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:

• Governing differential equation and initial condition:

y(0) = y0

A×u0(t) y(t)System

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:

DC gain =

y(0) = 0

A×u0(t) y(t)System Dy

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

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 gain4. C so that =0.1 seconds5. 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 3Determine 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: ;