IPhO 1987 · Theory problem 3

Infinite LC-grid

Download PDFIPhO 1987

When sine waves propagate in an infinite LC-grid (see the figure below) the phase of the ac-voltage across two successive capacitors differs by \(\Phi\).

Figure 1

a) Determine how \(\Phi\) depends on \(\omega\), L and C (\(\omega\) is the angular frequency of the sine wave).

b) Determine the velocity of propagation of the waves if the length of each unit is \(\ell\).

c) State under what conditions the propagation velocity of the waves is almost independent of \(\omega\). Determine the velocity in this case.

d) Suggest a simple mechanical model which is an analogue to the above circuit and derive equations which establish the validity of your model.

Formulae:

\[ \cos\alpha-\cos\beta =-2\cdot\sin\left(\frac{\alpha+\beta}{2}\right) \cdot\sin\left(\frac{\alpha-\beta}{2}\right) \]

\[ \sin\alpha-\sin\beta =2\cdot\cos\left(\frac{\alpha+\beta}{2}\right) \cdot\sin\left(\frac{\alpha-\beta}{2}\right) \]

Official English text of the problem.

Answers

  • a) The dispersion relation is \[ \boxed{\omega^2LC=4\sin^2\frac{\Phi}{2}}. \] Taking the principal phase lag \(0\leq\Phi\leq\pi\), \[ \boxed{\Phi=2\arcsin\left(\frac{\omega\sqrt{LC}}{2}\right)}. \] A real phase lag requires \(0\leq\omega\leq\omega_c=2/\sqrt{LC}\).

  • b) The phase velocity of the sinusoidal wave is \[ \boxed{v_{\mathrm{ph}}=\frac{\omega\ell}{\Phi} =\frac{\omega\ell} {2\arcsin\left(\omega\sqrt{LC}/2\right)}}. \] For completeness, the group velocity of a narrow wave packet is \[ \boxed{v_{\mathrm g}=\frac{\ell}{\sqrt{LC}} \sqrt{1-\frac{\omega^2LC}{4}}}. \]

  • c) For \(\omega\sqrt{LC}\ll1\), equivalently \(\Phi\ll1\), the propagation velocity is approximately independent of frequency: \[ \boxed{v_{\mathrm{ph}}\simeq v_{\mathrm g}\simeq \frac{\ell}{\sqrt{LC}}}. \]

  • d) An infinite chain of identical masses \(m\), separated by \(\ell\) and joined by identical massless springs of stiffness \(\kappa\), is an analogue. Its equation \[ m\ddot x_n=\kappa(x_{n-1}-2x_n+x_{n+1}) \] matches the circuit equation \[ L\ddot q_n=\frac{1}{C}(q_{n-1}-2q_n+q_{n+1}), \] where \(\dot q_n\) is the current through the \(n\)-th inductor. Thus the correspondence is \(m\leftrightarrow L\), \(\kappa\leftrightarrow1/C\).

Solution

Assume ideal, identical inductors and capacitors and negligible resistance. Number the capacitors by the integer \(n\). Let \(u_n(t)\) be the voltage across capacitor \(n\), and let \(i_n(t)\) flow through the inductor from node \(n\) to node \(n+1\).

a)

Kirchhoff’s current law at node \(n\) gives \[ C\frac{du_n}{dt}=i_{n-1}-i_n. \] The voltage–current relation for each inductor gives \[ L\frac{di_n}{dt}=u_n-u_{n+1}. \]

Differentiating the first equation and substituting the inductor relations, \[ LC\frac{d^2u_n}{dt^2} =(u_{n-1}-u_n)-(u_n-u_{n+1}), \] so \[ \boxed{LC\ddot u_n=u_{n-1}-2u_n+u_{n+1}}. \]

For a sinusoidal travelling wave with phase lag \(\Phi\) per cell, write \[ u_n(t)=U\cos(\omega t-n\Phi). \] The neighbouring voltages satisfy \[ u_{n-1}+u_{n+1}=2u_n\cos\Phi, \] while \(\ddot u_n=-\omega^2u_n\). Therefore \[ -\omega^2LC=2\cos\Phi-2, \] or \[ \boxed{\cos\Phi=1-\frac{\omega^2LC}{2}}. \]

Using \(1-\cos\Phi=2\sin^2(\Phi/2)\), \[ \boxed{\omega^2LC=4\sin^2\frac{\Phi}{2}}. \] On the principal branch, \[ \boxed{\Phi=2\arcsin\left(\frac{\omega\sqrt{LC}}{2}\right)}. \] Reversing the propagation direction reverses the signed phase lag.

Since \(\sin^2(\Phi/2)\leq1\), real travelling-wave modes are possible only within the frequency band bounded by \[ \boxed{\omega_c=\frac{2}{\sqrt{LC}}}. \] Above this cutoff, there is no propagating mode with real \(\Phi\).

b)

A constant phase satisfies \[ \omega t-n\Phi=\text{constant}. \] Consequently, the time for the same phase to advance by one cell is \[ \Delta t=\frac{\Phi}{\omega}. \] As the cell length is \(\ell\), the phase velocity is \[ \boxed{v_{\mathrm{ph}}=\frac{\ell}{\Delta t} =\frac{\omega\ell}{\Phi} =\frac{\omega\ell} {2\arcsin\left(\omega\sqrt{LC}/2\right)}}. \]

Equivalently, the wave number is \(k=\Phi/\ell\), and \(v_{\mathrm{ph}}=\omega/k\).

The problem specifies sine waves, so this phase velocity is the requested velocity. If one instead follows the envelope of a narrow wave packet, its velocity is the group velocity. From \[ \omega(k)=\frac{2}{\sqrt{LC}}\sin\frac{k\ell}{2}, \qquad 0\leq k\ell\leq\pi, \] we obtain \[ \boxed{v_{\mathrm g}=\frac{d\omega}{dk} =\frac{\ell}{\sqrt{LC}}\cos\frac{\Phi}{2} =\frac{\ell}{\sqrt{LC}} \sqrt{1-\frac{\omega^2LC}{4}}}. \]

At the cutoff, adjacent capacitor voltages oscillate in antiphase, \(\Phi=\pi\), and the group velocity vanishes.

c)

When the phase change across one cell is small, \[ \Phi\ll1, \] we may use \[ \sin\frac{\Phi}{2}\simeq\frac{\Phi}{2}. \] The dispersion relation then becomes \[ \omega^2LC\simeq\Phi^2, \qquad \Phi\simeq\omega\sqrt{LC}. \] Thus the required frequency condition is \[ \boxed{\omega\sqrt{LC}\ll1}, \] and \[ \boxed{v_{\mathrm{ph}}\simeq\frac{\ell}{\sqrt{LC}}}. \]

The group velocity has the same limit. Physically, this is the long-wavelength regime: many cells lie within one wavelength, so the discrete grid behaves approximately as a continuous transmission line.

For example, the leading correction to the phase velocity is \[ v_{\mathrm{ph}} =\frac{\ell}{\sqrt{LC}} \left[1-\frac{\omega^2LC}{24} +O\!\left((\omega^2LC)^2\right)\right]. \] Hence its frequency dependence is small under the stated condition. Since \(LC\) has units of \(\mathrm{s^2}\), \(\ell/\sqrt{LC}\) has units of velocity.

d)

Consider an infinite straight chain of identical masses \(m\), with equilibrium separation \(\ell\). Connect each pair of neighbouring masses by a massless spring of stiffness \(\kappa\). The masses move longitudinally without friction, and the springs obey Hooke’s law.

Let \(x_n(t)\) be the displacement of mass \(n\) from equilibrium. The right-hand spring exerts the force \[ \kappa(x_{n+1}-x_n), \] and the left-hand spring exerts \[ -\kappa(x_n-x_{n-1}). \] Newton’s second law therefore gives \[ \boxed{m\ddot x_n =\kappa(x_{n-1}-2x_n+x_{n+1})}. \]

To establish a direct circuit correspondence, define \(q_n(t)\) as the charge that has passed through inductor \(n\), so that \[ i_n=\dot q_n. \] Taking quantities relative to equilibrium, charge conservation at capacitor \(n\) gives \[ Cu_n=q_{n-1}-q_n. \] The inductor equation consequently becomes \[ L\ddot q_n=u_n-u_{n+1} =\frac{q_{n-1}-2q_n+q_{n+1}}{C}. \] Thus \[ \boxed{L\ddot q_n =\frac1C(q_{n-1}-2q_n+q_{n+1})}, \] which has precisely the same form as the mechanical equation under \[ \boxed{x_n\leftrightarrow q_n,\qquad m\leftrightarrow L,\qquad \kappa\leftrightarrow\frac1C}. \]

The energy expressions confirm this correspondence: \[ E_{\mathrm{LC}} =\sum_n\left[ \frac12L\dot q_n^{\,2} +\frac{(q_{n-1}-q_n)^2}{2C} \right], \] \[ E_{\mathrm{mech}} =\sum_n\left[ \frac12m\dot x_n^{\,2} +\frac12\kappa(x_{n-1}-x_n)^2 \right]. \] Inductive energy corresponds to kinetic energy, and capacitor energy corresponds to spring potential energy.

Finally, substituting \(x_n=A\cos(\omega t-n\Phi)\) into the mechanical equation yields \[ \omega^2=\frac{4\kappa}{m}\sin^2\frac{\Phi}{2}, \] identical to the LC dispersion relation when \(\kappa/m=1/(LC)\).

The solution was checked against the official solution.