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Lecture 21•Review: Second order electrical circuits
• Series RLC circuit• Parallel RLC circuit
•Second order circuit natural response•Sinusoidal signals and complex exponentials•Related educational modules:
– Section 2.5.2, 2.5.3
Second order input-output equations
• In general, the governing equation for a second order system can be written in the form:
• Where• is the damping ratio ( 0)• n is the natural frequency (n 0)
Solution of second order differential equations
• The solution of the input-output equation is (still) the sum of the homogeneous and particular solutions:
• We will consider the homogeneous solution first:
Homogeneous solution (Natural response)
• Assume form of solution:
• Substituting into homogeneous differential equation:
• We obtain two solutions:
Homogeneous solution – continued• Natural response is a combination of the solutions:
• So that:
• We need two initial conditions to determine the two unknown constants:
• ,
Natural response – discussion
• and n are both non-negative numbers– 1 solution composed of decaying exponentials– < 1 solution contains complex exponentials
Sinusoidal functions
• General form of sinusoidal function:
• Where:– VP = zero-to-peak value (amplitude)
– = angular (or radian) frequency (radians/second)– = phase angle (degrees or radians)
Sinusoidal functions – graphical representation• T = period• f = frequency
• cycles/sec (Hertz, Hz)
• = phase• Negative phase shifts
sinusoid right
Complex numbers – Polar coordinates• Our previous plot was in
rectangular coordinates
• In polar coordinates:
• Where:
Sinusoids and complex exponentials – continued
• Unit vector rotating in complex plane:
• Socos t
time
t
Complex exponentials – summary• Complex exponentials can be used to represent
sinusoidal signals• Analysis is (nearly always) simpler with complex
exponentials than with sines, cosines
• Alternate form of Euler’s identity:
• Cosines, sines can be represented by complex exponentials
Classifying second order system responses
• Second order systems are classified by their damping ratio:• > 1 System is overdamped (the response consists of
decaying exponentials, may decay slowly if is large)• < 1 System is underdamped (the response will
oscillate)• = 1 System is critically damped (the response consists
of decaying exponentials, but is “faster” than any overdamped response)
Note on underdamped system response
• The frequency of the oscillations is set by the damped natural frequency, d