Name: rlc circuit transfer functions
Uploaded: 2024-09-26T13:17:18.000Z
Duration: 1 min 21 s
Channel: JoVE

Electrical networks are first represented by equivalent circuits consisting of three passive linear components: resistors, capacitors, and inductors.
These components are combined into circuits, with input and output decided, and the transfer function is found using Kirchhoff&#39;s laws.
In an RLC circuit, the transfer function that relates the voltage across the capacitor to the input voltage can be derived using Kirchhoff&#39;s voltage law.
This yields an integro-differential equation for the network, assuming zero initial conditions.
Variables are initially changed from current to charge, followed by applying the voltage-charge relationship for a capacitor.
Taking the Laplace transform of this equation and simplifying it leads to the transfer function for this circuit.
Impedance is a transfer function similar to resistance but applicable to capacitors and inductors.
Transfer functions can also be obtained using Kirchhoff&#39;s current law, using nodal analysis.
The currents in the system are composed of the current flowing through the capacitor and the current that circulates through the series resistor and inductor.
After simplification, the same outcome for the transfer function is achieved.

electrical systems

In electrical engineering, the analysis of networks composed of passive linear components &#8212; resistors (R), capacitors (C), and inductors (L) &#8212; is fundamental. These components are organized into circuits where the relationship between input and output can be analyzed using transfer functions. The transfer function of an RLC circuit, which relates the voltage across a capacitor to the input voltage, can be derived using Kirchhoff's laws.
To derive the transfer function, consider an RLC circuit with the components connected in series. Kirchhoff's Voltage Law (KVL) states that the sum of all voltages around a closed loop is zero. For an RLC circuit, this translates to the following integro-differential equation, assuming zero initial conditions:
<img alt="Voltage distribution formula: V(t) = VR(t) + VC(t) + VL(t); key in circuit analysis." src="/files/ftp_upload/16096/16096_PT%20EQUATION_1.svg" style="text-align: center; height: 32px;" data-altupdatedat="1762173717000" data-alt="Equation1" />
Here, V(t) is the input voltage, VR(t)=i(t)R is the voltage across the resistor, VC(t) = <img alt="Equation of charge over capacitance q(t)/c; related to transient charge analysis; mathematical formula." src="/files/ftp_upload/16096/16096_PT%20EQUATION_5.svg" style="text-align: center; height: 20px;" data-altupdatedat="1762173717000" data-alt="Equation5" /> is the voltage across the capacitor, where q(t) is the charge on the capacitor, and VL(t) = <img alt="Static equilibrium equations, L di(t)/dt, electrical circuit analysis formula" src="/files/ftp_upload/16096/16096_PT%20EQUATION_6.svg" style="text-align: center; height: 18px;" data-altupdatedat="1762173717000" data-alt="Equation6" /> is the voltage across the inductor.
Current i(t) is related to charge q(t) by <img alt="Rate of charge change \(dq(t)/dt\) equation; calculus concept; derivative representation." src="/files/ftp_upload/16096/16096_PT%20EQUATION_7.svg" style="text-align: center; height: 20px;" data-altupdatedat="1762173717000" data-alt="Equation7" />. Substituting these relationships into the KVL equation gives:
<img alt="LC circuit differential equation formula; analysis of electrical circuits; resonance dynamics." src="/files/ftp_upload/16096/16096_PT%20EQUATION_2.svg" style="text-align: center; height: 48px;" data-altupdatedat="1762173717000" data-alt="Equation2" />
Writing the equation in voltage form, and taking the Laplace transform of both sides, and assuming all initial conditions are zero, the following equation is obtained:
<img alt="Transfer function equation, V(s)=(LCs&#178;+RCs+1)Vc(s), relevant in control system analysis." src="/files/ftp_upload/16096/16096_PT%20EQUATION_3.svg" style="text-align: center; height: 30px;" data-altupdatedat="1762173717000" data-alt="Equation3" />
Rearranging terms to solve for the transfer function:
<img alt="Laplace transform equation for RLC circuit frequency response; educational diagram." src="/files/ftp_upload/16096/16096_PT%20EQUATION_4.svg" style="text-align: center; height: 46px;" data-altupdatedat="1762173717000" data-alt="Equation4" />
This transfer function represents the voltage across the capacitor in response to the input voltage in the frequency domain. The impedance, analogous to resistance but applicable to capacitors and inductors, plays a key role in defining the transfer function. Additionally, Kirchhoff's Current Law (KCL) can also be used to derive transfer functions via nodal analysis. This law states that the sum of currents entering a node must equal the sum of currents leaving the node. In the RLC circuit, the total current is the sum of the current through the capacitor and the current through the series combination of the resistor and inductor. By applying KCL and simplifying, one can achieve the same transfer function.
KVL and KCL are powerful tools in circuit analysis, enabling the derivation of transfer functions that succinctly describe the dynamic behavior of electrical networks in the frequency domain. These methods are essential for the design and analysis of complex circuits in various applications.

Electrical Systems

Electrical networks are first represented by equivalent circuits consisting of three passive linear components: resistors, capacitors, and inductors.
...

Education

JoVE Core

Electrical Engineering

Unit 1

Modeling in Time and Frequency Domain

KEY TERMS AND CONCEPTS

rlc circuit transfer functions

RLC circuits use resistors, capacitors, and inductors to show how electrical signals change over time. In circuit analysis, these passive linear components are arranged so the link between an input and an output can be described with a transfer function. For this topic, the output is the voltage across the capacitor, compared with the input voltage.

A series RLC circuit can be analyzed with Kirchhoff's Voltage Law, or KVL. KVL says that the sum of all voltages around a closed loop is zero. Using that rule, the circuit can be written as an integro-differential equation when the initial conditions are zero.

In the equation, the input voltage is V(t). The resistor voltage is VR(t) = i(t)R, where i(t) is the current. The capacitor voltage depends on the charge q(t), and the inductor voltage depends on the current change in the loop. Since current and charge are related, these expressions can be substituted into the KVL equation.

The same circuit can also be written in voltage form and converted with the Laplace transform. With all initial conditions set to zero, this step gives an equation in the frequency domain. Rearranging the terms produces the transfer function for the capacitor voltage response.

Impedance plays an important role in this result. It works like resistance, but it applies to capacitors and inductors as well. The transfer function shows how the capacitor voltage responds to the input voltage in the frequency domain.

Kirchhoff's Current Law, or KCL, can also be used to reach the same result through nodal analysis. KCL says that the sum of currents entering a node must equal the sum of currents leaving it. In the RLC circuit, the total current is the current through the capacitor plus the current through the series resistor and inductor, and simplifying this relationship gives the same transfer function.

RLC Circuit Transfer Functions

transfer function in control systems

Transfer Function from Input to Output

The transfer function describes how a linear time-invariant, or LTI, system maps an input to an output in the frequency domain. It gives a compact way to study system behavior without working only in the time domain. It also connects the differential equations of the system to a form that is easier to handle mathematically. For an nth-order LTI system, the behavior is written as a differential equation with constant coefficients. In that equation, r(t) is the input and c(t) is the output. The...

RLC circuits use resistors, capacitors, and inductors to show how electrical signals change over time. In circuit analysis, these passive linear components are arranged so the link between an input and an output can be described with a transfer function. For this topic, the output is the voltage across the capacitor, compared with the input voltage. A series RLC circuit can be analyzed with Kirchhoff's Voltage Law, or KVL. KVL says that the sum of all voltages around a closed loop is zero.

mechanical systems

Transfer Functions in Mechanical Motion

Mechanical motion can be modeled with springs, masses, and viscous dampers. In these models, springs behave like inductors and masses act like capacitors in an electrical network. A viscous damper works like a resistor because it removes energy from the system. A force acts on the mass in the direction of motion. It is balanced by the spring force, the damping force, and the force from the mass’s acceleration. Newton’s second law is used to describe this balance, and the sum of the forces on...

electro-mechanical systems

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DC motor modeling is a key part of electromechanical systems. These systems combine electrical parts and mechanical parts to produce a useful output. A DC motor is one common example because it turns electrical energy into mechanical motion. It is used in many places, from simple fans to complex robotic mechanisms. Inside the DC motor, the armature is a rotating circuit placed in a magnetic field. When current flows through the armature, it experiences a force from the magnetic field. That...

linear approximation in frequency domain

Linearizing Nonlinear RL Circuits

Linearizing nonlinear RL circuits helps turn a hard circuit problem into a simpler linear one. Linear systems follow superposition, which means the total response equals the sum of the responses to separate inputs. They also follow homogeneity, which means scaling the input scales the output by the same amount. Nonlinear systems do not automatically have those properties. For small changes around an operating point, though, a nonlinear system can often be treated as linear. This is done with a...

state space representation

RLC Circuit State Equations

RLC circuit state equations show how a circuit changes over time using state variables. This time-domain method is useful for linear, time-invariant systems, but it also helps when frequency-domain analysis is not enough. It is especially useful for nonlinear systems, time-varying systems, and multiple-input multiple-output systems. The state-space approach turns an nth-order system into a set of simultaneous first-order differential equations called state equations. For a common second-order...

transfer function to state space

Phase-Variable State Equations from Transfer Functions

Phase-variable state equations show how a transfer function can be rewritten for computer simulation. In state-space form, the system is described with state variables, which are the output and its first several derivatives. This makes it easier to model a physical system on a digital computer. The starting point is an nth-order linear differential equation with constant coefficients. Systems like an RLC circuit can be written this way. To build the state equations, the transfer function is...

state space to transfer function

Finding a Transfer Function from State Space

Finding a transfer function from state-space form is a key step in system analysis. It changes a time-domain model into a frequency-domain form. That makes it easier to study and design control systems. The process starts with the state-space equations. These include the state equation and the output equation. In these equations, x(t) is the state vector, u(t) is the input vector, y(t) is the output vector, and A, B, C, and D are the matrices that define the system dynamics. Next, the Laplace...

linear approximation in time domain

Linearizing Nonlinear State-Space Models

Linearizing nonlinear state-space models helps describe systems that change with time or operating conditions. State-space representation is often effective for this kind of modeling. It is especially useful for systems such as a simple pendulum or a translational mechanical system with nonlinear springs. A simple pendulum can be modeled by summing torques around the pivot point. In this case, the mass is evenly distributed along the pendulum, and the center of mass is located at half its...

key terms and concepts

Learn how KVL, KCL, and the Laplace transform are used to find the transfer function of a series RLC circuit and track capacitor voltage in the frequency domain.

RLC Circuit Transfer Functions | JoVE K12

JoVE Core Electrical Engineering offers easy-to-understand animated videos that cover fundamental concepts of electric circuits along with circuit analysis techniques. It incorporates first-order and second-order circuits using operational amplifiers and circuit transformation concepts. The inclusion of practical design examples and real-world applications through scientist-in-action videos adds significant educational value, making complex concepts accessible.

electrical engineering

RLC Circuit Transfer Functions

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