91Ó°ÊÓ

The \(L \cdot R-C\) Parallel Circuit. A resistor, inductor, and capacitor are connected in parallel to an ac source with voltage amplitude \(V\) and angular frequency \(\omega\) . Let the source voltage be given by \(v=V \cos \omega t .\) (a) Show that the instantaneous voltages \(v_{R}, v_{L},\) and \(v_{C}\) at any instant are each equal to \(v\) and that \(i=i_{R}+i_{L}+i_{C},\) where \(i\) is the current through the source and \(i_{R},\) \(i_{L},\) and \(i_{C}\) are the currents through the resistor, the inductor, and the capacitor, respectively. (b) What are the phases of \(i_{R}, i_{L},\) and \(i_{C}\) with respect to \(v ?\) Use current phasors to represent \(i, i_{R}, i_{L},\) and \(i_{C}\) In a phasor diagram, show the phases of these four currents with respect to \(v\) . (c) Use the phasor diagram of part (b) to show that the current amplitude \(I\) for the current \(i\) through the source is given by \(I=\sqrt{I_{R}^{2}+\left(I_{C}-I_{L}\right)^{2}}\) (d) Show that the result of part (c) can be written as \(I=V / Z,\) with \(1 / Z=\sqrt{1 / R^{2}+(\omega C-1 / \omega L)^{2}}\)

Step by step solution

01

Express Instantaneous Voltages

For a parallel circuit, the voltage across all components is the same. Therefore, at any instant, the voltages across the resistor \( v_R \), inductor \( v_L \), and capacitor \( v_C \), are each equal to the source voltage \( v \). This means that: \[ v_R = v_L = v_C = V \cos \omega t \]

02

Summation of Currents in Parallel Circuits

In a parallel circuit, the total current \( i \) is the sum of the currents through each component: the resistor \( i_R \), the inductor \( i_L \), and the capacitor \( i_C \). Thus, \[ i = i_R + i_L + i_C \]

03

Determine Phases of Component Currents

The current through the resistor \( i_R \) is in phase with the voltage. The current through the inductor \( i_L \) lags the voltage by 90 degrees. The current through the capacitor \( i_C \) leads the voltage by 90 degrees. This phase relationship is fundamental to understanding AC circuit behavior.

04

Draw Phasor Diagram

Using current phasors, represent \( i, i_R, i_L, \) and \( i_C \) in a diagram. In this diagram, \( i_R \) is horizontal and in phase with \( v \). \( i_L \) is downward, indicating a 90-degree lag, and \( i_C \) is upward, indicating a 90-degree lead.

05

Derive Current Amplitude Expression

Using the phasor diagram, apply the Pythagorean theorem to find the total current amplitude \( I \). Since \( i_C \) and \( i_L \) are on opposite sides of the voltage phasor, their difference is considered. Therefore, \[ I = \sqrt{I_R^2 + (I_C - I_L)^2} \] This represents the total current in the circuit.

06

Rewrite the Expression for Total Current Amplitude

Recognize that \( I = V / Z \) for an AC circuit, where \( Z \) is the impedance. Relate this to the given expression: \[ \frac{1}{Z} = \sqrt{\left(\frac{1}{R}\right)^2 + (\omega C - \frac{1}{\omega L})^2} \] This confirms that the expression for \( I \) derived from the phasor diagram aligns with the impedance relationship.

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Key Concepts

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

Phasor Diagram

In AC circuit analysis, phasor diagrams are a crucial tool for visualizing the relationship between different electrical quantities. In the context of an LRC parallel circuit, a phasor diagram helps us understand the phase relationships between the voltage across circuit components and the currents flowing through them.
A phasor is a vector that represents the magnitude and phase angle of a sinusoidal function. In our circuit, the voltage phasor is typically taken as the reference. In the diagram:

• The current through the resistor, \(i_R\), is in-phase with the voltage across it.
• The inductor's current, \(i_L\), lags the voltage by 90 degrees due to the inductive reactance.
• The capacitor's current, \(i_C\), leads the voltage by 90 degrees because of its capacitive reactance.

These phase shifts are represented in the phasor diagram by arrows:

• \(i_R\) is a horizontal vector aligned with the voltage vector.
• \(i_L\) is a downward vertical vector.
• \(i_C\) is an upward vertical vector.

Understanding the phasor diagram helps us visualize the phase differences, which is essential for calculating the total current and impedance.

AC Circuit Analysis

AC circuit analysis involves understanding how alternating current (AC) behaves within an electrical circuit. In an LRC parallel circuit, key concepts such as phase angle, reactance, and resonance must be considered.
In AC systems, currents and voltages are often represented as sine and cosine functions, depending on their phases:

• The voltage is continuous across all parallel components, meaning the voltage across the resistor \(v_R\), inductor \(v_L\), and capacitor \(v_C\) is the same, \(v = V \cos \omega t\).
• The total current \(i\) in a parallel circuit equals the sum of the individual currents: \(i = i_R + i_L + i_C\).

By applying Kirchhoff's Current Law and considering phase relationships, we calculate the overall effect of these currents. Each type of component (resistor, inductor, capacitor) affects the current differently due to its unique phase interaction with the voltage. This is why, despite the constant voltage, the current through each component can have a different phase, leading to complex current calculations.

Impedance Calculation

Impedance is a key parameter in AC circuit analysis that abstracts and combines the effects of resistance, capacitance, and inductance in a circuit. In the LRC parallel circuit, impedance determines how much current flows for a given voltage.
The formula for impedance \(Z\) in parallel circuits takes into account both resistive and reactive components:

• The reciprocal of the impedance for the circuit is given by: \( \frac{1}{Z} = \sqrt{\left( \frac{1}{R} \right)^2 + ( \omega C - \frac{1}{\omega L} )^2 } \).
• This equation highlights that impedance depends on the angular frequency \(\omega\), which impacts how capacitive and inductive reactances change.

Impedance simplifies the complex relationship between voltage and current to a single value, showing how the total circuit 'resists' the flow of AC. It's important because calculating the total current amplitude, \(I = \frac{V}{Z}\), provides insight into the performance of the circuit when subject to an AC source. Hence, understanding and calculating impedance in parallel circuits is central to effective AC circuit design and analysis.