Chapter 8 Second-Order Circuits Study Notes

Introduction to Second-Order Circuits

• Definition: Circuits containing two storage elements of different types (e.g., one inductor and one capacitor) or the same type (e.g., two inductors or two capacitors) are referred to as second-order circuits.
• Typical Circuit Types:
  • $RLC$ circuits.
  • $RC$ and $RL$ circuits that result in a second-order response.
• Mathematical Representation: These circuits are characterized by a second-order differential equation.
• Classification:
  • Source-free circuits: These rely on energy stored in elements and may include dependent sources.
  • Independent source circuits: Circuits containing independent voltage or current sources.
• Typical Applications:
  • Automobile ignition systems.
  • Smoothing circuits for power supplies or signal processing.

Finding Initial and Final Values

• Objective: This process involves determining values for $v(0)$, $i(0)$, $\frac{dv(0)}{dt}$, $\frac{di(0)}{dt}$, $i(\infty)$, and $v(\infty)$.
• Key Principles:
  • Polarity and Direction: It is critical to carefully handle the polarity of the voltage $v(t)$ across the capacitor and the direction of the current $i(t)$ through the inductor.
  • Continuity Constraints:
    • The capacitor voltage is always continuous: $v(0^+) = v(0^-)$.
    • The inductor current is always continuous: $i(0^+) = i(0^-)$.
• Steady-State Behavior:
  • Inductor $\rightarrow$ short circuit.
  • Capacitor $\rightarrow$ open circuit.

Example 8.1 Analysis:

• Find $i(0^+)$, $v(0^+)$, $\frac{di(0^+)}{dt}$, $\frac{dv(0^+)}{dt}$, $i(\infty)$, and $v(\infty)$.
• Initial values (at $t = 0^-$): $i(0) = 0$, $v(0^-) = 12\,V$.
• At $t = 0^+$:
  • $i(0^+) = i(0^-) = 0\,A$.
  • $v(0^+) = v(0^-) = 12\,V$.
  • $\frac{di(0^+)}{dt} = \frac{v_L(0^+)}{L} = \frac{12 - 4(0) - 12}{L} = 0\,A/s$.
  • $\frac{dv(0^+)}{dt} = \frac{i_C(0^+)}{C} = \frac{0}{0.1} = 0\,V/s$.
• Final values (at $t \rightarrow \infty$):
  • $i(\infty) = 12 / 4 = 3\,A$.
  • $v(\infty) = 12\,V$.

Example 8.2 Analysis:

• Calculate $i_L(0^+)$, $v_C(0^+)$, $v_R(0^+)$, $\frac{di_L(0^+)}{dt}$, $\frac{dv_C(0^+)}{dt}$, $\frac{dv_R(0^+)}{dt}$, $i_L(\infty)$, $v_C(\infty)$, and $v_R(\infty)$.
• For $t < 0$ (steady state with $20u(-t)$ source):
  • $i_L(0^-) = 2\,A$, $v_C(0^-) = 20\,V$, $v_R(0^-) = 0\,V$.
• At $t = 0^+$:
  • $i_L(0^+) = 2\,A$, $v_C(0^+) = 20\,V$, $v_R(0^+) = -20\,V$.
• Derivatives at $t = 0^+$:
  • $\frac{dv_C(0^+)}{dt} = \frac{i_C(0^+)}{C} = \frac{1}{0.5} = 2\,V/s$.
  • $\frac{di_L(0^+)}{dt} = \frac{v_L(0^+)}{L} = \frac{v_R(0^+)}{L} = \frac{-20}{0.2} = -100\,A/s$.
  • $\frac{dv_R(0^+)}{dt} = 15\,V/s$.
• For $t \rightarrow \infty$:
  • $v_C(\infty) = 20 - 3 \times \frac{4}{4+2} \times 2 = 12\,V$.
  • $i_L(\infty) = \frac{4 \times 2}{4 + 2} = \frac{4}{3}\,A$.

The Source-Free Series RLC Circuit

• Circuit Excitation: The circuit is excited by energy initially stored in the capacitor (represented by initial voltage $V_0$) and the inductor (represented by initial current $I_0$).
• Differential Equation:     $L \frac{d^2i}{dt^2} + R \frac{di}{dt} + \frac{1}{C} i = 0$
• Characteristic Equation: Obtained by assuming a solution in exponential form $i = Ae^{st}$, leading to:     $s^2 + \frac{R}{L} s + \frac{1}{LC} = 0$
• Natural Frequencies ($s_1, s_2$):     $s_{1,2} = -\alpha \pm \sqrt{\alpha^2 - \omega_0^2}$
  • $\alpha$ (Neper frequency/damping factor): $\alpha = \frac{R}{2L}$.
  • $\omega_0$ (Resonant/undamped natural frequency): $\omega_0 = \frac{1}{\sqrt{LC}}$.

Damping Conditions

• Overdamped Case ($\alpha > \omega_0$): Roots are real and unequal.
  • The response $i(t)$ decays monotonically toward zero without oscillation.
  • $i(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}$.
  • The resistance is large enough to dissipate energy without allowing the system to overshoot.
• Critically Damped Case ($\alpha = \omega_0$): Roots are real and equal ($s_1 = s_2 = -\alpha$).
  • The resistance provides the minimum damping required to prevent oscillation.
  • The solution takes the form: $i(t) = (A_1 t + A_2) e^{-\alpha t}$.
• Underdamped Case ($\alpha < ω_0$): Roots are complex conjugates ($s_{1,2} = -\alpha \pm j\omega_d$).
  • Damping frequency $\omega_d = \sqrt{\omega_0^2 - \alpha^2}$.
  • The circuit releases energy through periodic oscillations with a decaying amplitude.
  • $i(t) = e^{-\alpha t} (A_1 \cos(\omega_d t) + A_2 \sin(\omega_d t))$.

Key Observations

1. Damping: Refers to the gradual depletion of energy, causing response amplitude to decrease. The rate is determined by $\alpha$, which depends on resistance $R$.
2. Loss-less Circuit: If $R = 0$, then $\alpha = 0$. The response becomes a pure oscillation (resonance) at frequency $\omega_0 = \frac{1}{\sqrt{LC}}$.
3. Oscillators: Real circuits have natural losses in $L$ and $C$. Electronic components called oscillators are designed to generate ideal sinusoidal responses.

The Source-Free Parallel RLC Circuit

• Differential Equation:     $C \frac{d^2v}{dt^2} + \frac{1}{R} \frac{dv}{dt} + \frac{1}{L} v = 0$
• Characteristic Parameters:
  • $\alpha = \frac{1}{2RC}$.
  • $\omega_0 = \frac{1}{\sqrt{LC}}$.
• Response Forms:
  • Overdamped: $v(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}$.
  • Critically Damped: $v(t) = (A_1 + A_2 t) e^{-\alpha t}$.
  • Underdamped: $v(t) = e^{-\alpha t} (A_1 \cos(\omega_d t) + A_2 \sin(\omega_d t))$, where $\omega_d = \sqrt{\omega_0^2 - \alpha^2}$.

Step Response Analysis

• Definition: The step response is the circuit's behavior following the sudden application of a DC source.
• Solution Structure: The total response consists of the transient response ($x_t(t)$) and the steady-state response ($x_{ss}(t)$).     $x(t) = x_{ss}(t) + x_t(t)$
  • Steady-state response: $v(\infty)$ or $i(\infty)$.
  • Transient response: Same form as the source-free response (overdamped, underdamped, or critically damped).

Step Response of Series RLC

• Differential Equation: $LC \frac{d^2v}{dt^2} + RC \frac{dv}{dt} + v = V_s$
• Constants: $A_1$ and $A_2$ are found using initial conditions $v(0)$ and $\frac{dv(0)}{dt}$.
  • $i = C \frac{dv}{dt}$.
  • $v_L = L \frac{di}{dt}$.

Step Response of Parallel RLC

• Differential Equation: $LC \frac{d^2i}{dt^2} + \frac{L}{R} \frac{di}{dt} + i = I_s$
• Constants: Found using $i(0)$ and $\frac{di(0)}{dt}$.
  • $v = L \frac{di}{dt}$.
  • $i_C = C \frac{dv}{dt}$.

General Second-Order Circuit Procedure

Steps to find the complete response $x(t)$:

1. Determine Initial Conditions: Find $x(0)$, $\frac{dx(0)}{dt}$, and the final steady-state value $x(\infty)$.
2. Find Transient Response: Turn off independent sources (voltage sources to short, current sources to open). Solve the characteristic equation of the resulting source-free circuit to find $x_t(t)$.
3. Define Steady-State Response: $x_{ss}(t) = x(\infty)$.
4. Combine and Solve: Let $x(t) = x_t(t) + x_{ss}(t)$. Use the initial conditions from Step 1 to determine the integration constants.

Second-Order Op Amp Circuits

• Active circuits containing an Op Amp and two storage elements (usually capacitors).
• Analyzed using Nodal Analysis at the input terminals of the Op Amp.
• The nodal equations yield a second-order differential equation relating input voltage $v_s$ to output voltage $v_o$.

Duality in Circuits

• Concept: Duality is a parallelism between pairs of characterizing equations and circuit variables. Using duality can save time by allowing the solution of one circuit to be applied to its dual.
• Dual Pairs:
  • Voltage ($v$) $\rightleftharpoons$ Current ($i$).
  • Resistance ($R$) $\rightleftharpoons$ Conductance ($G$).
  • Capacitance ($C$) $\rightleftharpoons$ Inductance ($L$).
  • Node $\rightleftharpoons$ Mesh.
  • Series $\rightleftharpoons$ Parallel.
• Graphical Technique to Construct Dual Circuits:
  1. Place a node at the center of each mesh of the original circuit. Place a reference node outside.
  2. Draw lines between nodes crossing each element. Replace the crossed element with its dual.
  3. Polarity Rule: A voltage source producing positive mesh current corresponds to a dual current source flowing from ground to the non-reference node.

Applications

Automobile Ignition System

• Functions as a voltage generating system using the properties of second-order circuits to create high sparks.
• Example 8.16: Involves finding $v_L$ for $t > 0$. Initial current $i(0^-) = 3\,A$. When the switch opens, energy is transferred, producing high voltage.
• Resulting voltage: $v_L(t) = 268 e^{-250t} \sin(11180t)\,V$.

Smoothing Circuits

• Used in Digital-to-Analog (D/A) converters to smooth out staircase output functions into continuous signals.