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**Lecture - 9 Second order circuits **

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**Outline Second order circuits. The characteristic equation.**

The form of the natural and step responses. Responses of second-Order Circuit. The natural response of second order circuit. The step response of second order circuit.

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Second order circuits Second order circuits have two independent energy storage elements (inductors and capacitors). Analysis of a second order circuit yields a second order differential equation (DE). A second order differential equation has the form: Solution of a second order differential equation requires two initial conditions: x(0) and x’(0) Since second order circuits have two energy-storage types, the circuits can have the following forms: A) Series RLC circuit. B) Parallel RLC circuit.

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**The characteristic equation**

The characteristic equation for both the parallel and series RLC circuits has the form: where α is the neper frequency α = 1/2RC for the parallel circuit. α = R/2L for the series circuit. w0 is the resonant frequency w0 2 = 1/LC for both the parallel and series circuits. The roots of the characteristic equation are:

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**The form of the natural and step responses**

The form of the natural and step responses of series and parallel RLC circuits depends on the values of α and w0. Such responses can be over-damped, under-damped, or critically damped depending on the roots of the characteristic equation which be real and distinct, repeated, or complex. These terms describe the impact of the dissipative element (R) on the response. The neper frequency, α , reflects the effect of R.

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**Responses of Second-Order Circuit**

The response of the second order circuit can be: Over-damped: roots are real and distinct the voltage or current approaches its final value without oscillation (α2 ˃ w20). Under-damped: roots are complex the voltage or current oscillates about its final value (α2 ˂w20). Critically damped: roots are repeated the voltage or current is on the verge of oscillating about its final value (α2 = w20).

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**The natural response of second order circuit**

We first determine whether it is over-, under-, or critically damped, and then we solve the appropriate equations as shown in the table. The unknown coefficients (i.e., the As, Bs, and Ds) are obtained by evaluating the circuit to find the initial value of the response, x(0), and the initial value of the first derivative of the response, dx(0)/dt.

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**The step response of second order circuit**

We apply the appropriate equations depending on the damping, as shown in the table. The unknown coefficients (i.e., the As, Bs, and Ds) are obtained by evaluating the circuit to find the initial value of the response, x(0), and the initial value of the first derivative of the response, dx(0)/dt.

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Example 1

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Example 1

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Example 2

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Example 2

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Example 3

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Summary Second order circuits have two independent energy storage elements (inductors and capacitors). The form of the natural and step responses of series and parallel RLC circuits depends on the values of α and w0. The response of the second order circuit can be over-damped, under-damped, or critically damped. Natural and step response are obtained by first determine whether it is over-, under-, or critically damped, and then solving the appropriate equations.

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Let us examine this LC circuit mathematically. To do this let us examine the energy of the system. Conservation of Energy 2 nd order differential equation.

Let us examine this LC circuit mathematically. To do this let us examine the energy of the system. Conservation of Energy 2 nd order differential equation.

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