EE 42/100Lecture 18: RLC Circuits

ELECTRONICSRev A 3/17/2010 (3:48 PM)

Prof. Ali M. Niknejad

University of California, Berkeley

Copyright c© 2010 by Ali M. Niknejad

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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 approximatesmechanical resonance (mass-spring, pendulum), molecular resonance, microwavecavities, transmission lines, buildings, bridges, ...

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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 seriesRLC circuit.

• Note that the solution depends on the initial charge on the capacitor and the initialflux (current) through the inductor.

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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 thesame.

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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 thefriction of the system.

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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 tankdischarges, the water is pushed into the second tank due to the inertia of theturbine (and the water itself!). Then eventually the right hand side tank fills up andthe direction of flow is reversed.

• Without friction, the system would never settle to a DC solution and the steadystate solution would be an oscillatory one.

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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 theinductance is increases, we find the rate of the step response changes. We alsoeventually observe overshoot and ringing.

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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 representsoscillatory behavior.

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Characteristic Equation• Plugging in Aest into the 2nd order equation, we arrive at the following

characteristic equation.

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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 standardcanonical form.

• The entire system is described by three constants: R, L, and C. The generalequation is parameterized by two constants: ω0 and Q.

• We shall show that ω0 represents the natural frequency of the system, or thefrequency at which the system would oscillate on its own.

• The Q factor, or the “quality” factor, is a measure of how quickly this oscillationdecays to zero (how much energy is lost per cycle).

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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

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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 thecapacitor discharges, it losses too much energy to the resistor. We can almostignore the inductance!

• In some applications, this is desired. The only problem is that the rate can be tooslow, similar to RC circuits.

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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.

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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 thefastest and never overshoots.

• You can imagine situations where this is exactly what you want – say your cruisecontrol in your automobile!

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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 (slowerdecay rate).

• In high Q circuits, the energy sloshes back and forth between the inductor andcapacitor and only a small fraction of the energy is lost per cycle to the resistor.

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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, butthe longer it takes it to settle due to “ringing”.

• Think about the shock absorbers in your car! Or imagine transmitting a signal fromone part of your circuit to another. The L and C would represent the parasiticinductances in your circuit.

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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 detectthe presence of the pen, even without touching. It can use the distance informationto estimate the “thickness” of lines that I draw.

• RLC circuits are used in radios that “tune” the signal to a particular frequency andreject 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 intothe tank.

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