# Resistance, Capacitance, Inductance, Impedance and Reactance

This is another post about basic concepts of electricity. This time I will talk about resistors, capacitors, inductors in CC and AC.

### Resistance

Resistance is a value that measures how much the component “resist” the passage of electrical current, the value is measured in ohms (Ω). One way to calculate resistance:

$R=\rho \frac{L}{A}$

• $\rho$ is the resistivity, a material’s property.

Another form to calculate the resistance is applying Ohm’s law.

$R=\frac{V}{I}$

• $V$ is the voltage and $I$ is the current.

This is the resistor, the component with a defined resistance and the resistor’s color code.

Resistors in series, the resistance is summed.

$R_{total}=R1+R2$

Resistors in parallel:

$\frac{1}{R_{total}}=\frac{1}{R1}+\frac{1}{R2}$

In AC circuits with very high frequency, the resistance even in resistors varies, passive components in high frequency stays to another post.

### Capacitance

Capacitance is the capacity to store energy in a capacitor, is measured in farads (F), these are capacitors.

Capacitance is calculated in this form:

$C=\frac{Q}{V}$

• $Q$ is the charge and $V$ is the voltage.

The capacitance in a capacitor with parallel plates.

$C=\epsilon \frac{A}{d}$

• $\epsilon$ is the electric permissiveness;
• $A$ is the plate area;
• $d$ is the distance between plates.

Capacitance in a cylindrical capacitor.

$C=2\pi \epsilon \frac{L}{ln(\frac{b}{a})}$

• $L$ is the cylinder’s length.

Capacitance in a spherical capacitor.

$C=4\pi \epsilon \frac{ab}{b-a}$

Capacitance in a isolated sphere.

$C=4\pi \epsilon R$

Capacitors in series:

$\frac{1}{C_{total}}=\frac{1}{C_{1}}+\frac{1}{C_{2}}$

Capacitors in parallel:

$C_{total}=C_{1}+C_{2}$

### Inductance

While the capacitor stores energy in an electrical field, the inductor stores energy in a magnetic field. Inductance is the inductor’s capacity to resist variation of electric current and is measured in henries (H). The inductor is nothing more than a rolled wire in spirals which can have a nucleus inside to increase the magnetic field and the inductance. Here are various types of inductors. Sometimes inductors are called solenoids.

The formula to calculate inductance in a solenoid:

$L=\frac{\mu N^{2}A}{l}$

• $\mu$ is the magnetic permeability;
• $N$ is the number of spires or turns in the inductor;
• $A$ is the section area;
• $l$ is the length.

The equation to calculate the inductance of a toroid, ring-shaped coil. With a square cross section.

$L=\frac{\mu N^{2}h}{2\pi }ln\frac{b}{a}$

If the cross section area is circular.

The inductor’s association in series and parallel is equal to the resistors and the total inductance is calculated in the same way.

### Impedance and Reactance

In the alternated current, the value of resistance in the passive components (resistor, capacitor, and inductor) is called impedance, which is formed by reactances. In the resistor, the impedance is equal to the resistance value in the CC. In capacitors and inductors, the reactance is an imaginary number and are called respectively capacitive reactance and inductive reactance.

Capacitive reactance.

$X_{c}=\frac{1}{\omega C}$

• $C$ is the capacitance and $\omega$ is the circuit’s frequency in radians/s.

Inductive reactance.

$X_{l}=\omega L$

This graphic shows the impedance as Z, reactances as $X_{c}$ and $X_{l}$ in the imaginary axis and the resistance in the real numbers axis.

Impedances of capacitor and inductor are respectively.

$Z_{c}=-j\frac{1}{\omega C}=\frac{1}{j\omega C}$

$Z_{l}=j\omega L$

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