3.0 Second Order Circuits A second order circuit is a circuit with two effective energy storage elements, either two capacitors, two inductors, or one of each. (In some circuits it may be possible to combine multiple capacitors or inductors into one equivalent capacitor or inductor ) We begin this section with the derivation of the governing differential equation for various second order circuits. At this point we will focus on circuits that we can put into a standard form. Once we have covered Laplace transforms we will analyze different types of second order circuits. This standard second order form will again allow us to easily determine physical characteristics of the circuit and predict the time response. We then solve the differential equations for the case of a constant input. 3.1 Governing Differential Equations for Second Order Circuits: Standard Form In this section we derive the governing differential equations that model various RL, RC, and RLC circuits. We then put the governing second order differential equations into a standard form, which allows us to read off descriptive information about the system very easily. The standard form we will use is 222()((())2)nn nddytdt dytyt Ktxtζω ω ω+=+ or 221()2() ()()nnddytytytdt dxttKζωω+=+ Here we assume the system input is()xt and the system output is . ()ytnω is the system natural frequency, which indicates the frequency at which the system will oscillate if there is no dampling. The natural frequencynω has units of radians/second. ζis the damping ratio, which indicates how much damping there is in the system. A damping ratio of zero indicates there is no damping at all. The damping ratioζis dimensionaless. K is the static gain of the system. For a constant input of amplitudeA (() ()xtAut=, where is the unit step function), in steady state we have()ut()0tdydt=, 22()t0dydt= , and . Hence the static gain lets us easily compute the steady state value of the output. To determine the units of the static gain we use () KA=()Kxt=yt [units of y] = [units of K][units of x] or [units of K] = [units of y]/[units of x] Note that not all second order circuits can be modeled by a differential equation of this form. While we can always write the left hand side of the differential equation in this form, for some circuits the right hand side of the differential equation may contain ©2009 Robert D. Throne 1derivatives of the inputs. In addition, this form may not always be the best way to write the differential equation. Example 3.2.1. Consider the RLC circuit shown in Figure 3.1. ()xt Figure 3.1. Circuit for Example 3.2.1. The input, ()xt, is the applied voltage and the output, , is the voltage across the capacitor. If we denote the current flowing in the circuit as , then applying Kirchhoff’s voltage law around the single loop gives us the equation ()yt()it ()())((di tyt itRdtxt L ++−+ )0= We can also relate the voltage across the capacitor with the current flowing through the capacitor ()()dy tit Cdt= Substituting this equation into our first expression we get 22() ()0() ()dyt dytyt RCdt dtxt LC ++−=+ or 22() ()() ()dyt dyLCtRC yddttxtt++= Comparing this expression with our standard form we get natural frequency: 21nLCω=, or 1nLCω= damping ratio: 2nRCζω=, or 2RCLζ= static gain:1K= ()yt+ - -+ ()itR©2009 Robert D. Throne 2Example 3.2.2. Consider the RLC circuit shown in Figure 3.2. ()xt R()yt•*vL C Figure 3.2. Circuit used in Example 3.2.2. The input, ()xt, is the applied current and the output, , is the current through the inductor. If we denote the node voltage at the top of the circuit as , then applying Kirchhoff’s current law give us ()yt*()vt**() ()() y(t) + Cv t dv txtRdt=+ We can also relate the voltage across the inductor with the current flowing through the inductor *()()dy tvt Ldt= Substituting this equation into our first expression we get 22() ()() y(t) + LCLdyt d ytxtRdt dt=+ or 22() ()LC y(t) = x(t)d y t L dy tdt R dt++ Comparing this expression with our standard form we get natural frequency: 21nLCω=, or 1nLCω= damping ratio:2nLRζω=, or 12LCRζ= static gain: 1K= ©2009 Robert D. Throne 3Example 3.2.3. Consider the RLC circuit shown in Figure 3.3. R Figure 3.3. Circuit used in Example 3.2.3. The input, ()xt, is the applied current and the output, , is the current through the inductor. If we denote the node voltage at the top of the circuit as , then applyingKirchhoff’s current law give us ()yt*()vt *()() y(t) + Cdv txtdt= We can then determine the node voltage as *()vt *()() ()dy tvt Ryt Ldt=+ Substituting this equation into our first expression we get 22() () ()() () [ () ] ()ddyt dytdxt yt C Ryt L yt RC LCdt dt dt dt=+ + =+ +yt or 22() ()() ()dyt dytLC RC y t x tdt dt++= Comparing this expression with our standard form we get natural frequency: 21nLCω=, or 1nLCω= damping ratio:2nRCζω=, or 2RCLζ= static gain:1K= C•*vL()xt ()yt©2009 Robert D. Throne 4Example 3.2.4. Consider the RLC circuit shown in Figure 3.4. Figure 3.4. Circuit used in Example 3.2.4. The input,()xt, is the applied voltage and the output, , is the voltage across resistor ()ytbR. Node voltages and are as shown in the figure. Applying Kirchhoff’s current law gives at node gives ( )avt ( )bvt( )tav () () () () () ()0aabaaavt xt vt vt vt dvtCRRRdt−−+++ = which we can simplify as ()3() () ()aabadv tvt xt vt RCdt−− =− Summing the currents into the negative terminal of the op amp gives us () ()0abbvt dvtCRdt+= or ()()babdv tvt RCdt=− Substituting this expression into our simplified equation above we get () ()3()()bbbbabdv t dv tdRC x t v t RC RCdt dt dt⎡⎤ ⎡⎤−−−=−−⎢⎥ ⎢⎥⎣⎦ ⎣⎦ or + - -+ RRRaRbRbCaC ()xt ()yt+ - • •bv av ©2009 Robert D. Throne 5222() ()3()bab bbbd t dv tCC RC v t xtdt dtvR ++=()− Finally, we have )(()bbabRtyvRRt =+ or () ()abbbRRvt ytR+= resulting in the differential equation 222() ()3()ab babbRd t dy tCC RC yt xtdt dt RyRR++=−+() Comparing this expression with our standard form we get natural frequency:221abnRCCω=, or 1nabRCCω= damping ratio:23bnRCζω=, or 32baCCζ= static gain: babRKRR=−+ 3.3 Solving Second Order Differential Equations in Standard Form In this section we will solve second order differential equations the standard form 2222()((())2)nn nddytdt dytyt Ktxtζω ω ω+=+ for a constant (step) input.