$ $$R_{eq1}=\frac{R_{2}R_{3}}{R_{2}+R_{3}}\qquad R_{eq2}=\frac{R_{1}R_{eq1}}{R_{1}+R_{eq1}}$$ $$\frac{1}{C_{eq}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}+\frac{1}{C_{3}}$$ $$J_{eq}=J_{1}+J_{2}-J_{3}$$

## Node & Junction

A node is a point at which two or more components or devices are connected together. The node can be called a junction point in the case of connecting of three or more wires.

## Branch & Loop

A branch is a part of a circuit containing one element or several elements connected in series.Actually a element with two terminals can be seen as a branch. A loop is a closed path formed by starting at a node, passing through a set of nodes, and returning to the starting node without passing through any node more than once.

## Star and delta connections

\left\{ \begin{aligned} & R_a&= & \frac{R_{ab}R_{ac}}{R_{ab}+R_{ac}+R_{bc}}&\\ & R_b&= & \frac{R_{ba}R_{bc}}{R_{ab}+R_{ac}+R_{bc}}&\\ & R_c&= & \frac{R_{cb}R_{ca}}{R_{ab}+R_{ac}+R_{bc}}&\\ \end{aligned} \right. \Leftrightarrow \left\{ \begin{aligned} & R_{ab}&= & \frac{R_aR_b+R_aR_c+R_bR_c}{R_c}&\\ & R_{ac}&= & \frac{R_aR_b+R_aR_c+R_bR_c}{R_b}&\\ & R_{bc}&= & \frac{R_aR_b+R_aR_c+R_bR_c}{R_a}&\\ \end{aligned} \right.