388. RLC circuits8.1. Series RLC Circuits. Electric circuits provide an important ex-ample of linear, time-invariant differential equations, alongside mechan-ical systems. We will consider only the simple series circuit picturedbelow.CoilResistorCapacitorVoltageSourceFigure 5. Series RLC CircuitThe Mathlet Series RLC Circuit exhibits the behavior of this sys-tem, when the voltage source provides a sinusoidal signal.Current flows through the circuit; in this simple loop circuit the cur-rent through any two points is the same at any given moment. Currentis denoted by the letter I, or I(t) since it is generally a function oftime.The current is created by a force, the “electromotive force,” whichis determined by voltage differences. The voltage drop across a com-ponent of the system except for the power source will be denoted by Vwith a subscript. Each is a function of time. If we orient the circuitconsistently, say clockwise, then we letVL(t) denote the voltage drop across the coilVR(t) denote the voltage drop across the resistorVC(t) denote the voltage drop across the capacitorV (t) denote the voltage increase across the power source39“Kirchoff’s Voltage Law” then states that(1) V = VL+ VR+ VCThe circuit components are characterized by the relationship be-tween the current flowing through them and the voltage drop acrossthem:(2)Coil : VL= L˙IResistor : VR= RICapacitor :˙VC= (1/C)IThe constants here are the “inductance” L of the coil, the “resistance”R of the resistor, and the inverse of the “capacitance” C of the capac-itor. A very large capacitor, with C large, is almost like no capacitorat all; electrons build up on one plate, and push out electrons on theother, to form an uninterrupted circuit. We’ll say a word about theactual units below.To get the expressions (2) into comparable form, differentiate thefirst two. Differentiating (1) gives˙VL+˙VR+˙VC=˙V , and substitutingthe values for˙VL,˙VR, and˙VCgives us(3) L¨I + R˙I + (1/C)I =˙VThis equation describes how I is determined from the impressedvoltage V . It is a second order linear time invariant ODE. Comparingit with the familiar equation(4) m¨x + b ˙x + kx = Fgoverning the displacement in a spring-mass-dashpot system reveals ananalogy between the two types of system:Mechanical ElectricalMass CoilDamper ResistorSpring CapacitorDriving force Time derivative ofimpressed voltageDisplacement Current408.2. A word about units. There is a standard system of units calledthe International System of Units, SI, formerly known as the mks(meter-kilogram-second) system. In terms of those units, (3) is cor-rect when:inductance L is measured in henries, Hresistance R is measured in ohms, Ωcapacitance C is measured in farads, Fvoltgage V is measured in volts, also denoted Vcurrent I is measured in amperes, ABalancing units in the equation shows thathenry · amperesec2=ohm · amperesec=amperefarad=voltsecThus one henry is the same as one volt-second per ampere.The analogue for mechanical units is this:mass m is measured in kilograms, kgdamping constant b is measured in kg/secspring constant k is measured in kg/sec2applied force F is measured in newtons, Ndisplacement x is measured in meters, mHerenewton =kg · msec2so another way to describe the units in which the spring constant ismeasured in is as newtons per meter—the amount of force it produceswhen stretched by one
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