resistive_circuits.md

Resistive Circuits

#Resistivity - Ohm's Law

The resistance of a material can be calculated as:

$$R = \rho \frac{\ell}{A} \quad \Omega $$

where \(\rho\) is the resistivity of the material, \(\ell\) is the length, \(A\) is the area. Remember to use SI units (m and m2) to get correect values.

Voltage across a resistor is proportional to current it is carrying.

$$Voltage = Current \times Resistance$$

$$V=IR$$

A short circuit is the case where the resistance between terminals is zero.

An open circuit is the case where the resistance between terminals is infinity. #Series Resistors The equivalent resistance of any number of resistors connected in series is the sum of the individual resistances.

$$R_{eq}=R_1+R_2+R_3 ... R_n$$

#Parallel Resitors The equivalent resistance of two parallel resistors is equal to the product of their resistances divided by their sum.

$$R_{eq}=\frac{R_1 R_2}{R_1 + R_2}$$

#Delta-Wye Transformations

Instead of two terminal equivalents, three terminal equivalents are used.

Delta Connections:

The following resistances are equal (Node C can be reduced to one point):

##Wye Connections:

Transformation

Delta to Wye

$$R_1=\frac{R_bR_c}{R_a + R_b + R_c}$$

$$R_2=\frac{R_aR_c}{R_a + R_b + R_c}$$

$$R_3=\frac{R_aR_b}{R_a + R_b + R_c}$$

Wye to Delta

$$R_a = \frac{R_1R_2 + R_2R_3+R_3R_1}{R_1}$$

$$R_b = \frac{R_1R_2 + R_2R_3+R_3R_1}{R_2}$$

$$R_c = \frac{R_1R_2 + R_2R_3+R_3R_1}{R_3}$$