Kirchoff's Laws

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  • Topic: Voltage drop, Electrical impedance, Kirchhoff's circuit laws
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  • Published : December 5, 2012
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Kirchhoff’s Laws
Gustav Kirchhoff, (1824-1887), was a German physicist who contributed greatly to the understanding of electrical circuits, spectroscopy, and the emission of black-body radiation. Kirchhoff formulated his famous circuit laws in 1845, while still a student at the University of Konigsberg, East Prussia. He completed this study as a seminar exercise, and it later became his doctoral dissertation.

Kirchhoff’s Laws include two basic principles. Kirchhoff’s First Law, also called Kirchhoff’s Current Law (KCL), states that at any node (junction) in an electrical circuit, the sum of all currents (Ik) flowing into that node is equal to the sum of currents flowing out of that node. In other words, the algebraic sum of all currents in a network of conductors meeting at a point is zero (Eq 1). (Eq 1)

Kirchhoff’s Second Law, also called Kirchhoff’s Voltage Law (KVL), states that the sum of the voltage gains in any closed loop is equivalent to the sum of the voltage drops in that loop. In other words, the algebraic sum of the voltages (Vk) in a closed loop is equal to zero (Eq 2). (Eq 2)

Equations (1) and (2) are interrelated, and this can be demonstrated through modeling a basic electrical circuit involving a second order, linear differential equation. For our basic circuit, we will use a single closed loop involving one voltage source, one resistor, one capacitor, and one inductor. First, it is important to establish some basic electrical circuit properties. The current, I, is related to the charge, Q, of a voltage source, and is a function of time (Eq 3). The voltage drop across a resistor is IR (Eq 4), where R is the resistance value. The voltage drop across a capacitor is Q/C (Eq 5), where C is the capacitance value. The voltage drop across an inductor is L(dI/dt) (Eq 6), where L is the inductance value. Also, the impressed voltage, E, is a function of time, and by Kirchhoff’s Second Law is equal to the sum of the voltage drops in...
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