Forced sinusoidal electrical oscillations

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1. 1. The RLC circuit driven by a generator

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

   We will build this step by step, from the circuit to resonance, with drawings and worked examples.

   Let's start with the circuit itself: a resistor, a coil, and a capacitor, all driven by a generator.

   This is called forced sinusoidal oscillations, that is, the generator pushes the circuit to oscillate at its own frequency.

   In other words, the circuit no longer chooses its own rhythm, the generator imposes it.

   We connect a resistor of resistance R, a coil of inductance L, and a capacitor of capacitance C.

   The generator delivers a voltage that changes with time in a sine pattern.

   Here, U_m is the maximum voltage, and omega is the angular frequency, in radians per second.

   The current in the circuit also follows a sine pattern, but it can lag behind the voltage.

   This lag is called the phase difference, written as the Greek letter phi.

   Here is why the current can lag: the coil and the capacitor react slowly to changes in voltage.

   The coil opposes a fast rise in current, and the capacitor needs time to charge.

   So the current's shape matches the voltage's shape, but shifted a little in time.

   Here is the voltage curve, a smooth sine wave over time.

   Here is the current curve, the same shape, but shifted to the right.

   When phi is zero, the current and the voltage move exactly together, in phase.

   Now let's work through a typical example from this part of the chapter.

   A generator drives a series circuit with R equal to 50 ohms, L equal to 0,2 henry, and C equal to 10 microfarads.

   The generator's voltage is u of t equals 10 sine of 100 pi times t, in volts.

   First, we read the maximum voltage directly from this formula.

   Next, we read the angular frequency, the number multiplying t inside the sine.

   From omega, we find the frequency f, using omega equals two pi f.

   This frequency is exactly what the generator forces onto the whole circuit.

   We also find the period T, the time for one full oscillation.

   So this circuit oscillates fifty times every second, each cycle lasting a tiny fraction of a second.

   Notice that the frequency here depends only on the generator, not on R, L, or C.

   The values of R, L, and C will instead control the amplitude and the phase shift.

   We will see exactly how in the next sections, using impedance.

   For now, remember the key idea: the generator imposes both the shape and the frequency of the oscillation.

   The circuit's own current just follows along, with its own amplitude I_m and its own phase lag phi.

   This whole situation, a generator forcing a sine wave onto an RLC circuit, is the subject of this entire chapter.

   What does the generator impose on the RLC circuit's current?

2. 2. Impedance and Ohm's law in AC

   Now let's move to impedance, the quantity that links voltage and current in this circuit.

   This quantity is called electrical impedance, that is, it measures how much the circuit resists the changing current.

   Impedance plays the same role here that resistance plays in a simple direct current circuit.

   This relation is the AC version of Ohm's law, linking the maximum voltage and the maximum current.

   Impedance is measured in ohms, the same unit as resistance.

   For the series RLC circuit, impedance depends on R, on L, on C, and on the angular frequency omega.

   The term L omega comes from the coil, and the term 1 over C omega comes from the capacitor.

   Here is why this formula looks this way: the coil and the capacitor act oppositely on the current.

   The coil's effect grows with frequency, but the capacitor's effect shrinks with frequency.

   So their two effects partly cancel out inside the difference, L omega minus 1 over C omega.

   Only the resistor's effect, R, never changes with frequency at all.

   Now let's apply this to a worked example close to a real bac exercise.

   Take a series circuit with R equal to 100 ohms, L equal to 0,5 henry, and C equal to 20 microfarads.

   The generator's angular frequency is omega equals 200 radians per second.

   First, we compute the coil's term, L times omega.

   Next, we compute the capacitor's term, one over C times omega.

   We subtract these two terms, since they oppose each other inside the circuit.

   Now we square this difference and add it to R squared.

   Finally, the impedance Z is the square root of that total.

   If the generator's voltage amplitude U_m is 20 volts, we find the current amplitude I_m from Ohm's law.

   This gives I_m equal to 20 volts divided by Z, in amperes.

   So impedance lets us predict the current's amplitude without ever drawing the full curve.

   Remember: impedance Z combines R, L, C, and omega into one single number, in ohms.

   And Ohm's law in AC links U_m and I_m through this single number Z.

   Next, we will see how to find phi, the phase lag, using a geometric method.

   What unit is impedance measured in?

3. 3. Fresnel construction

   Now let's look at a geometric tool to handle these sine signals more easily.

   It is called the Fresnel construction, that is, we represent each sine quantity as a rotating arrow, called a vector.

   In other words, instead of drawing curves over time, we draw arrows at one fixed instant.

   Each arrow's length equals the signal's amplitude, and its angle equals the signal's phase.

   We draw the voltage across the resistor along the same direction as the current, since they are always in phase.

   We draw the voltage across the coil turned ninety degrees ahead of the current.

   We draw the voltage across the capacitor turned ninety degrees behind the current.

   Here is why these ninety degree turns appear: the coil and the capacitor shift voltage and current differently from a resistor.

   The resistor's voltage always matches the current's direction exactly, with no turn at all.

   The coil and the capacitor each introduce their own fixed turn, one forward and one backward.

   Adding all three vectors together, tip to tail, gives the generator's total voltage vector.

   Now let's apply this to a worked example, continuing our earlier circuit values.

   Take R equal to 100 ohms, L omega equal to 100,0 ohms, and 1 over C omega equal to 250,0 ohms.

   With a current amplitude I_m of 0,111 ampere, we find each voltage amplitude using Ohm's law on each part.

   The resistor's voltage amplitude is R times I_m.

   The coil's voltage amplitude is L omega times I_m.

   The capacitor's voltage amplitude is 1 over C omega times I_m.

   Now the phase angle phi comes from the right triangle formed by these vectors.

   We plug in our values, giving tan phi equal to negative 150 over 100.

   Taking the inverse tangent gives us the phase angle phi, in radians or degrees.

   A negative phi here means the current actually leads the voltage, since the capacitor's effect dominates.

   This diagram, built from simple vectors, replaces long calculations with time functions.

   Remember: the Fresnel construction turns sine signals into fixed arrows with a length and an angle.

   The angle between the current arrow and the total voltage arrow is exactly phi.

   Next, we will use this same diagram to explain resonance.

   In the Fresnel diagram, what does the length of each vector represent?

4. 4. Current resonance

   Now let's study what happens when we change the generator's frequency.

   This special situation is called current resonance, that is, the current reaches its biggest possible amplitude.

   In other words, for one special frequency, the circuit lets the current flow as strongly as possible.

   Resonance happens exactly when the coil's term cancels the capacitor's term inside the impedance.

   At this special angular frequency, omega zero, impedance becomes as small as it can be.

   When impedance equals only R, the current amplitude reaches its maximum value, U_m over R.

   We find omega zero by solving L omega zero equals 1 over C omega zero.

   Here is why this frequency is special: the coil and capacitor's opposite effects fully cancel only here.

   Away from omega zero, one of the two effects dominates, and impedance grows larger than R.

   So the current amplitude is largest exactly at omega zero, and smaller everywhere else.

   This bell-shaped curve shows current amplitude against angular frequency, peaking at omega zero.

   At resonance, phi also becomes zero, meaning current and voltage move perfectly together.

   Now let's solve a worked example about finding this resonance frequency.

   Take a series circuit with L equal to 0,5 henry and C equal to 20 microfarads.

   We compute L times C first, before taking the square root.

   Omega zero equals one over the square root of this product.

   From omega zero, we also find the resonance frequency f zero, using omega equals two pi f.

   If R equals 100 ohms and U_m equals 20 volts, the maximum current is U_m over R.

   So, tuning the generator to exactly 50,3 hertz gives this circuit its strongest possible current.

   Remember: resonance happens only at one frequency, omega zero equal to one over the square root of L C.

   At that frequency, impedance drops to its minimum, equal to R alone.

   Next, we will see how this strong current relates to power and energy.

   At resonance, what is the impedance Z equal to?

5. 5. Power factor and energy

   Now let's study the electrical power that this circuit actually uses.

   The average power delivered to the circuit depends on U_m, I_m, and the phase angle phi.

   The term cosine of phi here is called the power factor, that is, it tells how much of the power is really used.

   In other words, a power factor close to one means almost all the delivered power does useful work.

   Only the resistor actually uses electrical power, turning it into heat.

   Here, I_eff is the effective current, a steady value equivalent to the real changing current.

   Here is why only R consumes power: the coil and the capacitor only store and return energy.

   Over one full cycle, the coil and the capacitor give back exactly what they received.

   So their average power is always zero, and only the resistor's heat remains.

   At resonance, phi is zero, so cosine of phi equals one, and power is largest.

   Now let's work through an example using our earlier resonance values.

   Take R equal to 100 ohms, U_m equal to 20 volts, and I_m equal to 0,200 ampere, at resonance.

   First, we find the effective current, I_m divided by the square root of two.

   Then, we find the power using P equals R times I_eff squared.

   We can check this using the other formula, with cosine phi equal to one at resonance.

   Both methods agree closely, confirming that at resonance, power reaches its maximum value.

   Remember: real power only depends on the resistor, through R times I_eff squared.

   The power factor, cosine of phi, shows how efficiently the generator's energy turns into heat.

   At resonance, this factor reaches its biggest value, equal to one.

   What is the value of the power factor at resonance?

6. 6. Quality factor and sharpness

   Now let's finish with how sharp or wide the resonance peak can be.

   This sharpness is described by the quality factor, also called the overvoltage factor.

   That is, it compares the voltage across the coil at resonance to the generator's own voltage.

   A large Q means the coil's voltage can be much bigger than the generator's voltage.

   This formula also uses L, omega zero, and R, all quantities we already found earlier.

   Here is why Q controls the peak's shape: a small R lets the current rise very high at resonance.

   A small R also makes the bell-shaped curve narrow, since the current drops quickly away from omega zero.

   A large R flattens the curve, so the peak becomes wide and less pronounced.

   Here is a sharp curve, for a small resistance R.

   Here is a wide curve, for a larger resistance R.

   Now let's solve a worked example using our earlier circuit values.

   Take L equal to 0,5 henry, R equal to 100 ohms, and omega zero equal to 316,2 radians per second.

   We multiply L by omega zero first, before dividing by R.

   Then we divide this result by R to find the quality factor Q.

   A Q value above one here means the coil's voltage exceeds the generator's own voltage at resonance.

   This whole chapter connects a generator, an RLC circuit, impedance, resonance, power, and this sharpness factor.

   Remember: Q equals L omega zero over R, and small R gives a sharp, narrow resonance peak.

   Large Q also means a large overvoltage across the coil compared to the generator.

   This finishes our tour of forced sinusoidal electrical oscillations in the RLC circuit.

   Does a small resistance R make the resonance peak sharp or wide?

   That completes our chapter on forced sinusoidal electrical oscillations in the RLC circuit.

   We went from the driven circuit, through impedance and Fresnel vectors, to resonance, power, and quality factor.

   Practice the two exercises to make these formulas and methods fully your own.