RLC Circuit Transient Analysis: Understanding Overshoot, Settling Time, and Damping Ratio, Lab Reports of Microelectronic Circuits

Download RLC Circuit Transient Analysis: Understanding Overshoot, Settling Time, and Damping Ratio and more Lab Reports Microelectronic Circuits in PDF only on Docsity! Laboratory 5 CEET 3010 Fall 2008 RLC Transient Circuits Purpose: Introduce RLC circuits to develop a familiarity with rise time, overshoot, and settling time. Equipment and Components: • Prototyping board, Multimeter, Signal Generator, Oscilloscope. • Resistors: 500 Ω, 10 kΩ, 10 turn 20 kΩ potentiometer (low inductance). • Inductor: 100 mH • Capacitor: 10 nF Additional Information: There are many ways to look at a 2nd order circuit. For a RLC circuit this could include classifying the transient as over damped, under damped, or crucially damped. There is also the Neper frequency that measures the decay of a signal, the resonance frequency that the circuit would oscillate at it there was no damping, and the damped frequency that under damped circuit oscillate at. All of these depend upon the relationship of the resistor, inductor, and capacitor values. If one is concerned about the rate at which a 2nd order circuit will respond, then the rise time, percent overshoot and settling time can play an important role. These too are dependent upon the RLC values. The general 2nd order equation that results from NODE or MESH analysis can be rewritten in its “standard form” as where the damping ratio (ζ) and the undamped natural/resonance frequency (ωo) are equal to series RLC => and parallel RLC => and . The characteristic roots equation then becomes . This simplifies the description of the circuit down to effectively one parameter. For ζ > 1 there are two distinct real roots, and the circuit is over damped. For ζ = 1 there are two real equal roots, and the circuit is critically damped. For ζ < 1 there are two complex conjugate roots, and the circuit is under damped. Typically L and C are chosen to set the undamped natural frequency and R is selected to adjust the damping ratio. Furthermore, the under damped responses (ζ < 1) have two distinctive features. The response will rise above the final value in a phenomenon called “overshoot” and then oscillate about and decay to the final value in a phenomenon called “ringing”. Overshoot Overshoot is defined as the difference between peak and final value over the final value. It can also be rewritten in terms of the damping ratio Overshoot = € Vpeak −Vfinal Vfinal = . For ζ > 1 (i.e. over or critically damped), the overshoot becomes complex, looses it’s meaning and becomes difficult to defined. However for 0 < ζ < 1, the smaller the damping ratio the larger the overshoot, with it approaching 1 or (100%) as ζ → 0, the undamped and pure oscillating condition. Ringing The time it takes for an under damped response to settle within a given band Vfinal(1+ε), where ε is some small error term defined by the user, is called the “settling time” and is denoted as ts. It can be approximated by the envelope of the ringing From this it can also be seen that the smaller ζ becomes the longer the settling time, the more ringing that will occur for an under damped circuit. Preliminary: 1. For the series circuit of Figure 5.1, find the resonant frequency ωo and calculate the size of the resistance R1 that will make the circuit critically damped.