Category: Part 4: Oscillations & Waves

Problem Statement Analyze the circuit: Analyze the circuit: Compare series and parallel resonance: in which is the impedance minimum vs. maximum? In which does current maximize? Series RLC resonance ($\omega = \omega_0 = 1/\sqrt{LC}$): Impedance $|Z|$ is minimum (= $R$) Current from voltage source is maximum ($I = V/R$) Voltages across $ Given Information $\omega…

Problem Statement Analyze the circuit: Analyze the circuit: Define RMS (root-mean-square) values for AC quantities and find the RMS of $i = I_0\cos(\omega t)$. The RMS value of $f(t)$ is defined as: $$f_{\rm rms} = \sqrt{\langle f^2\rangle} = \sqrt{\frac{1}{T}\int_0^T f^2(t)\,dt}$$ For $i = I_0\cos(\omega t)$: $$i^2_{\rm rms} = \langle I Given Information See problem statement…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: Show that in a damped RLC circuit the energy decays as $W = W_0 e^{-2\beta t}$ and find the power loss. Total energy $W = Q^2/(2C) + LI^2/2$. With $Q = Q_0 e^{-\beta t}\cos(\omega_1 t + \phi)$, time-averaging over one cycle: $$\langle W\rangle = \frac{Q_0^2}{2C}e^…

Problem Statement Analyze the circuit: Analyze the circuit: Plot and describe how $X_L = \omega L$ and $X_C = 1/(\omega C)$ vary with frequency. At what frequency are they equal? Inductive reactance: $X_L = \omega L$ — increases linearly with $\omega$. At DC ($\omega=0$): $X_L = 0$ (inductor is a short). At high frequency: $X_L…

Problem Statement Analyze the circuit: Analyze the circuit: An ideal transformer has $n_1$ primary turns and $n_2$ secondary turns. Find the voltage ratio, current ratio, and impedance transformation ratio. For an ideal transformer (perfect coupling, no losses): $$\frac{V_2}{V_1} = \frac{n_2}{n_1} = n \quad \text{(turns ratio)}$$ Power c Given Information See problem statement for all given…

Problem Statement Analyze the circuit: Analyze the circuit: A parallel RLC circuit (ideal $L$, ideal $C$, real resistor $R$ in parallel) is driven by a current source $I = I_0\cos(\omega t)$. Find the impedance and anti-resonance frequency. Total admittance: $$Y = \frac{1}{R} + \frac{1}{i\omega L} + i\omega C = \frac{1}{R} + i\left(\omeg Given Information See…

Problem Statement Analyze the circuit: Analyze the circuit: For maximum power transfer from a source (internal impedance $Z_s$) to a load $Z_L$, what condition must $Z_L$ satisfy? With source $\mathcal{E}_s$, internal impedance $Z_s = R_s + iX_s$, and load $Z_L = R_L + iX_L$: $$I = \frac{\mathcal{E}_s}{Z_s + Z_L}, \qquad P_L = \frac{1}{2 Given Information…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: In a series RLC circuit, sketch and describe how the phase angle $\psi$ between current and EMF varies with frequency $\omega$. The phase angle (by which current leads the EMF): $$\tan\psi = -\frac{\omega L – 1/(\omega C)}{R} = -\frac{X_L – X_C}{R}$$ Behavior acro Given Information…

Problem Statement Analyze the circuit: Analyze the circuit: A series RLC circuit ($R = 10$ $\Omega$, $L = 35$ mH, $C = 80$ $\mu$F) is driven by $\mathcal{E}_0 = 100$ V. Find the current at resonance and the $Q$-factor. $$\omega_0 = \frac{1}{\sqrt{LC}} = \frac{1}{\sqrt{35\times10^{-3}\times80\times10^{-6}}} = \frac{1}{\sqrt{2.8\times10^{- Given Information See problem statement for all given quantities.…

Problem Statement Solve the oscillation/wave problem: Solve the oscillation/wave problem: A forced RLC circuit has resonance frequency $\omega_0$ and quality factor $Q$. Find the frequencies at which the current falls to $1/\sqrt{2}$ of its maximum value. Current amplitude: $I_0(\omega) = \mathcal{E}_0/\sqrt{R^2+(\omega L-1/(\omega C))^2}$ Maximum at $ Given Information See problem statement for all given quantities.…