Electrical Circuitsadvmath.docx

Electrical Circuits Electrical circuit is an interconnection of electrical network elements, such as resistances, capacitances, inductances, independent and dependent voltages and current sources. Each network element is associated with two variables: the voltage variable v(t ) and the current variable i(t ). The determination of voltages and current associated with the elements of an electric circuit using numerical methods with the application of the governing laws of electrical circuit or the two postulates of circuit theory – the kirchoff’s voltage and current laws and of the other laws such as Ohm’s law, Faraday’s law and lenz’s law enables the method of solving tough problem in a simpler computational method. a. Kirchhoff’s Laws: 1. The sum of all voltage changes around any closed loop is zero 2. The sum of all currents at any node is zero. b. Ohm’s law: The voltage drop ER across a resistor is proportional to the current I passing through the resistor. ER = RI The proportionality constant R is called the resistance. c. Faraday’s law and Lenz’s law: The voltage drop EL across an inductor is proportional to the instantaneous rate of 𝑑𝐼 change of the current I. EL = L 𝑑𝑡 the proportionality constant L is called the inductance. An RLC circuit is a circuit having only constant resistors, capacitors, and inductors as elements and an electromotive driving force E(t). The current i (t) and charge q(t) are related by i(t)=q’(t). The voltage drop across a resistor having resistance R is iR, the drop across a capacitor having capacitance C is q/C, and the drop across an inductor having inductance L is Li’. The capacitance C(in Farads), resistance R(in ohms), and inductance L(in Henrys) are all constants.

Differential equations can be constructed for circuits by using Kirchhoff’s law. Now according to Kirchhoff's second law, the impressed voltage E(t) on a closed loop must equal the

sum of the voltage drops in the loop. This means that the total current entering the junction must balance the current leaving it (conservation of energy). The voltage law states that the algebraic sum of the potential rises and drops around any closed loop in a circuit is zero. Since current i(t) is related to charge q(t) on the capacitor by i = dq/dt, by adding the voltage drops on each element where: 𝑑2 𝑞

𝑑𝑖

Inductor: 𝐿 𝑑𝑡 = 𝐿 𝑑𝑡 2

Resistor: iR =R

𝑑𝑞 𝑑𝑡

1

Capacitor: 𝐶 𝑞 Equating the sum to the impressed voltage, a second-order differential equation can be obtained. 𝑑2 𝑞

𝐿 𝑑𝑡 2 + R

𝑑𝑞 𝑑𝑡

1

+ 𝐶 𝑞 = E(t)

This differential equation can also be applied on RC, LC, and RL circuits.