# Second-Order Circuits (6.3)

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Second-Order Circuits (6.3)
Dr. Holbert November 20, 2001 ECE201 Lect-23

2nd Order Circuits Any circuit with a single capacitor, a single inductor, an arbitrary number of sources, and an arbitrary number of resistors is a circuit of order 2. Any voltage or current in such a circuit is the solution to a 2nd order differential equation. ECE201 Lect-23

Important Concepts The differential equation
Forced and homogeneous solutions The natural frequency and the damping ratio ECE201 Lect-23

A 2nd Order RLC Circuit i (t) R + vs(t) C L The source and resistor may be equivalent to a circuit with many resistors and sources. ECE201 Lect-23

Applications Modeled by a 2nd Order RLC Circuit
Filters A lowpass filter with a sharper cutoff than can be obtained with an RC circuit. ECE201 Lect-23

The Differential Equation
vr(t) i (t) + R + + vc(t) vs(t) C vl(t) + L KVL around the loop: vr(t) + vc(t) + vl(t) = vs(t) ECE201 Lect-23

Differential Equation
ECE201 Lect-23

The Differential Equation
Most circuits with one capacitor and inductor are not as easy to analyze as the previous circuit. However, every voltage and current in such a circuit is the solution to a differential equation of the following form: ECE201 Lect-23

Important Concepts The differential equation
Forced and homogeneous solutions The natural frequency and the damping ratio ECE201 Lect-23

The Particular Solution
The particular (or forced) solution ip(t) is usually a weighted sum of f(t) and its first and second derivatives. If f(t) is constant, then ip(t) is constant. If f(t) is sinusoidal, then ip(t) is sinusoidal. ECE201 Lect-23

The Complementary Solution
The complementary (homogeneous) solution has the following form: K is a constant determined by initial conditions. s is a constant determined by the coefficients of the differential equation. ECE201 Lect-23

Complementary Solution
ECE201 Lect-23

Characteristic Equation
To find the complementary solution, we need to solve the characteristic equation: The characteristic equation has two roots-call them s1 and s2. ECE201 Lect-23

Complementary Solution
Each root (s1 and s2) contributes a term to the complementary solution. The complementary solution is (usually) ECE201 Lect-23

Important Concepts The differential equation
Forced and homogeneous solutions The natural frequency and the damping ratio ECE201 Lect-23

Damping Ratio () and Natural Frequency (0)
The damping ratio is . The damping ratio determines what type of solution we will get: Exponentially decreasing ( >1) Exponentially decreasing sinusoid ( < 1) The natural frequency is w0 It determines how fast sinusoids wiggle. ECE201 Lect-23

Roots of the Characteristic Equation
The roots of the characteristic equation determine whether the complementary solution wiggles. ECE201 Lect-23

Real Unequal Roots If  > 1, s1 and s2 are real and not equal.
This solution is overdamped. ECE201 Lect-23

Overdamped ECE201 Lect-23

Complex Roots If  < 1, s1 and s2 are complex.
Define the following constants: This solution is underdamped. ECE201 Lect-23

Underdamped ECE201 Lect-23

Real Equal Roots If  = 1, s1 and s2 are real and equal.
This solution is critically damped. ECE201 Lect-23

Example i (t) 10W + vs(t) 769pF 159mH This is one possible implementation of the filter portion of the IF amplifier. ECE201 Lect-23

More of the Example For the example, what are z and w0? ECE201 Lect-23

Even More Example z = 0.011 w0 = 2p455000
Is this system over damped, under damped, or critically damped? What will the current look like? ECE201 Lect-23

Example (cont.) The shape of the current depends on the initial capacitor voltage and inductor current. ECE201 Lect-23

Slightly Different Example
i (t) 1kW + vs(t) 769pF 159mH Increase the resistor to 1kW What are z and w0? ECE201 Lect-23

More Different Example
z = 2.2 w0 = 2p455000 Is this system over damped, under damped, or critically damped? What will the current look like? ECE201 Lect-23

Example (cont.) The shape of the current depends on the initial capacitor voltage and inductor current. ECE201 Lect-23

Damping Summary ECE201 Lect-23

Class Examples Learning Extension E6.9 Learning Extension E6.10
ECE201 Lect-23