ENGINEERING 11, 2004

ELECTRIC CIRCUIT ANALYSIS

Lab 4 - BACKGROUND

 Second –order systems:

  An RLC circuit is shown below.

 

           

The response of this system can be solved for by using node equations.  Assume a voltage, V’ between the resistor R and inductor L.  Now writing the node equations at V’ and V₀:

 

          (1)

 

and

 

Solving the other node equation, , for V’ and substituting into equation (1), yields

 

 

Second order systems are often characterized as follows:

 

           ₍₂₎

 

where  is the natural frequency and  is the damping ratio.  These terms result from the fact that if the damping ratio is zero, the system is an oscillator, and a unit step input would cause an output that is a unit sinusoidal varying about 1 with frequency radians/second.  Shown below is the step response

  

 

For a damping ratio between 0 and 1, the output is a damped sinusoidal and the rate of damping increases as the damping ratio increases.  This situation is referred to as an underdamped system.  If the natural frequency is unchanged, but the damping ratio is increased to say 0.3, the step response looks like

 

If the damping ratio is increased to 0.7, the step response is:

If the damping ratio is larger than 1, the system is said to overdamped.  Below is the result with a damping ratio of 1.5;

 The solution to the above equation for the underdamped case with a unit step driving function, is

 

where

 

The time to peak is given by:

 

This is found by differentiating y(t) and setting it to zero. 

 

The peak overshoot is found by evaluating V₀(t) at the time to peak and subtracting 1. 

Maximum overshoot as percentage is

 

 

The damping ratio for a particular %OS is