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

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: – V P = 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

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