For the circuit in Fig. 8.98, find v(t) for t > 0. Assume that v(0^+)=4 V and i(0^+)=2 A.

Name: Problem 49, Chapter 8: Fundamentals of Electric Circuits
Uploaded: 2022-02-25T19:16:38-08:00
Duration: 4 min 17 s
Channel: Kajal Gautam
Description: For the circuit in Fig. 8.98, find v(t) for t > 0. Assume that v(0^+)=4 V and i(0^+)=2 A.

For the circuit in Fig. 8.98, find v(t) for t

Key Concepts

Exponential Response of First-Order Circuits

In many transient analyses, the response of first-order circuits can be described by an exponential function. This exponential behavior captures how the voltage or current in the circuit transitions from its initial value to its final steady-state value over time, following a form that typically includes a decay factor related to the time constant.

Time Constant

The time constant is a parameter that characterizes the speed at which a circuit's response decays or rises towards its steady state. In first-order circuits, which commonly arise in transient analysis, the time constant is determined by the circuit elements such as resistance and capacitance (or inductance). It provides a measure of how quickly the transient response diminishes over time.

Transient Response

Transient response refers to the part of a circuit’s behavior that changes over time immediately after a sudden change in circuit conditions, such as switching. It is the response of the circuit that eventually dies out as the stored energy in reactive components (capacitors and inductors) is dissipated. Understanding transient response is crucial for predicting how circuits reach their new steady state after a disturbance.

Initial Conditions in Circuit Analysis

Initial conditions are the values of the circuit variables (such as voltage across capacitors or current through inductors) just at the moment after a switching event. They are essential for solving the differential equations that describe the circuit's behavior during transient analysis, and they ensure that the solution is physically consistent with the state of the circuit at the start of the transient period.

Transcript

00:01 Hello everyone, in the following question we have to find the value for vt.

00:04 Here we will apply that at the top node.

00:08 We will apply that at the top node.

00:10 Well, a equation of kcl.

00:13 It can be written as i by 4 plus i is equals to c1, dv1 by d t.

00:20 Here c1 is given as 0 .1 ferret.

00:24 Here, so it will write further.

00:26 This is 5 i by 4.

00:27 This is c1 d b by d t c1 is 0 .1 d b by d t so from here we will get value of i is 0 .08 d b by d t this is equation 1 but next if we'll write the value for potential v is equal to minus 2 i plus 1 by c2 1 by c2 integration of i is 5 into d t here c2 is given as 0 .5 ferret or it will write value for d b by d t this can be written as 2 d i 2 di by d t plus 2 i this is the equation for d b by d t now this we will say this is equation number 2 next we will substitute 1 into 2.

01:29 We will substitute 1 into 2.

01:31 So here we will get value of minus d b b by d t is equal to 0 .16 d b squared by d t square plus 0 .16 d b by d t this equation can be further written as 0 .16 d b square by d t square plus 0 .16 d b squared by d t square plus 0.

01:52 D b by d t plus d b by d t is equal to zero or we will solve this equation as d b square by d t squared plus 7 .25 d b by d t is equal to zero.

02:11 If we will solve this equation we will say that this equation led to we will say which led to s square plus 7 .5 plus 7...

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