5.7
Circuitos de segundo orden
La integración de dos elementos fundamentales de almacenamiento de energía en los circuitos eléctricos da como resultado circuitos de segundo orden, q…
Un circuito eléctrico que comprende dos elementos de almacenamiento de energía irreducibles se denomina circuito de segundo orden.
Algunos ejemplos incluyen circuitos RLC, así como circuitos RC y RL con condensadores duales e inductores, respectivamente.
Estos circuitos se caracterizan por una ecuación diferencial de segundo orden que correlaciona las señales de entrada y salida del circuito.
Las señales de entrada generalmente se originan en fuentes de voltaje o corriente, mientras que la salida suele ser el voltaje a través del condensador y/o la corriente a través del inductor.
Considere un circuito RLC, donde la energía inicial almacenada en el condensador y el inductor impulsa el circuito.
Aplicando la ley de voltaje de Kirchhoff al circuito y tomando la derivada del tiempo se obtiene una ecuación diferencial de segundo orden.
Los coeficientes de esta ecuación, expresados en términos de resistencia, capacitancia e inductancia, se conocen como coeficiente de amortiguamiento y frecuencia de resonancia.
El coeficiente de amortiguación indica la amortiguación debida a la resistencia y determina la tasa de disipación de energía en el sistema.
La frecuencia de resonancia representa la frecuencia de oscilación natural del circuito a medida que se intercambia energía entre el inductor y el condensador.
Q1: What makes a circuit a second-order circuit?
A second-order circuit contains two irreducible energy storage elements, such as capacitors and inductors. Examples include RLC circuits and RC or RL circuits with dual capacitors or inductors. These circuits are characterized by a second-order differential equation that correlates input and output signals, where inputs typically originate from voltage or current sources.
Q2: How do you derive the differential equation for a second-order RLC circuit?
Apply Kirchhoff's voltage law to the RLC circuit, then take the time derivative of the resulting equation. This yields a second-order differential equation whose coefficients are expressed in terms of resistance, capacitance, and inductance. These coefficients manifest as the damping coefficient and resonant frequency, which characterize the circuit's behavior.
Q3: What is the damping coefficient in a second-order circuit?
The damping coefficient indicates the damping caused by the resistor and determines the rate of energy dissipation in the system. It directly influences how fast energy is lost due to resistance. A higher damping coefficient means faster energy loss, while a lower coefficient allows energy to persist longer in the circuit.
Q4: What does resonant frequency represent in second-order circuits?
Resonant frequency represents the natural oscillation frequency of the circuit as energy is exchanged between the inductor and capacitor. It measures how quickly energy shifts between these two storage elements, illustrating the circuit's inherent tendency to oscillate at a particular frequency independent of external driving forces.
Q5: How do damping coefficient and resonant frequency interact in second-order circuits?
The damping coefficient controls the rate of energy loss due to resistance, while the resonant frequency highlights the circuit's natural oscillation speed. Together, these two factors determine the overall response of the circuit. The damping coefficient dictates how quickly oscillations decay, whereas resonant frequency determines the frequency at which oscillations occur.
Q6: What are the typical input and output signals in a second-order circuit?
Input signals generally originate from voltage or current sources. Output signals are usually the voltage across the capacitor and/or the current through the inductor. In circuits with initial energy stored in these elements, the stored energy drives the circuit response without external input.
Q7: What are common examples of second-order circuits?
Common examples include RLC circuits, which contain a resistor, inductor, and capacitor. Other second-order circuits include RC circuits with dual capacitors and RL circuits with dual inductors. Understanding types of responses of series RLC circuits helps predict how these configurations behave under different conditions.