E11 Lab #5
Pre-Lab
2008


Second Order Time Domain Response


A Second Order Circuit:

We will be analyzing the behavior of a second order circuit; it has both an inductor and a capacitor.  The circuit that we will be using in lab is shown below.

The circuit is shown below.

RLC Circuit

The analysis of this circuit is in some ways similar to the analysis of a first order circuit.  We find a homogeneous response and a particular response.  There are several ways to find the form of the homogeneous response.  In class you will use the "impedance method."  So, for the pre-lab we will derive the differential equation, and proceed from there.

We can write the following two equations assuming current is flowing clockwise (note that this is neither the node method nor the loop method - with many circuits you can simply examine the circuit and write a sufficient number of equations to solve).

First set of equations

We can substitute the second equation into the first and rearrange.

Final DE

Particular Solution

We will  only be dealing with constant inputs (for t≥0).  As with the first order case, we assume the particular solution is a constant.  If vi(t)=K (a constant), then we can find the particular solution by simply letting the derivatives go to zero (since the particular solution is constant).

particular solution

Homogeneous Solution

To find the homogeneous response we assume the form of the response is Aest and plug it into the equation with the input equal to zero.

Equations

Before we look at the solution to the homogeneous problem, let's rewrite the homogeneous equation in a standard form used by engineers

2nd order

Where ζ is called the damping ratio and ω0 is the resonant frequency (rad/sec).  The reason for this notation becomes apparent when we look at the roots of the equation.  Note that we will find two values for s, while in first order circuits we only obtained one.

Roots of Second order C.E.

From this equation it is clear that zeta; determines the nature of the two roots (we will assume for now that 0≤ζ≤∞):

zeta ranges

We will only consider the overdamped and underdamped cases (you'll do the critically damped case - while it is interesting mathematically, it doesn't behave much differently than the over- or underdamped case with ζ near 1).

Let's examine the two cases:

  Overdamped Underdamped
Homogeneous Response Overdamped Underdamped 
Particular Response
(for 1 volt input)
Particular  Particular
Complete Response
(Find coefficients from initial conditions)
Complete Overdamped  Complete underdamped

Some typical responses

ζ varies:

Note how as the damping ratio increases from undamped (ζ=0) to overdamped (ζ=2) that the system oscillates less and less (and not at all for ζ≥1)

Zeta varies

ω0 varies:

As ω0 increases the shape of the response remains the same, but its speed changes.

Omega_0 varies

Summary:

The utility of representing the roots in terms of the parameter ζ and ω0 (as opposed to α and ωd is that ζ represents the amount of damping (shaped of the response) while ω0 represents the speed.  In the lab it turns out that it is actually easier to measure α (the time constant of the decay envelope) and ωd (the frequency of oscillation).  See the graphs below.

Measuring frequency 

From this graph we can find α and ωd

and from α and ωd we can find ζ and ωn.

 


On to the lab.


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