ES 2210 Electric Circuit Analysis - First-Order RL and RC Circuits

Chapter 7—Objectives

• Be competent in analyzing the following circuits:

  • Natural response of RL and RC circuits.

  • Step response of RL and RC circuits.

  • Sequential switching of first-order circuits.

What do we mean by “first-order” circuits?

• Definition: A circuit that includes:

  • Only resistors and inductors (RL)

  • Only resistors and capacitors (RC)

• Characteristics:

  • Such circuits are considered first-order because they can be described using a first-order differential equation (easy to solve).

• Note on Second-Order Circuits:

  • If a circuit includes all three (RLC), the solution requires a second-order differential equation, which is more complex.

  • Second-order circuits are covered in EE 2220 (Circuits II), hence
    Chapter 8 is excluded from this course.

Importance of Prior Knowledge

• Previous circuit analysis tools (from Chapters 1-4 and exam 1) must be retained, as they apply to inductors and capacitors:

  • Ohm’s Law, Kirchhoff’s Voltage Law (KVL), Kirchhoff’s Current Law (KCL), voltage division, current division, source transformations, node voltage analysis, mesh current analysis, Thévenin equivalent, etc.

  • The new knowledge of inductors and capacitors builds upon this existing knowledge base.

RL Natural Response

• Problem Statement:

  • Given an initial current, i(0), through an inductor at t = 0, determine i(t) for t ≥ 0.

  • Note: This response is termed “natural” as there are no independent sources active in the system at t ≥ 0; the response is generated solely by the energy stored in the inductor.

Differential Equation Formulation

• The governing equation is a:

  • First-order (highest derivative is first)

  • Homogeneous (right-hand side is 0)

  • Ordinary differential equation with constant coefficients.

  • Voltage across the inductor + Voltage across the resistor:
    $L rac{di(t)}{dt} + Ri(t) = 0$

Solution Steps for RL Circuit

1. Rearranging the equation and multiplying by dt gives:
   $L rac{di(t)}{dt} = -Ri(t)$
   Rearranged to:
   $rac{di}{i} = - rac{R}{L} dt$

2. Integrating this equation:
   $ext{Integrate: } ln(i(t)) - ln(i(0)) = - rac{R}{L} t$
   Leading to:
   $i(t) = I_s e^{- rac{R}{L} t}, ext{ for } t orall 0$

3. Where I_s represents the initial current, i(0).

Circuit Example for RL Response

• For t < 0, replace the inductor accordingly to analyze the circuit and establish initial conditions:

  • Choices for replacement include:

  • A. An open circuit

  • B. A short circuit

  • C. A resistor.

RL Natural Response Summary

• Formula for i(t):

  • For t < 0: Initial response established as a function of stored current.

  • For t ≥ 0:
    $i(t) = I_s e^{- rac{R}{L} t}$

RC Natural Response

• Problem Statement:

  • Find v(t) for t ≥ 0, knowing that the circuit voltage v(0) establishes initial voltage across capacitor C at t = 0.

  • RC circuits are the dual of RL circuits, with their response being similarly exponential.

Differential Equation Formulation for RC Circuits

• The governing equation:

  • First-order, homogeneous ordinary differential equation with constant coefficients:

  • Current through capacitor + Current through the resistor:
    $rac{dv(t)}{dt} + rac{1}{RC}v(t) = 0$

Solution Steps for RC Circuit

1. Rearranging gives:
   $rac{dv}{v} = - rac{1}{RC} dt$

2. Integration results in:
   $ln(v(t)) - ln(v(0)) = - rac{t}{RC}$
   Leading to:
   $v(t) = V_g e^{- rac{t}{RC}}, ext{ for } t ≥ 0$

Key Concepts on Time Constants

Time Constants for RL and RC Circuits

• In RL circuits:

  • Time constant = $au_{RL} = rac{L}{R}$

• In RC circuits:

  • Time constant = $au_{RC} = R imes C$

Terminology and Concepts

• Long Time: Defined as time ≥ 5 time constants after a switch changes positions.

• Transient Response: Describes circuit response from 0 to 5 time constants.

• Steady-State Response: Describes circuit behavior after 5 time constants.

General Solution Method for RL and RC Response

1. Identify variable of interest (current for RL, voltage for RC).

2. Determine the initial value for - i(0) or v(0).

3. Calculate the time constant $au$ for the circuit.

4. Write expression for variable of interest:

   • For RL: $i(t) = I<em>f + (I</em>0 - I_f)e^{-t/ au}$

   • For RC: $v(t) = V<em>f + (V</em>0 - V_f)e^{-t/ au}$

5. Conduct analyses for any additional variables.

Sequential Switching for RL and RC Circuits

• In circuits with multiple switching events:

  • Each switch affects circuit behavior and can exhibit natural or step response.

• Analysis involves breaking down the switching sequence into distinct time segments and analyzing each using foundational concepts.

  • Shift the exponential function for subsequent segments reflecting the previous change in time.

Examples Covered in Sequential Switching

• Problems specifying switch actions over time and requiring calculations of circuit responses for each segment.

Conclusion

• Mastery of first-order RL and RC circuits requires understanding of differential equations representing their responses, knowledge of time constants, and experience in applying circuit analysis techniques. Sequential switching adds complexity but utilizes the same fundamental principles.