Lecture 20•First order circuit step response

• Nonzero initial conditions and multiple sources• Steady-state response and DC gain• Bias points and nominal operating conditions

•Introduction to second order systems•Related educational modules:

– Section 2.5.1

First order system step response• Block diagram:

• So far, we have considered only circuits which are initially relaxed y(0) = 0

• We now consider circuits with non-zero initial conditions

y(0) = y0

A×u0(t) y(t)System

Example 1 • The switch moves from A to B at time t=0

• Find v(t), t>0

• Sketch input function on previous slide

Example 1 – initial condition

Example 1 – Differential equation for t>0

Example 1 – Check , steady-state response

Example 1 – circuit response

• Differential equation:

• Initial, final conditions: ,

• Form of solution:

Example 1 – sketch input, output

Alternate representation of example 1

• The circuit of example 1 can be written as:

• Now determine the response using superposition

• Annotate previous slide to show input function

Example 1 – superposition approachResponse to (constant) 2V source

Example 1 – superposition approach (cont’d)Response to 3V step input

• Input-output equation:

Example 1 – superposition approach (cont’d)Response to 3V step input

• Governing equation:

• Form of solution:

• Initial condition:

• Final condition:

Example 1 – superposition approach (cont’d)Overall response

Note on overall approach• Both the input and output can be decomposed into

a constant value and a time-varying value

• It is sometimes convenient to analyze these components independently• For example, the DC gain of the system applies to both

the constant input and the time varying input

Graphical interpretation

• The system DC gain =

t

u(t)

t=0

U

Why is this approach useful?• Decomposing the input and output into constant and

time-varying components can simplify analysis and interpretation of results• The constant part of the input and output is the bias point

or nominal operating point• The system dynamic response is often characterized by the

time-varying part of the input-output relationship• A nonlinear system, for example, can be

approximated as a linear system with a bias point

Introduction to second order systems• Second order systems are governed by second

order differential equations• Input-output relation contains a second order derivative

term, but no derivatives higher than second order• The physical system has two independent energy storage

elements• The natural response of a second order system can

oscillate with time (but doesn’t necessarily have to)• The response can overshoot its final value

Introduction to second order systems – continued• The oscillations in the natural response are due to

energy being traded between the energy storage elements• Increasing energy dissipation reduces the amplitude of

the oscillations (the system is said to be more highly damped)

• If energy dissipation is above a critical value, the response will no longer oscillate

• In general, increasing the energy dissipation will also cause the system to respond to changes more “slowly”

• On previous slide, talk about damping and energy dissipation– Example: suspension system in car

Example: Series RLC circuit• Write the differential equation governing iL(t)

Series RLC circuit – continued

Example: Parallel RLC circuit• Write the differential equation governing vC(t)

Parallel RLC circuit – continued