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Lecture 19•Review:
•First order circuit step response•Steady-state response and DC gain•Step response examples•Related educational modules:
–Section 2.4.5
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First order system step response• Block diagram:
• Governing differential equation and initial condition:
y(0) = y0
A×u0(t) y(t)System
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First order system step response• Solution is of the form:
• Initial condition:
• Final condition:
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• 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
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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)
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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
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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
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DC gain – graphical interpretation• Input and output signals:• Block Diagram:
DC gain =
y(0) = 0
A×u0(t) y(t)System Dy
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Suggested Overall Approach
1. Write governing differential equation2. Determine initial condition3. Determine final condition (from circuit or diff. eqn)4. Check differential equation
• Check time constant (circuit vs. differential equation)• Check final condition (circuit vs. differential equation)
5. Solve the differential equation
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Example 1• The circuit below is initially relaxed. Find vL(t) and iL(t) , t>0
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• Determine initial AND final conditions on previous slide.
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Example 1 – continued
• Circuit for t>0:
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Example 1 – checking results
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Example 1 – continued again
• Apply initial and final conditions to determine K1 and K2
Governing equation:
Form of solution:
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Example 1 – sketch response
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Example 1 – Still continued…• Now find vL(t).
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Example 2 (alternate approach to example 1)• Find vL(t) , t>0
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Example 2 – continued
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Example 3 (still another approach to example 1)
• Find iL(t) , t>0
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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 gain4. C so that =0.1 seconds5. v(t), t>0 for the value of
C determined above
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Example 4 – Part 1• Determine the differential equation governing v(t)
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Example 4 – Parts 2 and 3Determine the initial and final values for v(t) and the circuit’s DC gain
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Example 4 – Checking differential equation• Governing differential
equation (Part 1):
• Final Condition (Part 2):
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Example 4 – Part 4 Determine C so that =0.1 seconds
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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: ;
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