Lecture 19
•Review:
•First order circuit step response
•Steady-state response and DC gain
•Step response examples
•Related educational modules:
–Section 2.4.5
First order system step response
• Block diagram:
y(0) = y0
A×u0(t)
System
y(t)
• Governing differential equation and initial
condition:
First order system step response
• Solution is of the form:
• Initial condition:
• Final condition:
• Note on previous slide that we can determine the
solution without ever writing the governing
differential equation
– Only works for first order circuits; in general we need
to write the governing differential equations
– We’ll write the governing equations for first order
circuits, too – give us valuable practice in our overall
dynamic system analysis techniques
Notes on final condition
• Final condition can be determined from the circuit
itself
• For step response, all circuit parameters become constant
• Capacitors open-circuit
• Inductors short circuit
• Final conditions can be determined from the
governing differential equation
• Set
and solve for y(t)
2. Checking the final response
• These two approaches can be used to double-check
our differential equation
1. Short-circuit inductors or open-circuit capacitors and
analyze resulting circuit to determine y(t)
2. Set
in the governing differential equation and
solve for y(t)
• The two results must match
Steady-state step response and DC gain
• The response as t is also called the steady-state
response
• The final response to a step input is often called the
steady-state step response
• The steady state step response will always be a constant
• The ratio of the steady-state response to the input
step amplitude is called the DC gain
• Recall: DC is Direct Current; it usually denotes a signal
that is constant with time
DC gain – graphical interpretation
• Input and output signals:
• Block Diagram:
y(0) = 0
A×u0(t)
DC gain =
System
y(t)
Dy
Suggested Overall Approach
1.
2.
3.
4.
Write governing differential equation
Determine initial condition
Determine final condition (from circuit or diff. eqn)
Check differential equation
• Check time constant (circuit vs. differential equation)
• Check final condition (circuit vs. differential equation)
5. Solve the differential equation
Example 1
• The circuit below is initially relaxed. Find vL(t) and iL(t) , t>0
• Determine initial AND final conditions on
previous slide.
Example 1 – continued
• Circuit for t>0:
Example 1 – checking results
Example 1 – continued again
• Apply initial and final conditions to determine K1 and K2
Governing equation:
Form of solution:
Example 1 – sketch response
Example 1 – Still continued…
• Now find vL(t).
Example 2 (alternate approach to example 1)
• Find vL(t) , t>0
Example 2 – continued
Example 3 (still another approach to example 1)
• Find iL(t) , t>0
Example 4
• For the circuit shown:
determine:
1. The differential equation
governing v(t)
2. The initial (t=0+) and final
(t) values of v(t)
3. The circuit’s DC gain
4. C so that =0.1 seconds
5. v(t), t>0 for the value of
C determined above
Example 4 – Part 1
• Determine the differential equation governing v(t)
Example 4 – Parts 2 and 3
Determine the initial and final values for v(t) and the circuit’s
DC gain
Example 4 – Checking differential equation
• Governing differential
equation (Part 1):
• Final Condition (Part 2):
Example 4 – Part 4
Determine C so that =0.1 seconds
Example 4 – Part 5
Determine v(t), t>0 for the value of C determined in part 3
Governing equation, C = 0.01F:
Form of solution:
Initial, final conditions:
;