MTH101 Coursework
Phase Analysis of Parallel RLC Circuits:
A Theoretical Exploration
Dec. 2023
1. Background
RLC circuit is a kind of circuit structure composed of resistance (R), induct-or (L) and capacitance (C). The name of the circuit comes from the letter used to represent the circuit component, where the component order may be different from the RLC. The RLC circuit is also known as a second order circuit. The voltage or current in a circuit is a solution to a second order differential equation, and its coefficient is determined by the circuit structure. The phase difference between the current and the supply voltage is 0 which are in the R-route. The phase difference between the current and the supply voltage is 90 which are in the L-route and the current lags. The phase difference between the current branch and the supply voltage is 90 degrees which are in C-route, and the current is advanced. As these pictures showing, the RLC circuits are divided into parallel and series.
Fig. 1
2. Description of the models
Consider the following model:
Fig. 2
The above diagram shows a parallel RLC circuit model. Among them, R, L, and C represent resistors, inductors, and capacitors respectively. The currents on these components are represented by IR, IL and IC. IS represents the main circuit current.
According to Ohm's law, VR = RIR. According to the inductance characteristics, VL = jωLIL. According to the capacitance characteristics, where j is the imaginary unit, ω Is the angular frequency.
According to Ohm's law, where Z is the total resistance of the circuit. According to the following formula:
Through the complex analysis method, we can analyze the frequency response, phase difference, power and other characteristics of parallel RLC AC circuit. This method can be more convenient for calculation and analysis, and can get a more comprehensive description of circuit behavior.
In AC circuit, the current and voltage are represented by sine function, so there is a phase relationship between current and voltage. In RLC circuit, the phase relationship between current and voltage depends on the circuit frequency and component characteristics. When the circuit is in a stable state, there is a certain phase difference between the two. The current voltage phase relationship of each element in the above circuit will be analyzed below.
3. Explanation of solutions
In an actual AC power circuit, the phase difference between current and voltage is a crucial aspect that depends on the circuit frequency and the characteristics of the circuit components. Let’s delve deeper into the parallel RLC circuit mentioned earlier, which comprises a resistance (R), inductance (L), and capacitor (C) circuit.
Beginning with the resistance element (R), it’s important to emphasize that the current and voltage are in phase, meaning there is no phase difference between them. This implies that the resistance element does not introduce any phase shift between the current and voltage.
Moving on to the inductance element (L), we observe that the current lags behind the voltage by 90 degrees. This effect suggests that in a parallel RLC circuit, the inductive element introduces a phase difference between the current and voltage, causing the current to lag behind the voltage.
Conversely, the capacitor element (C) leads the voltage by 90 degrees, resulting in a phase difference between the current and voltage. Therefore, in the parallel RLC circuit, the capacitive element introduces a leading phase difference between the current and voltage.
Crucially, it’s important to note that the actual phase difference between current and voltage in a parallel RLC circuit is influenced not only by the characteristics of each individual component but also by the circuit’s operating frequency. Furthermore, in certain scenarios, the phase differences introduced by the inductive and capacitive elements can either cancel each other out or superimpose, thus affecting the overall phase difference of the circuit.
When considering the circuit as a whole, the current and voltage in each branch of the parallel RLC circuit are determined by the characteristics of the entire circuit. The phase relationships between the components are intricately influenced by the operating frequency and the impedance characteristics of the components. Therefore, a comprehensive analysis of the phase relationship in a circuit necessitates a consideration of these factors.
In summary, in the parallel RLC circuit, the resistance element (R) does not cause a phase difference between current and voltage. The inductive element (L) introduces a lagging phase difference, while the capacitive element (C) introduces a leading phase difference. The overall phase difference between current and voltage depends on the sum of the phase differences introduced by all components, as well as the operating frequency of the circuit. This understanding is paramount for effectively analyzing and designing parallel RLC circuits.
4. Conclusion
From the above discussion it can be inferred that for the parallel RLC circuit, the value of phase difference between current and voltage has a huge correlation with the types of components: resistance element (R) does not introduce any phase shift between the current and voltage; when we focus on the inductance element (L), it introduces a phase difference between the current and voltage, causing the current lags behind the voltage by 90 degrees; contrarily for the capacitor element (C), it introduces a leading phase difference of 90 degrees between the current and voltage.
In conclusion, this research report has examined the parallel RLC circuit in detail, focusing on the phase difference of the circuit structure. By solving second order differential equations, we analyzed the resistance, inductance, and capacitance elements, confirming the understanding of the correlation between the resistance element (R), inductance element (L), capacitor element (C) and the phase difference between current and voltage, as well as the impact of circuit operating frequency.