Transcript Lecture 1 - Digilent Inc.

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 materials:
–Chapter 8.2, 8.3
Summary: Series & parallel RLC circuits
• Series RLC circuit:
• Parallel RLC circuit
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
• Complex numbers have
real and imaginary
parts:
• Where:
Complex numbers – Polar coordinates
• Our previous plot was in
rectangular coordinates
• In polar coordinates:
• Where:
Complex exponentials
• Polar coordinates are
often expressed as
complex exponentials
• Where
Sinusoids and complex exponentials
• Euler’s Identity:
Sinusoids and complex exponentials – continued
• Unit vector rotating in
complex plane:
• So
cos t
t
time
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
Second order system natural response
• Now we can interpret our previous result
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