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How do you compare and contrast the natural and forced responses of RL, RC, and RLC circuits?

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RL, RC, and RLC circuits are common types of linear circuits that involve resistors, inductors, and capacitors. These circuits can have different behaviors depending on the type and source of input voltage or current. In this article, you will learn how to compare and contrast the natural and forced responses of these circuits, and how they relate to the transient and steady state analysis of circuit dynamics.

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1 Natural response

The natural response of a circuit is the voltage or current that exists in the circuit when the input source is removed or set to zero. The natural response is determined by the initial conditions and the energy stored in the inductors and capacitors. For example, an RL circuit will have a natural response that decays exponentially, as the inductor releases its stored energy as current. An RC circuit will have a natural response that also decays exponentially, as the capacitor releases its stored energy as voltage. An RLC circuit will have a natural response that can be underdamped, overdamped, or critically damped, depending on the values of the resistance, inductance, and capacitance.

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Brian Dibble
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The natural response of a system corresponds to the homogeneous solution. Setting the input to zero and analyzing the system response resulting from the initial conditions only allows for characterization of stability, quality, damping and resonance.

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2 Forced response

The forced response of a circuit is the voltage or current that is induced by the input source. The forced response is determined by the frequency and amplitude of the input source and the impedance of the circuit. For example, an RL circuit will have a forced response that is proportional to the input voltage and inversely proportional to the inductive reactance. An RC circuit will have a forced response that is proportional to the input current and inversely proportional to the capacitive reactance. An RLC circuit will have a forced response that depends on the resonance frequency and the quality factor of the circuit.

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Brian Dibble
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Isolating the forced portion of a systems response involves analyzing the system independent of initial conditions. Typically the natural response solution will be known as well as the general form of the input forcing function. The particular solution can be found using calculus methods and from this the forced response can be extrapolated and interpreted.

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3 Transient response

The transient response of a circuit is the voltage or current that changes over time as the circuit reaches its steady state. The transient response is the sum of the natural and forced responses. For example, an RL circuit will have a transient response that starts from the initial current and approaches the forced current as time goes on. An RC circuit will have a transient response that starts from the initial voltage and approaches the forced voltage as time goes on. An RLC circuit will have a transient response that can oscillate, decay, or grow depending on the damping ratio and the input frequency.

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Brian Dibble
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Performance parameters regarding the systems response to spontaneous disturbances are defined by looking at the transients involved with the corresponding natural solution to a unit step input. This Heaviside function plays a large role in Controls. The transfer function can be built which fundamentally builds the basis of controlling and conditioning these parameters to obtain a desired, stable system output. PID controllers for instance

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4 Steady state response

The steady state response of a circuit is the voltage or current that remains constant after the transient response has died out. The steady state response is equal to the forced response. For example, an RL circuit will have a steady state response that is equal to the input voltage divided by the inductive reactance. An RC circuit will have a steady state response that is equal to the input current divided by the capacitive reactance. An RLC circuit will have a steady state response that depends on the impedance of the circuit at the input frequency.

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Brian Dibble
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The steady state response defines system parameters which will remain as time continues infinitely following a transient event. These are time invariant parameters resulting from the systems particular solution. The final value theorem shows is that these steady state time parameters are equivalent to the zero frequency parameters. The Steady state solution may involve both dc and periodic waveforms and for our Basic input basic output (BIBO) systems stability is defined when both the initial system condition and final value approach finite real values.

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5 Phasor analysis

One way to compare and contrast the natural and forced responses of RL, RC, and RLC circuits is to use phasor analysis. Phasor analysis is a method of representing sinusoidal voltages and currents as complex numbers that have magnitude and phase. By using phasor analysis, you can simplify the differential equations that describe the circuit dynamics and solve for the steady state response. You can also use phasor analysis to find the impedance, power, and power factor of the circuit.

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6 Impulse response

Another way to compare and contrast the natural and forced responses of RL, RC, and RLC circuits is to use impulse response. Impulse response is the voltage or current that results from applying a unit impulse to the circuit. By using impulse response, you can find the transfer function of the circuit, which relates the output to the input in terms of frequency. You can also use impulse response to find the frequency response, which shows how the magnitude and phase of the output vary with frequency.

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Brian Dibble
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The response of a system to an impulse will provide great insight in terms of characterizing the systems natural harmonics. The Dirac delta function is an instantaneous excitation of the system defined only at t=0 seconds with infinite magnitude. For all other time this function is equal to zero. The “sifting property” of the impulse function is realized by the Integral property of the impulse function and its application as a Fourier transform pair to the time domain unity function. The integral area under the curve for this function will be unity for any range capturing the impulse. By this definition the impulse response provides many useful applications when looking at our system in the frequency domain. Vibrational test is easy

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7 Here’s what else to consider

This is a space to share examples, stories, or insights that don’t fit into any of the previous sections. What else would you like to add?

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How do you compare and contrast the natural and forced responses of RL, RC, and RLC circuits?

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1 Natural response

2 Forced response

3 Transient response

4 Steady state response

5 Phasor analysis

6 Impulse response

7 Here’s what else to consider

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