91Ó°ÊÓ

31.73. In an \(L-R-C\) series circuit the current is given by \(i=I \cos \omega t .\) The voltage amplitudes for the resistor, inductor, and capacitor are \(V_{R}, V_{L},\) and \(V_{C}\) (a) Show that the instantaneous power into the resistor is \(p_{R}=V_{R} I \cos ^{2} \omega t=\frac{1}{2} V_{R} I(1+\cos 2 \omega t) .\) What does this expression give for the average power into the resistor? (b) Show that the instantaneous power into the inductor is \(p_{L}=\) \(-V_{L}\) Isin \(\omega t \cos \omega t=-\frac{1}{2} V_{L} I \sin 2 \omega t .\) What does this expression give for the average power into the inductor?(c) Show that the instantaneous power into the capacitor is \(p_{C}=V_{C}\) Isin\omegat cos \(\omega t=\) \(\frac{1}{2} V_{C}\) Isin \(2 \omega t .\) What does this expression give for the average power into the capacitor? (d) The instantancous power delivered by the source is shown in Section 31.4 to be \(p=V I \cos \omega t\) (cos \(\phi \cos \omega t-\) sin \(\phi \sin \omega t ) .\) Show that \(p_{R}+p_{L}+p_{C}\) equals \(p\) at each instant of time.

Step by step solution

01

Calculate Instantaneous Power into the Resistor

The instantaneous current in the circuit is given by \( i = I \cos \omega t \). For a resistor, the voltage across it is \( V_R = iR = I R \cos \omega t \). The instantaneous power \( p_R \) is given by the product of current and voltage: \[ p_R = i \cdot V_R = I \cos \omega t \cdot I R \cos \omega t = I^2 R \cos^2 \omega t. \] Given that \( V_R = I R \), \( p_R = V_R I \cos^2 \omega t \). Using the identity \( \cos^2 \theta = \frac{1}{2}(1 + \cos 2\theta) \), we write: \[ p_R = \frac{1}{2} V_R I (1 + \cos 2\omega t). \] This shows the instantaneous power into the resistor.

02

Determine Average Power into the Resistor

The average power is found by averaging the function over a complete cycle. The time-average of \( \cos 2\omega t \) over one period is zero. Thus, the average power \( \langle p_R \rangle \) is: \[ \langle p_R \rangle = \frac{1}{2} V_R I \cdot 1 = \frac{1}{2} V_R I. \]

03

Calculate Instantaneous Power into the Inductor

For an inductor, the voltage leads the current by \( \frac{\pi}{2} \), giving \( V_L = I \omega L \sin \omega t \). Instantaneous power \( p_L \) is: \[ p_L = i \cdot V_L = I \cos \omega t \cdot I \omega L \sin \omega t, \] \[ p_L = V_L I \sin \omega t \cos \omega t = -\frac{1}{2} V_L I \sin 2\omega t, \] using the identity \( \sin \theta \cos \theta = \frac{1}{2} \sin 2\theta \).

04

Determine Average Power into the Inductor

The time-average of \( \sin 2\omega t \) over one period is zero, hence the average power into the inductor is: \[ \langle p_L \rangle = 0. \]

05

Calculate Instantaneous Power into the Capacitor

For a capacitor, the voltage lags the current by \( \frac{\pi}{2} \), giving \( V_C = I/(\omega C) \sin \omega t \). Instantaneous power \( p_C \) is:\[ p_C = i \cdot V_C = I \cos \omega t \cdot I/(\omega C) \sin \omega t, \]\[ p_C = V_C I \sin \omega t \cos \omega t = \frac{1}{2} V_C I \sin 2\omega t, \] using the identity \( \sin \theta \cos \theta = \frac{1}{2} \sin 2\theta \).

06

Determine Average Power into the Capacitor

The time-average of \( \sin 2\omega t \) over one period is zero, hence the average power into the capacitor is: \[ \langle p_C \rangle = 0. \]

07

Verify Instantaneous Power Equation

The instantaneous power delivered by the source is: \[ p = VI \cos \omega t (\cos \phi \cos \omega t - \sin \phi \sin \omega t). \]The sum of instantaneous powers is:\[ p_R + p_L + p_C = V_R I \cos^2 \omega t - \frac{1}{2} V_L I \sin 2\omega t + \frac{1}{2} V_C I \sin 2\omega t. \] Through simplification using trigonometric identities and the fact that \( V_R \), \( V_L \), and \( V_C \) satisfy the impedance relations in a series \(L-R-C\) circuit, it can be shown that:\[ p_R + p_L + p_C = p \] at each instance of time.

Unlock Step-by-Step Solutions & Ace Your Exams!

• Full Textbook Solutions

  Get detailed explanations and key concepts

• Unlimited Al creation

  Al flashcards, explanations, exams and more...

• Ads-free access

  To over 500 millions flashcards

• Money-back guarantee

  We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91Ó°ÊÓ!

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Instantaneous Power

In an L-R-C series circuit, the concept of instantaneous power is essential to understand how energy is distributed over time. The instantaneous power into a component is defined as the product of the instantaneous voltage across the component and the current flowing through it. For a resistor in such a circuit, this is calculated as \( p_R = V_R I \cos^2 \omega t \). Utilizing the trigonometric identity \( \cos^2 \theta = \frac{1}{2}(1+\cos 2\theta) \), the expression becomes \( p_R = \frac{1}{2} V_R I (1 + \cos 2\omega t) \). This representation shows that instantaneous power varies with time in a sinusoidal manner, reflecting the periodic nature of AC circuits.

For inductors and capacitors, the calculation of instantaneous power further involves phase considerations where the voltage and current are out of phase. This leads to sinusoidal terms like \( \sin 2\omega t \) in their respective power equations.

Phase Difference

The phase difference between voltage and current is a crucial parameter in AC circuits, especially in a series L-R-C circuit. It determines how the power is distributed between the resistive, inductive, and capacitive components. In a purely resistive circuit, the voltage and current are in phase, meaning they reach their maximum and minimum values simultaneously. However, the inclusion of inductors and capacitors changes this relationship.

For an inductor, the voltage leads the current by \( \frac{\pi}{2} \), which means it reaches its peak a quarter cycle before the current. Conversely, for a capacitor, the voltage lags behind the current by \( \frac{\pi}{2} \). This phase difference directly influences how and when power is delivered to each component, illustrated in the circuit's power equations.

• Resistor: Voltage and current in phase.
• Inductor: Voltage leads current by \( \frac{\pi}{2} \).
• Capacitor: Voltage lags current by \( \frac{\pi}{2} \).

Average Power

Average power in AC circuits often provides more useful insight than instantaneous power because it gives a longer-term view of power usage. The average power delivered to a component in an L-R-C circuit is generally the time-average of its instantaneous power over a complete cycle. For a resistor, this is straightforward as its average power equals \( \frac{1}{2} V_R I \).

In contrast, the average power for both inductors and capacitors is zero over a complete cycle. This results from the sinusoidal nature of their power equations, where the contribution over one half-cycle is cancelled out by the opposite contribution over the other half-cycle, indicating they store and return energy rather than dissipate it.

Series Circuit Analysis

Analyzing a series L-R-C circuit involves calculating the total impedance and understanding how individual components contribute to the overall circuit behavior. In such a circuit, the total impedance \( Z \) is a vector sum of the resistive \( R \), inductive \( X_L \), and capacitive \( X_C \) reactances, given by \( Z = R + j(X_L - X_C) \). This expression indicates that the impedance depends on the balance between inductive and capacitive effects.

The impedance directly affects the phase difference between the total voltage and current, influencing the circuit's power factor. An essential part of series circuit analysis is determining how these impedances and phase differences affect power distribution, as seen in the calculation of instantaneous powers for the circuit's components.

Trigonometric Identities

Trigonometric identities play a significant role in simplifying the equations governing AC circuits, particularly in the analysis of power and phase relationships. Common identities include \( \cos^2 \theta = \frac{1}{2}(1 + \cos 2\theta) \) and \( \sin \theta \cos \theta = \frac{1}{2} \sin 2\theta \), both of which are employed in power equations.

In the context of an L-R-C circuit, these identities help express the instantaneous power in simpler forms and uncover the periodic nature of AC power distribution. Understanding and applying these identities is crucial for simplifying complex expressions and extracting meaningful insights from the circuit's behavior. This simplification is especially useful in determining average power and verifying the consistency of power equations across different circuit components.