A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 1/19 EE 42/100 Lecture 18: RLC Circuits ELECTRONICS Rev A 3/17/2010 (3:48 PM) Prof. Ali M. Niknejad University of California, Berkeley Copyright c 2010 by Ali M. Niknejad
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 2/19 RLC Circuits The RLC circuit circuit is one of the most important and fundamental circuits. As we shall see, it has natural resonant frequencies. The physics describes not only physical RLC circuits, but also approximates mechanical resonance (mass-spring, pendulum), molecular resonance, microwave cavities, transmission lines, buildings, bridges,...
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 3/19 Series RLC Circuit As we shall demonstrate, the presence of each energy storage element increases the order of the differential equations by one. So for an inductor and a capacitor, we have a second order equation. Using KVL, we can write the governing 2nd order differential equation for a series RLC circuit. Note that the solution depends on the initial charge on the capacitor and the initial flux (current) through the inductor.
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 4/19 Parallel RLC Circuit A Parallel RLC circuit is the dual of the series. In other words, the role of voltage/current and inductance/capacitance are swapped but the equation is the same.
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 5/19 Mechanical Analog: Mass-Spring-Damper System Recall that using Newton s law and Hook s law, we arrive at a second order differential equation for a mass-spring system. A damper is used to model the friction of the system.
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 6/19 Fluid Flow Analog We can construct an analog to the RLC circuit by modeling the capacitors and inductors using water tanks and turbines. As you might expect, the water flow would slosh back and forth. When the tank discharges, the water is pushed into the second tank due to the inertia of the turbine (and the water itself!). Then eventually the right hand side tank fills up and the direction of flow is reversed. Without friction, the system would never settle to a DC solution and the steady state solution would be an oscillatory one.
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 7/19 Step Response: Steady-State Solution The steady-state solution is easy to find. If d/dt = 0, then the equation reduces to a simple DC equation. For small L, we know the output exponentially climbs toward the solution. As the inductance is increases, we find the rate of the step response changes. We also eventually observe overshoot and ringing.
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 8/19 Second Order Equations: Homogeneous Solution For any second order homogeneous system, the solution is an exponential function. The amplitude and the argument of the exponential must be selected to satisfy the differential equations. We shall see that the arguments can become complex, which represents oscillatory behavior.
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 9/19 Characteristic Equation Plugging in Ae st into the 2nd order equation, we arrive at the following characteristic equation.
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 10/19 Canonical Characteristic Equation Since we will meet this same equation over and over, we solve it once and carefully categorize the solution. To do this we put the equation into a standard canonical form. The entire system is described by three constants: R, L, and C. The general equation is parameterized by two constants: ω 0 and Q. We shall show that ω0 represents the natural frequency of the system, or the frequency at which the system would oscillate on its own. The Q factor, or the quality factor, is a measure of how quickly this oscillation decays to zero (how much energy is lost per cycle).
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 11/19 I - Over Damped Response In the over damped case, both roots of the characteristic equation are real and different. The quality factor of the circuit Q is low: Q < 1 2
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 12/19 I - Step Response The system is described by two decaying time constants. There is so much loss in the system that we do not observe any kind of oscillation. For example, as the capacitor discharges, it losses too much energy to the resistor. We can almost ignore the inductance! In some applications, this is desired. The only problem is that the rate can be too slow, similar to RC circuits.
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 13/19 II - Critically Damped Response If the quality factor of the circuit Q is exactly 1/2, we say the circuit is critically damped. Q = 1 2 Then there is only one root to the characteristic equation.
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 14/19 II - Step Response This has the desirable characteristic that the circuit step response settles to a DC state the fastest without overshooting. In other words, the transient response is the fastest and never overshoots. You can imagine situations where this is exactly what you want say your cruise control in your automobile!
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 15/19 III - Underdamped Response When Q > 1/2, the circuit is said to be underdamped. That means the natural response is oscillatory in nature. The higher the Q, the longer it oscillates (slower decay rate). In high Q circuits, the energy sloshes back and forth between the inductor and capacitor and only a small fraction of the energy is lost per cycle to the resistor.
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 16/19 III - Step Response As evident in the plot, the quality factor plays an important role in the step response. The higher the Q, the faster the circuit reaches the desired output, but the longer it takes it to settle due to ringing. Think about the shock absorbers in your car! Or imagine transmitting a signal from one part of your circuit to another. The L and C would represent the parasitic inductances in your circuit.
A. M. Niknejad University of California, Berkeley EE 100 / 42 Lecture 18 p. 19/19 Applications of RLC Circuits RLC circuits are ubiquitous. This tablet PC uses a pen that has an RLC resonant circuit in the pen. Using near field magnetic coupling, the screen is able to detect the presence of the pen, even without touching. It can use the distance information to estimate the thickness of lines that I draw. RLC circuits are used in radios that tune the signal to a particular frequency and reject other frequencies. They are also used to generate signals (oscillators, voltage controlled oscillators, clocks) in circuits by designing high quality tanks. The decay in the tank is compensated by adding a bit of energy per cycle back into the tank.