Free electrical oscillations

What Nour says

1. 1. The RLC series circuit

   Today, the whole chapter: "Free electrical oscillations", Physics, Year 4, Mathematics section.

   We will look at a circuit with a coil, a resistor and a capacitor, and watch how its charge and current behave over time.

   We move step by step: the circuit itself, then damped oscillations, then energy, then the ideal undamped case and its period.

   Let's start with the RLC series circuit.

   This is a circuit with three parts: a coil, a resistor, and a capacitor, all in one loop.

   That is, the same current flows through every part, one after another.

   You already know the capacitor: it stores charge q and the voltage across it.

   You also know the coil: it resists sudden changes in current and has inductance L.

   The resistor R simply uses up electrical energy, turning it into heat.

   To study free oscillations, we first charge the capacitor using the battery E.

   Then we switch the circuit so the capacitor is alone with the coil and resistor.

   No more battery: the capacitor now drives the circuit by itself.

   This is why we call it free: nothing pushes the circuit except its own stored charge.

   The capacitor pushes current out, the coil pushes back, and the resistor slows everything down.

   This battle between the three parts makes the charge and current oscillate before dying out.

   This equation just says: going around the loop, the voltages add up to zero.

   Let's write each voltage using known quantities: charge q, current i, and their derivatives.

   Remember that current i is the rate of change of charge q over time.

   Putting it all together gives the differential equation of the RLC series circuit.

   This single equation controls how q changes with time in the whole circuit.

   Let's see a worked example with real values from a typical circuit.

   We plug these three values into our general equation.

   This describes exactly how the charge on this capacitor evolves once we close the switch.

   Physicists often simplify this by dividing every term by the inductance L.

   This form is the one we will use again in the sections ahead.

   The key result: the RLC series circuit obeys one differential equation linking charge, current and their rates.

   This equation is the starting point for everything we study in this chapter.

   The exact shape of the solution, oscillating or not, depends on how big R is.

   What stores the charge that drives the circuit once the battery is disconnected?

2. 2. Damped free oscillations

   Now let's talk about damped free oscillations.

   That is, the charge oscillates back and forth, but each swing gets smaller than the last.

   This happens because the resistor R always takes energy away from the circuit.

   Let's draw the charge q as a function of time t.

   The curve swings up and down, crossing zero many times, like a wave.

   Notice how each peak is lower than the one before it.

   An invisible envelope shrinks the oscillation over time, pulling it toward zero.

   This is why we call these oscillations damped: the resistor slowly destroys the motion.

   If R is small, the circuit oscillates many times before stopping, almost like the ideal case.

   If R is large, the charge barely moves before settling straight to zero.

   In that case, there is no oscillation at all, just a slow return to rest.

   Let's look at a worked example to see this more concretely.

   Suppose we test two resistors: one with R equal to 10 ohms, one with R equal to 2000 ohms.

   With the small resistor R1, the circuit swings several times before the charge dies out.

   With the large resistor R2, the energy disappears so fast there is barely any swing.

   Engineers choose R carefully, depending on whether they want oscillations or a quick, smooth stop.

   We can also see this damping on an oscilloscope, by measuring the voltage across the resistor.

   The screen shows the same shrinking wave pattern we just drew on the board.

   The key result: in a real RLC circuit, resistance always causes the oscillations to fade over time.

   The amplitude shrinks following the envelope, while the oscillation frequency stays roughly the same.

   Next we will see exactly where all that lost energy goes.

   What causes the oscillation amplitude to shrink over time in this circuit?

3. 3. Total energy and its non-conservation

   Now let's talk about total energy and its non-conservation.

   That is, the circuit's total stored energy does not stay constant, it keeps decreasing.

   The total energy has two parts: the capacitor's energy and the coil's energy.

   The capacitor stores energy through its charge, the coil stores energy through its current.

   During oscillation, energy keeps passing back and forth between these two parts.

   But every time current flows through the resistor, some energy turns into heat.

   This heat leaves the circuit forever; it never comes back as electrical energy.

   That is why the total energy E keeps falling, instead of staying the same.

   The rate of energy loss equals exactly the power burned by the resistor.

   Let's draw the total energy as a function of time, to see this clearly.

   The curve starts high and drops in a wavy but steadily decreasing way.

   Each little bump is energy moving between capacitor and coil, but the overall trend falls.

   Let's see a worked example with numbers from our earlier circuit.

   Suppose at some instant the charge is 4 microcoulombs and the current is 2 milliamperes.

   We compute the capacitor's energy using charge and capacitance, and the coil's energy using current.

   This small number tells us how much energy remains stored right at this moment.

   A moment later this value will be a little smaller, because of the resistor.

   The key result: total energy in a real RLC circuit always decreases, because of resistance.

   This loss of energy is exactly what makes the oscillations fade, as we saw before.

   Without any resistance at all, this story would be completely different.

   Where does the circuit's lost energy actually go?

4. 4. Undamped free oscillations

   Now let's talk about undamped free oscillations.

   That is, we imagine a circuit with no resistance at all, so nothing ever fades.

   This is an ideal case, useful for understanding the pure behavior of L and C together.

   We take our earlier equation and simply remove the resistor term.

   Setting R to zero, we get a simpler equation with only L and C.

   This equation describes a charge that oscillates forever, with constant amplitude.

   Let's draw this charge q as a function of time t.

   The curve is a perfect, steady wave, repeating the same pattern again and again.

   Unlike before, there is no shrinking envelope: the peaks stay the same height forever.

   The solution of this equation has a known mathematical form, using cosine.

   Here Qm is the maximum charge, and T0 is the time for one full oscillation.

   Phi is the phase, which depends on the charge's value at the very start.

   Let's see a worked example with actual numbers for this ideal circuit.

   Suppose at time zero the charge equals 5 microcoulombs and the current is zero.

   Since the current is zero at the start, the cosine's phase must be zero too.

   So the maximum charge Qm simply equals the starting charge of 5 microcoulombs.

   The key result: without resistance, charge oscillates forever with the same unchanging amplitude.

   This perfect, lasting oscillation is why we call it undamped, unlike the damped case before.

   Next we will find exactly how fast this oscillation repeats itself.

   What stays constant in this ideal circuit, unlike in the damped case?

5. 5. The period of undamped oscillations

   Now let's talk about the period of undamped oscillations.

   The period T0 is the time it takes for the charge to repeat one full cycle.

   It depends only on the two parts left in our ideal circuit: L and C.

   This is called the period formula, linking inductance L and capacitance C to time.

   A bigger inductance or a bigger capacitance both make the period longer.

   In other words, the circuit swings more slowly when L or C increases.

   Let's draw one full cycle of the charge, marking the period clearly.

   One full swing, from peak to peak, takes exactly one period T0.

   Let's see a worked example using our circuit values from before.

   We plug L and C directly into the period formula.

   This gives a period of about 0.00397 seconds.

   This tiny period means the charge oscillates very fast, many times each second.

   The key result: the period T0 only depends on L and C, not on the amplitude.

   Even a much bigger starting charge gives the exact same period T0.

   Next we will see how energy itself moves during one of these cycles.

   If you double the capacitance C, does the period T0 get longer or shorter?

6. 6. Energy exchange in undamped oscillations

   Now let's talk about energy exchange in undamped oscillations.

   In this ideal circuit, energy never leaves, it just moves between the capacitor and the coil.

   The total energy stays exactly the same at every single instant.

   When the charge is at its maximum, all the energy sits in the capacitor.

   At that same moment, the current is zero, so the coil holds no energy.

   A little later, the charge crosses zero, but the current reaches its maximum value.

   Now all the energy has moved into the coil, and the capacitor is empty.

   The two curves show this trade: when one goes up, the other goes down.

   Their sum, the total energy, stays perfectly flat and constant the whole time.

   Let's see a worked example using our familiar circuit values.

   At maximum charge, the total energy equals the capacitor's energy alone.

   This same total energy later appears entirely as the coil's energy, at maximum current.

   The key result: total energy stays constant, only shifting between electric and magnetic forms.

   This perfect exchange, without any loss, is only possible because there is no resistor.

   Real circuits always lose a little energy, as we saw earlier in this chapter.

   At the moment the current is maximum, where is all the circuit's energy stored?

   That completes our chapter on free electrical oscillations, from the damped RLC circuit to the ideal undamped case.

   Keep in mind how resistance changes everything: it drains energy and fades the oscillations away.

   Practice both worked exercises to prepare well for your bac exam.