## Rlc circuit differential equation calculator

### RLC Circuit Frequency Calculator

Underdamped Overdamped Critically Damped. The formulas on this page are associated with a series RLC circuit discharge since this is the primary model for most high voltage and pulsed power discharge circuits. Each of the following waveform plots can be clicked on to open up the full size graph in a separate window. The circuit schematic for the underdamped case is shown below.

In this specific model, the resistance is an order of magnitude 10X less than the value required for a critically damped circuit. The results of the circuit model are shown below. V 1 is the voltage on the 1 m F capacitor as it discharges in an oscillatory mode.

## Analyze a Series RC Circuit Using a Differential Equation

V 3 is the voltage on the load resistor, in this case a 0. The circuit schematic for the overdamped case is shown below. In this specific model, the resistance is an order of magnitude 10X more than the value required for a critically damped circuit. V 3 is the voltage on the load resistor, in this case a 20 ohm value.

In this case, once the switch closes and the voltage on the load resistor rises to match the capacitor voltage, both waveforms then essentially overlap and decay at the same rate since the voltage across the inductor is minimal. The circuit current is graphed in the second, lower plot.

### Analyze an RLC Second-Order Parallel Circuit Using Duality

The circuit schematic for the critically damped case is shown below. In this specific model, the resistance is exactly equal to the value required for a critically damped circuit. One can see that the resistor voltage also does not overshoot. This circuit is often desirable if possible with high voltage, energy storage capacitors since voltage reversals can frequently decrease the lifetime of the capacitor.

V 0 is the initial voltage on the capacitor V L is the circuit inductance H R is the circuit resistance W.RLC circuit frequency calculator is an online tool for electrical and electronic circuits to measure the resonant frequency, series damping factor, parallel damping factor and bandwidth.

An electrical circuit consists of three major electric components of a resistor, an inductor and a capacitor connected in series or in parallel. The characteristics of these components in the manner resistance R, inductance L and capacitance C caused to bring this name RLC circuit. When it comes to online calculation, this RLC circuit frequency calculator can assist you to calculate bandwidth in hertz, resonant frequency in hertz, series and parallel damping factor.

Home Engineering Electronics RLC circuit frequency calculator is an online tool for electrical and electronic circuits to measure the resonant frequency, series damping factor, parallel damping factor and bandwidth.

Close Download. Continue with Facebook Continue with Google. By continuing with ncalculators. You must login to use this feature! Privacy Terms Disclaimer Feedback.Using the Laplace transform as part of your circuit analysis provides you with a prediction of circuit response. Analyze the poles of the Laplace transform to get a general idea of output behavior. Real poles, for instance, indicate exponential output behavior.

Apply the Laplace transformation of the differential equation to put the equation in the s -domain. Apply the inverse Laplace transformation to produce the solution to the original differential equation described in the time-domain. To get comfortable with this process, you simply need to practice applying it to different types of circuits such as an RC resistor-capacitor circuit, an RL resistor-inductor circuit, and an RLC resistor-inductor-capacitor circuit.

Here you can see an RLC circuit in which the switch has been open for a long time. Next, formulate the element equation or i-v characteristic for each device. Substituting the element equations, v R tv C tand v L tinto the KVL equation gives you the following equation with a fancy name: the integro-differential equation :. The next step is to apply the Laplace transform to the preceding equation to find an I s that satisfies the integro-differential equation for a given set of initial conditions:.

The preceding equation uses the linearity property allowing you to take the Laplace transform of each term. For the first term on the left side of the equation, you use the differentiation property to get the following transform:. Because the switch is open for a long time, the initial condition I 0 is equal to zero.

To get the time-domain solution i tuse the following table, and notice that the preceding equation has the form of a damping sinusoid. For this RLC circuit, you have a damping sinusoid. The oscillations will die out after a long period of time. John M. Santiago Jr. During that time, he held a variety of leadership positions in technical program management, acquisition development, and operation research support.

About the Book Author John M.Second-order RLC circuits have a resistor, inductor, and capacitor connected serially or in parallel. To analyze a second-order parallel circuit, you follow the same process for analyzing an RLC series circuit. Here is an example RLC parallel circuit. The left diagram shows an input i N with initial inductor current I 0 and capacitor voltage V 0. The top-right diagram shows the input current source i N set equal to zero, which lets you solve for the zero-input response.

The bottom-right diagram shows the initial conditions I 0 and V 0 set equal to zero, which lets you obtain the zero-state response. With duality, you substitute every electrical term in an equation with its dual, or counterpart, and get another correct equation.

For example, voltage and current are dual variables.

RLC Circuits - Differential Equation Application

KCL says the sum of the incoming currents equals the sum of the outgoing currents at a node. Next, put the resistor current and capacitor current in terms of the inductor current. The current i L t is the inductor current, and L is the inductance. This constraint means a changing current generates an inductor voltage. Parallel devices have the same voltage v t.

Substitute the values of i R t and i C t into the KCL equation to give you the device currents in terms of the inductor current:. The RLC parallel circuit is described by a second-order differential equation, so the circuit is a second-order circuit. The unknown is the inductor current i L t. Compare the preceding equation with this second-order equation derived from the RLC series:.

The two differential equations have the same form. The unknown solution for the parallel RLC circuit is the inductor current, and the unknown for the series RLC circuit is the capacitor voltage. These unknowns are dual variables. With dualityyou can replace every electrical term in an equation with its dual and get another correct equation.

If you use the following substitution of variables in the differential equation for the RLC series circuit, you get the differential equation for the RLC parallel circuit. For a parallel circuit, you have a second-order and homogeneous differential equation given in terms of the inductor current:. The zero-input responses of the inductor responses resemble the form shown here, which describes the capacitor voltage.

When you have k 1 and k 2you have the zero-input response i ZI t. The solution gives you. You can find the constants c 1 and c 2 by using the results found in the RLC series circuit, which are given as.

Apply duality to the preceding equation by replacing the voltage, current, and inductance with their duals current, voltage, and capacitance to get c 1 and c 2 for the RLC parallel circuit:. After you plug in the dual variables, finding the constants c 1 and c 2 is easy.

Zero-state response means zero initial conditions. The second-order differential equation becomes the following, where i L t is the inductor current:. Adding the homogeneous solution to the particular solution for a step input IAu t gives you the zero-state response i ZS t :. You now apply duality through a simple substitution of terms in order to get C 1 and C 2 for the RLC parallel circuit:.

You finally add up the zero-input response i ZI t and the zero-state response i ZS t to get the total response i L t :. The solution resembles the results for the RLC series circuit.

Also, the step responses of the inductor current follow the same form as the ones shown in the step responses found in this sample circuit, for the capacitor voltage. John M. Santiago Jr. During that time, he held a variety of leadership positions in technical program management, acquisition development, and operation research support.

Based on the information given in the book I am using, I would think to setup the equation as follows:. The issue I am having is with the 12 volt battery that is connected. Maybe these notes will also be helpful. Sign up to join this community. The best answers are voted up and rise to the top. Home Questions Tags Users Unanswered. Asked 7 years, 1 month ago.

Active 5 years, 7 months ago. Viewed 62k times. Andrew M. Andrew M Andrew M 1 1 gold badge 2 2 silver badges 7 7 bronze badges. SE than here. If you're question is about solving that equation, the this is the place, but it looks like the first one. SE would be electronics. Active Oldest Votes. You would of course solve the homogeneous and non-homogeneous case. Make sense? Amzoti Amzoti Do you still have the notes?

## RLC Series Circuit

Or can you maybe find it again?A first-order RC series circuit has one resistor or network of resistors and one capacitor connected in series. First-order RC circuits can be analyzed using first-order differential equations. By analyzing a first-order circuit, you can understand its timing and delays. If your RC series circuit has a capacitor connected with a network of resistors rather than a single resistor, you can use the same approach to analyze the circuit.

The simple RC series circuit shown here is driven by a voltage source. Because the resistor and capacitor are connected in series, they must have the same current i t. Generating current through a capacitor takes a changing voltage. Zero current implies infinite resistance for constant voltage across the capacitor.

This gives you the voltage across the resistor, v R t :. You now have a first-order differential equation where the unknown function is the capacitor voltage. Knowing the voltage across the capacitor gives you the electrical energy stored in a capacitor.

In general, the capacitor voltage is referred to as a state variable because the capacitor voltage describes the state or behavior of the circuit at any time. A circuit reduced to having a single equivalent capacitance and a single equivalent resistance is also a first-order circuit. The circuit has an applied input voltage v T t. To find the total response of an RC series circuit, you need to find the zero-input response and the zero-state response and then add them together.

Here is an RC series circuit broken up into two circuits. The top-right diagram shows the zero-input response, which you get by setting the input to 0. The bottom-right diagram shows the zero-state response, which you get by setting the initial conditions to 0. John M. Santiago Jr. During that time, he held a variety of leadership positions in technical program management, acquisition development, and operation research support.

About the Book Author John M.Random converter. This series RLC circuit impedance calculator determines the impedance and the phase difference angle of a resistoran inductor and a capacitor connected in series for a given frequency of a sinusoidal signal. The angular frequency is also determined. This example shows the near-resonance impedance of about If you want to check the impedance at almost exact resonance, enter If you enter a slightly higher frequency of Enter the resistance, capacitance, inductance and frequency values, select the units and click or tap the Calculate button.

Try to enter zero or infinitely large values to see how this circuit behaves. Infinite frequency is not supported. To enter the Infinity value, just type inf in the input box. To calculate, enter the resistance, the inductance, the capacitance, and the frequency, select the units of measurements and the result for the RLC impedance will be shown in ohms and for the phase difference in degrees.

The Q factor, C and L reactance, and the resonant frequency will also be calculated. Click or tap Calculate at the resonant frequency to see what will happen at resonance. Like a pure series LC circuitthe RLC circuit can resonate at a resonant frequency and the resistor increases the decay of the oscillations at this frequency.

The resonance occurs at the frequency at which the impedance of the circuit is at its minimum, that is if there is no reactance in the circuit. In other words, if the impedance is purely resistive or real. This phenomenon occurs when the reactances of the inductor and the capacitor are equal and because of their opposite signs, they cancel each other the canceling can be observed on the right phasor diagram below.

The calculator defines the resonant frequency of the RLC circuit and you can enter this frequency or the value slightly above or below it to view what will happen with other calculated values at resonance. The calculator can also define the Q factor of the series RLC circuit — a parameter, which is used to characterize resonance circuits and not only electrical but mechanical resonators as well.

Damped and lossy RLC circuits with high resistance have a low Q factor and are wide-band, while circuits with low resistance have a high Q factor. For a series RLC circuit, the Q factor can be calculated using the formula above. In the series circuit, the same current flows through the resistor, the inductor, and the capacitor, but the individual voltages across the components are different. The phasor diagram shows the V T voltage of the ideal sine voltage source.

The voltage drop on the resistor V T is shown on the horizontal axis in phase with the current that flows through the circuit. The vector sum of the two opposing vectors can be pointed downwards or upwards depending on the voltage drop across the inductor and the capacitor. At the resonant frequency the capacitive and inductive reactances are equal and if we look at the equation for Z above, we will see that the effective impedance is equal to resistance R because the two opposing voltages simply cancel each other.

The current flowing through the inductor and capacitor is the same and the voltages across them are equal and opposing. So, at the resonant frequency, the current drawn from the source is limited only by the resistance because the ideal series LC circuit at the resonant frequency acts as a short circuit.

What if something goes wrong in this circuit? Click or tap a corresponding link to view the calculator in various failure modes:. Various direct current modes. Short circuit. Open circuit. 