Consider the circuit shown in Figure 7.1 .2 . Let I1, I1,...

Consider the circuit shown in Figure $7.1 .2 .$ Let $I_{1}, I_{1}, I_{2},$ and $I_{3}$ be the current through the capacitor, resistor, and inductor, respectively. Likewise, let $V_{1}, V_{2},$ and $V_{3}$ be the corresponding voltage drops. The arrows denote the arbitrarily chosen directions in which currents and voltage drops will be taken to be positive. (a) Applying Kirchhoff's second law the upper loop in the circuit, show that $$ V_{1}-V_{2}=0 $$ In a similar way, show that $$ V_{2}-V_{3}=0 $$ (b) Applying Kirchhoff's first law to either node in the circuit, show that $$ I_{1}+I_{2}+I_{3}=0 $$ (c) Use the current-voltage relation through each element in the circuit to obtain the equations $$ C V_{1}^{\prime}=I_{1}, \quad V_{2}=R I_{2}, \quad L I_{3}^{\prime}=V_{3} . $$ (d) Eliminate $V_{2}, V_{3}, I_{1},$ and $I_{2}$ among Eqs. (i) through (iv) to obtain $$ C V_{1}^{\prime}=-I_{3}-\frac{V_{1}}{R}, \quad L I_{3}^{\prime}=V_{1} $$ Observe that if we omit the subscripts in Eqs. (v), then we have the system (2) of the text.

Consider the circuit shown in Figure $7.1 .2 .$ Let $I_{1}, I_{1}, I_{2},$ and $I_{3}$ be the current through the capacitor, resistor, and inductor, respectively. Likewise, let $V_{1}, V_{2},$ and $V_{3}$ be the corresponding voltage drops. The arrows denote the arbitrarily chosen directions in which currents and voltage drops will be taken to be positive.
(a) Applying Kirchhoff's second law the upper loop in the circuit, show that
$$
V_{1}-V_{2}=0
$$
In a similar way, show that
$$
V_{2}-V_{3}=0
$$
(b) Applying Kirchhoff's first law to either node in the circuit, show that
$$
I_{1}+I_{2}+I_{3}=0
$$
(c) Use the current-voltage relation through each element in the circuit to obtain the equations
$$
C V_{1}^{\prime}=I_{1}, \quad V_{2}=R I_{2}, \quad L I_{3}^{\prime}=V_{3} .
$$
(d) Eliminate $V_{2}, V_{3}, I_{1},$ and $I_{2}$ among Eqs. (i) through (iv) to obtain
$$
C V_{1}^{\prime}=-I_{3}-\frac{V_{1}}{R}, \quad L I_{3}^{\prime}=V_{1}
$$
Observe that if we omit the subscripts in Eqs. (v), then we have the system (2) of the text.

Key Concepts

Kirchhoff's Voltage Law

This law states that the algebraic sum of all voltages in a closed loop in a circuit must be zero. It is used to analyze potential differences around loops in electrical circuits, ensuring that the energy supplied is balanced by the energy consumed around the loop.

Kirchhoff's Current Law

This law dictates that the total current entering a junction in a circuit must equal the total current leaving that junction. It is based on the conservation of charge and is crucial for analyzing how currents distribute at circuit nodes.

Capacitor Current-Voltage Relationship

In capacitors, the current is related to the time derivative of the voltage across the capacitor by the equation I = C(dV/dt), where C is the capacitance. This relationship is vital for understanding transient behaviors in circuits containing capacitors.

Resistor Ohm's Law

Ohm's Law establishes a linear relationship between voltage and current for resistors, given by V = IR, where R is the resistance. This fundamental principle is used to analyze steady-state and dynamic responses in circuits involving resistive elements.

Inductor Voltage-Current Relationship

For inductors, the voltage across the element is proportional to the rate of change of current flowing through it, expressed as V = L(dI/dt), where L is the inductance. This property is key to understanding the dynamic response in circuits with inductors.

System of Differential Equations in Circuit Analysis

Combining the individual element equations with Kirchhoff's laws yields a set of differential equations that describe the time-dependent behavior of the circuit. This system approach is essential for analyzing and predicting the transient responses in dynamic electrical circuits.

Transcript

00:01 Okay, so starting off with part 8, it wants us to apply kirkoff's second law to the upper loop.

00:06 So for the upper loop in the circuit, kirkoff's voltage law states that the sum of the potential differences around any closed loop is zero.

00:19 So for the first loop equation, if we trace the loop and apply kirkov's voltage law, we sum the voltage drops around the loop, which gives you the equation v1 minus v2 is equal to zero.

00:31 For the second loop equation, we apply kirkov's voltage law again, and we get v2 minus v3 equals zero.

00:41 So we have these two equations, v1 equals v2, and v2 is equal to v3.

00:50 For part b, we won't need to apply kirkov's first law to either node.

00:57 So kirkov's current law states that the sum of currents entering a node is equal to the sum of currents leaving the node.

01:04 So let's consider the node where the capacitor, resistor, and indicator are connected.

01:09 So we are tasked with the following i -2, sorry, i -1 plus i -2 plus i -3 equals 0.

01:19 So this means that the current through the capacitor, the current through the resistor, and the current through the indicator or inductor must sum to zero at the junction because there is no external current source or sync.

01:35 For part c, we need to determine the current voltage relations through each element.

01:41 So if we start off with the capacitor, the current through a capacitor is related to the rate of change of voltage across it by i1 is equal to c times dv1 over dt...

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