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:
y(0) = y0
A×u0(t)
System
y(t)
• So far, we have considered only circuits which are
initially relaxed  y(0) = 0
• We now consider circuits with non-zero initial
conditions
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 approach
Response 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
u(t)
U
t=0
t
• The system DC gain =
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