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:– 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:

• 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

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