# Electric Circuits

## Basic Circuits

Basic circuits are composed of energy sources (eg. batteries) linked to resistors. Ohm's Law (V = IR) states that Voltage (potential difference) divided by the total resistance in a circuit equals the total current running through any hypothetical circuit. Although this is a seemingly difficult concept to relate to, this concept can be compared to pressure in a pipe. The pressure (like the voltage) provides the driving force for a fluid to flow through a pipe, while the viscosity of a fluid (essentially resistance) impedes the flow. The combination of the pressure and viscosity (among other factors) contribute to the flow rate of the liquid (essentially the current)!

## RLC Circuits

Many of the common questions with circuits have to do with an RLC (or Resistor (R), Inductor (L) and Capacitor (C)) circuit. The circuit contains these 3 elements in series (the order doesn't matter) and it has several fundamental properties.

The undamped (R = 0) resonant frequency of the system is defined as: w_0 = (LC)^(-1/2)

w_0 (omega) is the resonant frequency, in radians / second

L is the inductance, in Henries

C is the capacitence, in Farads

Essentially, this tells you at what frequency the system oscillates (between the energy stored in the Capacitor and the energy stored in the conductor. It is also the frequency at which the current in the system oscillates).

If you have an RLC circuit that is charged at some point, then it will slowly uncharge, and that rate that it will uncharge is given by: w' = (w_0^2 - (R/2L)^2)^(1/2)

w' is the new frequency, in radians/second

w_0 is the old frequency (shown above), in radians/second

R is the resistance, in Ohms L is the inductance, in Henries

The full equation for the current in the circuit (which can be used to get the current as a function of time, which leads to many other elements) is: I = I_0 * e^(-(R * t)/(2 * L)) cos(w' * t + phi)

I is the current at time **t**, in Amps I_0 is a constant scaling factor, in Amps R is the resistance, in Ohms t is the time, in seconds L is the inductance, in Henries w' is the angular frequency (as defined above), in radians/second phi is a phase constant, in radians

The constants I_0 and phi describe the amplitude and the phase shift of the current, respectively. the e^(-(R * t)/(2 * L))factor is an exponential decay, showing that the current oscillates, but rapidly approaches zero with increasing time.

## CAPACITORS

A capacitor is an electrical/electronic device that can store energy in the electric field between a pair of conductors (called "plates"). The process of storing energy in the capacitor is known as "charging", and involves electric charges of equal magnitude, but opposite polarity, building up on each plate. Parallel Capacitors:

Capacitors in a parallel configuration each have the same potential difference (voltage). Their total capacitance (Ceq) is given by:

Ceq = C1+C2+...+Cn

The reason for putting capacitors in parallel is to increase the total amount of charge stored. In other words, increasing the capacitance also increases the amount of energy that can be stored. Its expression is:

E=0.5*C*V^2

Series Capacitors:

The current through capacitors in series stays the same, but the voltage across each capacitor can be different. The sum of the potential differences (voltage) is equal to the total voltage. Their total capacitance is given by:

1/Ceq = 1/C1+1/C2+...+1/Cn Electric Circuits for Engineers:

These pages are useful for engineers: [Modified Nodal Analysis] [Y-delta transform] [Equivalent resistance]

• Thevenin and Norton Equivalent**
```A Thevenin circuit is a circuit with a voltage source with a series resistance. A Norton circuit is one with a current source and a parallel resistance. We are often asked to convert a circuit from one form to another. The method of doing so is by Ohm's Law V = IR.
```
• Q* A circuit has a voltage source of 2V in series with two resistors, one with 100 ohms and the other with 300 ohms. The terminals of the circuit come out from the two sides of the 300 ohms resistor. Convert into Norton equivalent, and then to the Thevenin equivalent of this Norton circuit.
• A* We need to move the 100-ohm in parallel to the 300-ohm. First, we calculate the current through the 100-ohm. I = V/R = 2/100 = 20mA. This is the current of the current source. Then we find the equivalent resistance of the two parallel resistors: 100//300 = 100*300/(100 + 300) = 75 ohms. So the Norton equivalent has a 20mA source and a 75-ohm resistance in parallel.

To convert this into Thevenin circuit (note that the answer is different from the original circuit described in the question because the Thevenin and the Norton circuits are in their simplest form), we calculate the voltage by V = IR = 0.02*75 = 1.5V. The Thevenin circuit is thus a 1.5V source in series with a 75-ohm resistance.