Power in AC Circuits and LC Oscillations

Power In Ac Circuit: The Power Factor ($ P_{avg} = V_{rms}I_{rms}\cos\phi $)

In AC circuits, the instantaneous voltage and current are continuously varying. The power delivered to a circuit also varies with time. While instantaneous power is sometimes useful, we are often more interested in the average power delivered to the circuit over a complete cycle. This is the power that is actually consumed by the circuit and converted into other forms of energy (like heat, light, or mechanical work).

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Instantaneous Power

Let the instantaneous voltage across an AC circuit element or a circuit be $v(t) = V_0 \sin(\omega t)$. Let the instantaneous current through it be $i(t) = I_0 \sin(\omega t + \phi)$, where $V_0$ and $I_0$ are the peak voltage and current, $\omega$ is the angular frequency, and $\phi$ is the phase difference by which the current leads the voltage.

The instantaneous power $p(t)$ delivered to the circuit element at time $t$ is the product of the instantaneous voltage and current:

$ p(t) = v(t) i(t) = [V_0 \sin(\omega t)] [I_0 \sin(\omega t + \phi)] $

$ p(t) = V_0 I_0 \sin(\omega t) \sin(\omega t + \phi) $

Using the trigonometric identity $\sin A \sin B = \frac{1}{2}[\cos(A-B) - \cos(A+B)]$:

$ p(t) = V_0 I_0 \cdot \frac{1}{2} [\cos(\omega t - (\omega t + \phi)) - \cos(\omega t + (\omega t + \phi))] $

$ p(t) = \frac{V_0 I_0}{2} [\cos(-\phi) - \cos(2\omega t + \phi)] $

Since $\cos(-\phi) = \cos\phi$:

$ p(t) = \frac{V_0 I_0}{2} [\cos\phi - \cos(2\omega t + \phi)] $

$ p(t) = \frac{V_0 I_0}{2} \cos\phi - \frac{V_0 I_0}{2} \cos(2\omega t + \phi) $

This expression shows that the instantaneous power consists of two parts: a constant term $\frac{V_0 I_0}{2} \cos\phi$ and a time-varying term $\frac{V_0 I_0}{2} \cos(2\omega t + \phi)$ that oscillates at twice the angular frequency ($2\omega$).

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Average Power Over a Cycle

The average power over one complete cycle is the average value of $p(t)$. The average value of the cosine term $\cos(2\omega t + \phi)$ over a complete cycle is zero.

$ P_{avg} = \langle p(t) \rangle = \left\langle \frac{V_0 I_0}{2} \cos\phi \right\rangle - \left\langle \frac{V_0 I_0}{2} \cos(2\omega t + \phi) \right\rangle $

$ P_{avg} = \frac{V_0 I_0}{2} \cos\phi - 0 $

$ P_{avg} = \frac{V_0 I_0}{2} \cos\phi $

We can express this average power in terms of RMS values. Recall that $V_{rms} = V_0/\sqrt{2}$ and $I_{rms} = I_0/\sqrt{2}$. Thus $V_0 I_0 = (\sqrt{2} V_{rms}) (\sqrt{2} I_{rms}) = 2 V_{rms} I_{rms}$.

$ P_{avg} = \frac{(2 V_{rms} I_{rms})}{2} \cos\phi $

$ P_{avg} = V_{rms} I_{rms} \cos\phi $

This is the formula for the average power consumed in an AC circuit.

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The Power Factor ($\cos\phi$)

In the average power formula $P_{avg} = V_{rms} I_{rms} \cos\phi$, the term $\cos\phi$ is called the power factor of the circuit.

Here, $\phi$ is the phase angle between the voltage and the current. $\phi > 0$ means current lags voltage, $\phi < 0$ means current leads voltage. Note that $\cos(-\phi) = \cos\phi$, so the direction of phase difference doesn't affect the power factor magnitude, only the nature of the load (inductive or capacitive).

The power factor ranges between 0 and 1, as $\cos\phi$ is between 0 and 1 for $0 \le |\phi| \le 90^\circ$.

The power factor indicates how effectively the electrical power is being used in the circuit.

• If $\phi = 0^\circ$ (pure resistive circuit), $\cos\phi = \cos 0^\circ = 1$. Power factor is 1. $P_{avg} = V_{rms} I_{rms}$. All the power delivered is consumed and dissipated.
• If $\phi = \pm 90^\circ$ (pure inductive or pure capacitive circuit), $\cos\phi = \cos(\pm 90^\circ) = 0$. Power factor is 0. $P_{avg} = 0$. No power is consumed on average over a cycle. Energy is just exchanged with the source.
• For circuits containing a combination of R, L, and C, $0 < |\phi| < 90^\circ$, so $0 < \cos\phi < 1$. The power factor is between 0 and 1. The circuit consumes some power due to resistance, but also exchanges energy with reactive components (L and C).

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Power Consumption in Individual Components

Let's look at the average power consumed by individual pure components:

• Resistor: $v_R$ and $i$ are in phase ($\phi = 0$). $P_{avg, R} = V_{R, rms} I_{rms} \cos 0^\circ = V_{R, rms} I_{rms} = I_{rms}^2 R = V_{R, rms}^2/R$. Resistors consume and dissipate power.
• Inductor: $v_L$ leads $i$ by $90^\circ$ ($\phi = +90^\circ$). $P_{avg, L} = V_{L, rms} I_{rms} \cos 90^\circ = 0$. Pure inductors do not consume power on average.
• Capacitor: $v_C$ lags $i$ by $90^\circ$ ($\phi = -90^\circ$). $P_{avg, C} = V_{C, rms} I_{rms} \cos(-90^\circ) = 0$. Pure capacitors do not consume power on average.

In a series LCR circuit, the total average power consumed is the sum of the average power consumed by each component:

$ P_{avg, total} = P_{avg, R} + P_{avg, L} + P_{avg, C} = I_{rms}^2 R + 0 + 0 = I_{rms}^2 R $

This must be equal to the total average power delivered by the source, $V_{rms} I_{rms} \cos\phi$.

$ V_{rms} I_{rms} \cos\phi = I_{rms}^2 R $

Since $I_{rms} = V_{rms}/Z$:

$ V_{rms} \left(\frac{V_{rms}}{Z}\right) \cos\phi = \left(\frac{V_{rms}}{Z}\right)^2 R $

$ \frac{V_{rms}^2}{Z} \cos\phi = \frac{V_{rms}^2}{Z^2} R $

Assuming $V_{rms} \neq 0$:

$ \frac{1}{Z} \cos\phi = \frac{R}{Z^2} \implies \cos\phi = \frac{R}{Z} $

This is another important expression for the power factor in an LCR circuit (or any AC circuit with equivalent series impedance $Z$). The power factor is the ratio of the total resistance to the total impedance. Since $Z = \sqrt{R^2 + (X_L - X_C)^2}$, $\cos\phi = \frac{R}{\sqrt{R^2 + (X_L - X_C)^2}}$.

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Apparent Power and Real Power

In AC circuits, the product $V_{rms} I_{rms}$ is called the apparent power. It is the maximum possible power that the circuit *could* consume if the power factor were 1. The unit of apparent power is Volt-Ampere (VA).

The average power $P_{avg} = V_{rms} I_{rms} \cos\phi$ is called the real power or true power. This is the power actually consumed by the resistive part of the circuit and converted into heat or useful work. The unit of real power is Watt (W).

$ \text{Real Power} = \text{Apparent Power} \times \text{Power Factor} $

The term $V_{rms} I_{rms} \sin\phi$ is called the reactive power. It represents the power that flows back and forth between the source and the reactive components (L and C), not being consumed. The unit of reactive power is Volt-Ampere Reactive (VAR). Reactive power is related to the energy stored and released by inductors and capacitors.

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Significance of Power Factor

A low power factor ($\cos\phi < 1$, i.e., $|\phi| > 0$) means that for a given amount of real power consumed, the apparent power and thus the RMS current drawn from the source is higher than necessary.

$ I_{rms} = \frac{P_{avg}}{V_{rms} \cos\phi} $

For a fixed voltage $V_{rms}$ and desired power $P_{avg}$, a lower $\cos\phi$ requires a higher $I_{rms}$. This has several disadvantages in power transmission and distribution:

• Higher current leads to more power loss as heat in transmission lines ($P_{loss} = I_{rms}^2 R_{lines}$).
• Higher current requires thicker and more expensive wiring.
• Higher current demands larger capacities for transformers and generators.

Therefore, power utilities often penalise consumers (especially industrial ones) for operating with a low power factor. Power factor correction techniques (typically adding capacitors in parallel to inductive loads) are employed to bring the overall circuit power factor closer to 1.

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Lc Oscillations

An LC circuit is an electrical circuit consisting of an inductor ($L$) and a capacitor ($C$) connected together. In an ideal LC circuit (one with no resistance), electrical energy is continuously transferred back and forth between the electric field of the capacitor and the magnetic field of the inductor. This results in electrical oscillations, analogous to the mechanical oscillations of a mass-spring system.

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The LC Circuit and Energy Exchange

Consider a capacitor initially charged to a potential difference $V_0$ and holding a charge $Q_0 = C V_0$. At time $t=0$, the charged capacitor is connected across an inductor.

A simple LC circuit.

The process unfolds as follows:

1. Initial State ($t=0$): The capacitor has maximum charge ($Q_0$) and maximum stored electrical energy ($U_E = \frac{1}{2} \frac{Q_0^2}{C}$). The current in the circuit is zero, and the energy stored in the inductor's magnetic field is zero ($U_B = \frac{1}{2} L I^2 = 0$). Total energy is purely electrical.
2. Capacitor Discharging: The capacitor begins to discharge through the inductor. The charge on the capacitor decreases, and a current starts to flow through the inductor. As current increases, energy begins to be stored in the inductor's magnetic field ($U_B = \frac{1}{2} L I^2$), while the capacitor's electrical energy ($U_E = \frac{1}{2} q^2/C$) decreases.
3. Current Maximum: When the capacitor is fully discharged ($q=0$), its electrical energy is zero ($U_E = 0$). The current in the inductor reaches its maximum value ($I_0$), and the energy stored in the magnetic field is maximum ($U_B = \frac{1}{2} L I_0^2$). All the initial electrical energy has been transferred to the inductor's magnetic field.
4. Capacitor Charging (Reverse Polarity): Once the capacitor is discharged, the magnetic field in the inductor starts collapsing. The collapsing magnetic flux induces a back EMF that opposes the change in current (Lenz's Law). This induced EMF drives the current in the same direction for a while, charging the capacitor with the opposite polarity compared to the initial charge. The current magnitude decreases as energy is transferred back to the capacitor's electric field.
5. Current Zero, Capacitor Fully Charged (Reversed): When the current becomes zero, the magnetic field energy is zero ($U_B = 0$). The capacitor is now fully charged with the opposite polarity, storing maximum electrical energy ($U_E = \frac{1}{2} \frac{(-Q_0)^2}{C} = \frac{1}{2} \frac{Q_0^2}{C}$). All the energy is purely electrical again.
6. Oscillation Continues: The capacitor then starts discharging again, and the process repeats, transferring energy back and forth between the electric field in the capacitor and the magnetic field in the inductor. The charge and current in the circuit oscillate sinusoidally with time.

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Analogy with Mechanical Oscillations

The LC oscillation is analogous to the oscillation of a mass attached to a spring (a simple harmonic oscillator).

Mass-Spring System	LC Circuit
Displacement ($x$)	Charge ($q$)
Velocity ($v = dx/dt$)	Current ($i = dq/dt$)
Mass ($m$)	Inductance ($L$) - Inertia for charge/current
Spring constant ($k$)	Reciprocal of capacitance ($1/C$) - Stiffness for charge/voltage
Momentum ($p = mv$)	Magnetic flux linkage ($LI$)
Kinetic energy ($ \frac{1}{2}mv^2 $)	Magnetic energy ($ \frac{1}{2}Li^2 $)
Potential energy ($ \frac{1}{2}kx^2 $)	Electrical energy ($ \frac{1}{2}q^2/C $)
Total Energy ($ \frac{1}{2}mv^2 + \frac{1}{2}kx^2 $)	Total Energy ($ \frac{1}{2}Li^2 + \frac{1}{2}q^2/C $)
Friction/Damping	Resistance ($R$) - Energy dissipation
Natural frequency ($ \omega_0 = \sqrt{k/m} $)	Natural frequency ($ \omega_0 = \sqrt{1/LC} $)

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Mathematical Description of LC Oscillations

Consider the simple LC circuit. Applying Kirchhoff's Loop Rule, the sum of the voltage drops around the loop is zero. Let $q$ be the instantaneous charge on the capacitor. The voltage across the capacitor is $v_C = q/C$. The instantaneous current is $i = dq/dt$. The voltage across the inductor is $v_L = L di/dt = L d^2q/dt^2$.

$ v_L + v_C = 0 $

$ L \frac{d^2q}{dt^2} + \frac{q}{C} = 0 $

$ \frac{d^2q}{dt^2} + \frac{1}{LC} q = 0 $

This is a second-order linear differential equation, which is exactly the same form as the differential equation for simple harmonic motion ($ \frac{d^2x}{dt^2} + \omega_0^2 x = 0 $).

Comparing the two equations, we see that the charge $q$ oscillates harmonically with an angular frequency $\omega_0$ given by:

$ \omega_0^2 = \frac{1}{LC} $

$ \omega_0 = \frac{1}{\sqrt{LC}} $

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Solution for Charge and Current

The general solution for the charge $q(t)$ is:

$ q(t) = Q_0 \cos(\omega_0 t + \delta) $

Where $Q_0$ is the amplitude of oscillation (maximum charge on the capacitor), and $\delta$ is the phase constant, determined by the initial conditions. If the capacitor is initially charged to $Q_0$ and the circuit is closed at $t=0$, $q(0) = Q_0$ and $i(0) = (dq/dt)_{t=0} = 0$. This gives $\delta = 0$.

$ q(t) = Q_0 \cos(\omega_0 t) $

The instantaneous current $i(t)$ is the rate of change of charge:

$ i(t) = \frac{dq}{dt} = \frac{d}{dt} [Q_0 \cos(\omega_0 t)] = -Q_0 \omega_0 \sin(\omega_0 t) $

The maximum current amplitude is $I_0 = Q_0 \omega_0$.

$ i(t) = -I_0 \sin(\omega_0 t) $

Using $-\sin(\omega_0 t) = \cos(\omega_0 t + \pi/2)$ or $\sin(\omega_0 t - \pi/2) = -\cos(\omega_0 t)$:

$ i(t) = I_0 \cos(\omega_0 t + \pi/2) \quad \text{or} \quad i(t) = I_0 \sin(\omega_0 t - \pi/2) $

The charge and current oscillate sinusoidally with the same angular frequency $\omega_0$, but they are $90^\circ$ out of phase. When charge is maximum, current is zero, and when charge is zero, current is maximum.

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Energy in Ideal LC Oscillations

In an ideal LC circuit, there is no resistance, so no energy is lost as heat. The total energy in the circuit remains constant and is continuously exchanged between the capacitor's electric field and the inductor's magnetic field.

Instantaneous electrical energy in the capacitor: $U_E(t) = \frac{1}{2} \frac{q(t)^2}{C} = \frac{1}{2C} [Q_0 \cos(\omega_0 t)]^2 = \frac{1}{2C} Q_0^2 \cos^2(\omega_0 t)$.

Maximum electrical energy: $U_{E, max} = \frac{1}{2} \frac{Q_0^2}{C}$.

$ U_E(t) = U_{E, max} \cos^2(\omega_0 t) $

Instantaneous magnetic energy in the inductor: $U_B(t) = \frac{1}{2} L i(t)^2 = \frac{1}{2} L [-Q_0 \omega_0 \sin(\omega_0 t)]^2 = \frac{1}{2} L Q_0^2 \omega_0^2 \sin^2(\omega_0 t)$.

Substitute $\omega_0^2 = 1/LC$: $U_B(t) = \frac{1}{2} L Q_0^2 \left(\frac{1}{LC}\right) \sin^2(\omega_0 t) = \frac{1}{2C} Q_0^2 \sin^2(\omega_0 t)$.

Maximum magnetic energy occurs when $\sin(\omega_0 t) = \pm 1$, which corresponds to $\cos(\omega_0 t) = 0$, i.e., when $q=0$ and $|i|=I_0 = Q_0\omega_0$. $U_{B, max} = \frac{1}{2} L I_0^2 = \frac{1}{2} L (Q_0\omega_0)^2 = \frac{1}{2} L Q_0^2 (1/LC) = \frac{1}{2} Q_0^2/C = U_{E, max}$.

$ U_B(t) = U_{E, max} \sin^2(\omega_0 t) $

The total energy $U(t)$ in the circuit at any instant is the sum of electrical and magnetic energy:

$ U(t) = U_E(t) + U_B(t) = U_{E, max} \cos^2(\omega_0 t) + U_{E, max} \sin^2(\omega_0 t) $

$ U(t) = U_{E, max} [\cos^2(\omega_0 t) + \sin^2(\omega_0 t)] = U_{E, max} (1) = U_{E, max} $

$ U_{total} = \frac{1}{2} \frac{Q_0^2}{C} = \frac{1}{2} L I_0^2 $ (using $I_0 = Q_0 \omega_0 = Q_0/\sqrt{LC}$)

The total energy in the ideal LC circuit is constant. Energy oscillates between being purely electrical (when $q=\pm Q_0, i=0$) and purely magnetic (when $q=0, i=\pm I_0$).

Energy oscillates between electric and magnetic fields in an ideal LC circuit. Total energy is constant.

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Damped Oscillations (Real LC Circuit - RLC)

In a real circuit, there is always some resistance $R$, even if it's just the resistance of the connecting wires and the inductor coil. This resistance dissipates energy as heat ($I^2R$). Therefore, in a real LC circuit (an RLC circuit with no external driving source), the total energy decreases over time, and the oscillations gradually die out. This is called damped oscillation.

The equation for charge in a series RLC circuit (without external source) is:

$ L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{q}{C} = 0 $

This is the equation of a damped harmonic oscillator. The solution for $q(t)$ involves a decaying sinusoidal function, provided the resistance is not too large.

The rate of energy loss is $P_{loss} = I^2 R = (dq/dt)^2 R$.

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