Resonance in Series RLC Circuits

The Phenomenon of Resonance in a Series RLC Circuit (E16)

Introduction to Resonance

• General Significance: Resonance conditions are crucial in various fields of technology and research, beyond their everyday manifestations.

• Applications:

  • Resonance Absorption: Maximal absorption of vibrational energy from a signal source by an object when its natural frequency matches the source's frequency.

    • Applications include: investigations in gases, mechanics, nuclear physics, astro- and solid-state physics, and electrical engineering.

  • Signal Amplification/Filtering: Amplification of a signal or filtering of defined alternating voltages from a mix of frequencies when excitation and natural frequencies match.

• Experiment Focus: Investigating resonance phenomena in a series RLC (resistor, capacitor, inductor) circuit.

  • Key Quantities to Determine: Natural frequency ($\omega<em>0$), impedance ($Z$), quality factor ($Q$), and resonance frequency ($\omega</em>{\text{res}}$).

• Coupled Oscillating Circuits: When two oscillating circuits are coupled, a splitting of the resonance frequency into two natural frequencies is observed.

  • The magnitude of this splitting depends on the strength of the coupling.

  • Examples: This effect occurs in quantum optics and with string instruments (e.g., the 'Wolf tone').

Preparatory Tasks

For the circuit in Figure 1b, calculate:

• The resonance frequency ($\omega<em>0$) (for the undamped case): $\omega</em>0 = \frac{1}{\sqrt{LC}}$

• The value of R for the aperiodic critical damping case.

• For $R = 1 \text{k}\Omega$, calculate the frequency ($\omega$):
  $\omega = \sqrt{\frac{1}{LC} - \left(\frac{R}{2L}\right)^2}$

• For the preceding $\omega$ and R, calculate the logarithmic decrement ($\Lambda$) and quality factor ($Q$):
  $\Lambda = \ln \left( \frac{I(t)}{I(t+T)} \right) = \frac{\pi R}{\omega L}$
  $Q = \frac{\pi}{\Lambda}$

For forced oscillation, calculate:

• For $R = 100 \Omega$, calculate $\omega$, $\Lambda$, and $Q$.

• The voltage boosting at the inductor ($U<em>L(\omega</em>0)$) and capacitor ($U<em>C(\omega</em>0)$) at resonance, with $U<em>0 = 7 \text{V}$: $U</em>L(\omega<em>0) = \frac{\omega</em>0 L U<em>0}{R}$ $U</em>C(\omega<em>0) = \frac{1}{\omega</em>0 C} \frac{U_0}{R}$

• The phase shift ($\varphi$) when excited at the resonance frequency:
  $\varphi = \arctan \left( \frac{\omega<em>0 L - \frac{1}{\omega</em>0 C}}{R} \right)$

• What holds true for the impedance ($Z$) in this case?

Theoretical Background

Free Damped Oscillations

• Circuit Configuration: A closed circuit consisting of a resistor ($R$), capacitor ($C$), and inductor ($L$) forms an oscillating circuit.

• Process (Figure 1a):

  1. Charging: Capacitor is charged (Switch S at position 1).

  2. Discharge & Oscillation: Circuit is closed (Switch S at position 2), discharge current flows through the coil.

     • Self-Induction: The coil induces a voltage opposing the current, delaying the capacitor's discharge.

     • Current Maintenance: After full discharge, self-induction maintains the current, partially recharging the capacitor with reversed polarity.

     • Result: This leads to a damped oscillation of the current.

• Governing Equation: Applying Kirchhoff's loop rule and Ohm's law to the closed circuit yields:
  $U<em>R + U</em>C + U_L = 0 \Rightarrow R I + L \dot{I} + \frac{1}{C} \int I dt = 0 \quad (1)$

• Differential Equation: Differentiating with respect to time and dividing by $L$ gives: $\ddot{I} + \frac{R}{L} \dot{I} + \frac{1}{LC} I = 0 \quad (2)$

  • This equation is analogous to mechanical oscillation equations with restoring and damping terms.

• General Solution: The complete solution is: $I(t) = I<em>1 e^{\lambda</em>1 t} + I<em>2 e^{\lambda</em>2 t} \quad \text{where} \quad \lambda_{1,2} = - \frac{R}{2L} \pm \sqrt{\left(\frac{R}{2L}\right)^2 - \frac{1}{LC}} \quad (3)$

  • $I<em>1$ and $I</em>2$ are complex integration constants determined by initial conditions.

  • The nature of the roots ($\lambda_{1,2}$) depends on the damping term $(R/2L)^2$ relative to $1/LC$.

Cases of Damping (Determined by $(R/2L)^2$)

• Oscillation Case (Underdamped): \left(\frac{R}{2L}\right)^2 < \frac{1}{LC}

  • The square root in Eq. (3) is imaginary.

  • Current Profile: Represents a damped oscillation (Curve 1 in Figure 2):
    $I(t) = I_0 e^{-\frac{R}{2L} t} \cos(\omega t - \varphi) \quad (4)$

  • Behavior: Amplitude decays exponentially over time.

  • Angular Frequency: $\omega = \sqrt{\frac{1}{LC} - \left(\frac{R}{2L}\right)^2}$

  • Natural Frequency ($\omega<em>0$): The oscillation frequency for the undamped case ($R \to 0$):
    $\omega</em>0 = \sqrt{\frac{1}{LC}} \quad (5)$

  • Logarithmic Decrement ($\Lambda$):

    • Ratio of successive maxima (or minima) at $t$ and $t+T$ (where $T$ is the period):
      $\frac{I(t)}{I(t+T)} = e^{\frac{R}{2L} T} \quad (6)$

    • Defined as the natural logarithm of this ratio:
      $\Lambda = \ln \left( \frac{I(t)}{I(t+T)} \right) = \frac{R}{2L\nu} = \frac{\pi R}{\omega L} \quad (7)$

  • Quality Factor ($Q$):

    • Definition: The ratio of stored energy in the oscillating system to the energy loss per period:
      $Q = 2\pi \frac{E}{-\dot{E} T} \quad (8)$

    • Energy Storage: Energy oscillates between the capacitor's electric field and the inductor's magnetic field. At maximum current, all energy is in the inductor's magnetic field: $E = \frac{L I^2}{2}$.

    • Calculation: Given exponential decay of current amplitudes ($I<em>{\text{max}} = I</em>0 e^{-\frac{R}{2L} t}$):
      $Q = \frac{2\pi \nu L}{R} = \frac{\pi}{\Lambda} \quad (9)$

• Creep Case (Overdamped): \left(\frac{R}{2L}\right)^2 > \frac{1}{LC}

  • The roots $\lambda_{1,2}$ in Eq. (3) are real and negative.

  • Current Profile: A superposition of two decaying exponential functions (no oscillations).

  • Behavior: May exhibit one zero-crossing depending on initial conditions (Curves 2 or 3 in Figure 2).

• Aperiodic Critical Damping: $\left(\frac{R}{2L}\right)^2 = \frac{1}{LC}$

  • Degeneracy: Roots are degenerate ($\lambda<em>1 = \lambda</em>2 = -R/2L$).

  • General Solution: Requires a second linearly independent solution (e.g., via variation of constants):
    $I(t) = (I<em>0 + I</em>1 t) e^{-\frac{R}{2L} t} \quad (10)$

  • Behavior: No oscillation. Depending on initial conditions, $I(t)$ may pass through zero or not.

  • Special Significance: Achieves the fastest return of the system to its equilibrium position without oscillation. Used in pointer instruments.

  • Condition: Damping is adjusted to the natural frequency such that $\omega_0 = \frac{R}{2L}$.

Forced Oscillations

• Circuit with Voltage Source (Figure 3): Introducing a time-varying voltage source $U(t)$ leads to:
  $U<em>R + U</em>C + U_L = U(t) \quad (11)$

• Sinusoidal Excitation: For a sinusoidal voltage $U(t) = U<em>0 e^{i\omega t}$, the differential equation becomes: $L\ddot{I} + R\dot{I} + \frac{1}{C} I = i\omega U</em>0 e^{i\omega t} \quad (12)$

• Solution Components: The general solution is a superposition of:

  • The general solution of the homogeneous equation (Eq. 2), representing the transient (damped) behavior.

  • A particular solution of the inhomogeneous equation, representing the steady-state behavior (dominant after sufficient time).

• Steady-State Current Amplitude ($I<em>0$): Obtained by substituting $I(t) = I</em>0 e^{i\omega t}$ into Eq. (12):
  $I<em>0 = \frac{U</em>0}{R + i\left(\omega L - \frac{1}{\omega C}\right)} \quad (13)$

• Amplitude and Phase (Eq. 14): Representing $I<em>0$ by magnitude $|I</em>0|$ and phase $\varphi$:
  $I(t) = |I<em>0| e^{i(\omega t + \varphi)} \quad \text{where} \quad |I</em>0| = \frac{U_0}{\sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}} \quad \text{and} \quad \tan \varphi = \frac{\omega L - \frac{1}{\omega C}}{R} \quad (14)$

• Resonance Frequency ($\omega<em>0$): The amplitude $|I</em>0|$ reaches its maximum when the inductive and capacitive reactances cancel out ($\omega L - 1/\omega C = 0$).

  • This condition defines the resonance frequency: $\omega_0 = \frac{1}{\sqrt{LC}}$.

• Voltage Boosting at Resonance (Spannungsüberhöhung):

  • Amplitudes of voltages across the inductor and capacitor at resonance (Eq. 15):
    $|U<em>L(\omega</em>0)| = |L\dot{I}| = \frac{\omega<em>0 L U</em>0}{R} = Q U<em>0$ $|U</em>C(\omega<em>0)| = \left| \frac{1}{C} \int I dt \right| = \frac{1}{\omega</em>0 C} \frac{U<em>0}{R} = \frac{\omega</em>0 L U<em>0}{R} = Q U</em>0 \quad (15)$

  • Since $Q$ can be significantly greater than 1, $U<em>L$ and $U</em>C$ can exceed the source voltage $U_0$. This phenomenon is termed voltage resonance.

• Quality Factor ($Q$) in Forced Oscillations: Also crucial for resonance curve shape. $Q = \frac{\pi}{\Lambda} = \frac{\omega_0 L}{R} = \frac{1}{R} \sqrt{\frac{L}{C}} \quad (16)$

  • Higher $Q$ leads to narrower and steeper resonance curves (Figure 4).

• Half-Width ($\Delta \omega$):

  • A common parameter to characterize resonance curves and spectral lines.

  • Defined as the frequency difference between points where the signal is half of its resonance maximum. For oscillating circuits, it's usually defined where the current is $I_0/\sqrt{2}$ (corresponding to half power).

  • This occurs when $R = |\omega<em>{1,2} L - 1/\omega</em>{1,2} C|$.

  • Relation to Q (Eq. 17): For pronounced resonance ($\Delta \omega \ll \omega<em>0$), where the curve is symmetric around $\omega</em>0$: $\frac{1}{Q} \approx \frac{\Delta \omega}{\omega_0} \quad (17)$

    • The relative width of the resonance is inversely proportional to the quality factor.

• Complex Impedance ($Z$):

  • For an AC circuit, $U = |U| e^{i\omega t}$ and $I = |I| e^{i(\omega t + \varphi)}$.

  • Complex impedance is then time-independent: $Z = \frac{U}{I} = |Z| e^{-i\varphi}$

    • Magnitude: $|Z| = \sqrt{R^2 + \left(\omega L - \frac{1}{\omega C}\right)^2}$

  • The function $Z = Z(\varphi)$ is called the Z-locus curve; its reciprocal is the admittance (Y) locus curve.

Experimental Setup and Procedure

Part 1: Free Damped Oscillations

• Circuit Assembly: As in Figure 1b.

• Voltage Source: Rectangular pulse generator (RG) with adjustable duty cycle ($T<em>x / T</em>R$) (Figure 5).

  • The rectangular pulse simulates the switch: during the pulse, the capacitor charges (like Switch S position 1); after the pulse, the circuit oscillates freely (like Switch S position 2).

  • The period $T_R$ should be long enough for oscillations to decay before the next pulse.

• Current Measurement: Current $I$ is measured via the voltage drop across a small measurement resistor $R<em>m$. ($R</em>m = 100 \Omega$, sets minimum damping).

• Adjustable Resistance: A potentiometer (10 k$\Omega$) serves as the adjustable resistor $R$.

• Oscilloscope Use: Triggered by the rectangular pulses, displaying the periodic current waveform. Generator output should also be monitored on a second channel.

  • $T_x$ should be short but long enough for a detectable signal.

• Components: $C = 0.1 \mu F$ and $L \approx 120 \text{mH}$ (one half of a double coil, 1800 turns; exact value on the coil).

• Measurement Steps:

  1. Measure Natural Frequency ($\omega<em>0$): Determine $\omega</em>0$ using the oscilloscope at low damping.

  2. Measure Logarithmic Decrement ($\Lambda$): Set three different damping levels. Measure $\Lambda$ for each. Simultaneously, measure the total ohmic resistance $R$ in the circuit with a multimeter.

  3. Determine Aperiodic Critical Damping: Approach this condition from two directions:

     • Decrease $R$ gradually from an overdamped state.

     • Increase $R$ gradually from an underdamped state.

     • Measure $R$ for both approaches and estimate the error by comparing values. The aperiodic critical damping is identified by the fastest decay to zero current (though a zero-crossing is still possible).

     • Verification: Check if the condition $(R/2L)^2 = 1/LC$ is met.

Part 2: Forced Oscillations

• Circuit Assembly: As in Figure 3.

• Voltage Source: A PC-controlled