Mechatronics Engineering

Mechatronics Engineering
MT-144 NETWORK ANALYSIS Mechatronics Engineering (10)

TRANSIENT RESPONSE OF SECOND-ORDER CIRCUITS: (Chapter 9)
9.1 Natural Response of Series & Parallel RLC Circuits 9.2 Transient Response of Second Order Circuits 9.3 Step Response of Second Order Circuits 9.4 Second Order Op Amp Circuits 9.5 Transient Analysis Using Spice

Introduction. In this chapter we turn our attention to second-order circuits, that is, circuits containing two energy-storage elements that cannot be reduced to a single equivalent element via series/ parallel reductions. A second-order circuit may contain two capacitances, two inductances, or one of each. The last case is by far the most interesting because it may result in oscillatory behavior, a phenomenon found neither in first-order circuits nor in passive second-order circuits with two energy-storage elements of the same type. This phenomenon stems from the ability of energy to flow back and forth between the capacitance and the inductance, just as energy flows back and forth between the mass and the spring of a mechanical system.

Introduction… We begin by formulating the differential equations governing the series RLC and the parallel RLC circuits, and we find that the roots of the characteristic equation and, hence, the natural response, are characterized in terms of two parameters known as the under-damped natural frequency ω0 and the damping ratio ζ . Varying the damping ratio changes the location of the roots in the s plane as well as the damping characteristics of the response. Over-damped responses consist of exponentially decaying terms similar to those of first-order circuits; however, under-damped responses consist of decaying oscillations, a feature unique to higher-order passive circuits with mixed energy-storage element types. When subjected to a step function, an under-damped circuit exhibits overshoot and ringing, phenomena not possible with first-order circuits. Our mathematical analysis of the response under different damping conditions is of interest also to other fields, such as mechanical engineering and control.

Introduction… We then turn our attention to second-order Op Amp circuits and show that using an op-amp to provide a controlled amount of positive feedback, it is possible to position the roots of the characteristic equation anywhere in the s plane. This allows us to achieve not only damped oscillations using energy storage elements of the same type, but also diverging oscillations, a feature not possible with purely passive circuits. Moreover, this kind of behavior is achieved without the use of inductors, which are generally undesirable in modern design. Clearly, in this chapter we are witnessing some of the most intriguing properties of the op amp. We conclude by illustrating the use of SPICE to display the transient response of second-order circuits.

9.1 Natural Response of Series and Parallel RLC Circuits In this section we study the source-free or natural response of linear circuits containing a capacitance and an inductance either directly in series or in parallel with each other. Once the remainder of the circuit is replaced by its Thevenin or Norton equivalent, we are left with the basic configurations of Figure 9.1, known as the series RLC and the parallel RLC circuits. Figure 9.1 Basic second-order RLC circuits: (a) series and (b) parallel configurations.

9.1 Natural Response of Series and Parallel RLC Circuits Referring to the series circuit of Figure 9.1(a), we have, by KVL, vL + vR + vc = vS. Differentiating both sides with respect to time yields dvL/dt + dvR/dt + dvc/dt = dvS/dt. Letting vL = Ldi/dt, vR = Ri or dvR/dt =Rdi/dt and dvC/dt = i / C , and dividing through by L yields (9.1) : Figure 9.1 Basic second-order RLC circuits: (a) series and (b) parallel configurations.

9.1 Natural Response of Series and Parallel RLC Circuits Applying dual reasoning to the parallel circuit of Figure 9.1 (b) we have, by KCL, iC + iR + iL = iS. Differentiating both sides yields: dic/d t + diR/dt + diL/dt = diS/dt Letting iC = Cdv/dt, iR = v/R, and diL/dt = v/ L, and dividing through by C yields (9.2): Figure 9.1 Basic second-order RLC circuits: (a) series and (b) parallel configurations.

9.1 Natural Response of Series and Parallel RLC Circuits Both equations are second-order differential equations because this is the order of the highest derivative present. It is not surprising that the inclusion of an additional energy-storage element has increased the order of the circuit and its equation. Figure 9.1 Basic second-order RLC circuits: (a) series and (b) parallel configurations.

9.1 Natural Response of Series and Parallel RLC Circuits The Characteristic Equation. We wish to investigate the source-free or natural response of both circuits, that is, the response with vS = 0 or iS = 0. Since this makes dvS/dt = 0 or diS/dt = 0, both equations are of the type: Figure 9.1 Basic second-order RLC circuits: (a) series and (b) parallel configurations.

9.1 Natural Response of Series and Parallel RLC Circuits The Characteristic Equation. where y(t), representing either v or i , is the unknown variable; ω0 having the dimensions of radians/ second (rad/ s), is called the un-damped natural frequency; ζ (zeta) is a dimensionless parameter called the damping ratio. The reasons for this form and terminology will become apparent shortly. The expressions for ω0 and ζ are found by equating the corresponding coefficients. Thus, letting ω20 = 1/LC yields: ω0=1/√(LC) … (9.4) , both for the series and parallel circuits. Moreover, letting 2 ζ ω0 = R/L yields; ζ = (R/L) / (2ω0), or ζ = (½) R √(C/L) … (9.5), for the series circuit; likewise, letting 2ζω0 = 1/RC yields: ζ = 1/{2R √(C/L)} …(9.6) , for the parallel circuit. Note the duality of the two expressions.

9.1 Natural Response of Series and Parallel RLC Circuits The Characteristic Equation. Equation (9.3) states that a linear combination of the unknown function and its first and second derivatives must equal zero. The function and its derivatives must cancel somehow, suggesting an exponential solution of the type: y(t) = AeSt (9.7) Let us now seek suitable expressions for A and s that will make this solution work. Substituting into Equation (9.3) yields: s2Aest + 2ζω0sAest+ ω20Aest = 0, or (s2 + 2ζω0s + ω20 ) Aest = 0 Since we are seeking a solution A est ≠ 0, the expression within parentheses must vanish, s2 + 2ζω0s + ω20 = … (9.8) This equation is known as the characteristic equation because the parameters ω0 and ζ depend only on the elements R, L, and C and the way they are interconnected to form the circuit, regardless of the voltages or currents.

9.1 Natural Response of Series and Parallel RLC Circuits The Characteristic Equation. s2 + 2ζω0s + ω20 = … (9.8) Consequently, we expect it to provide information about the character of the natural behavior of the circuit. The roots of the characteristic equation (9.8) , variously called the natural frequencies, the characteristic frequencies, or the critical frequencies of the circuit, are readily found as s1,2= - ζω0 ± √(ζ2ω ω20 ) , or s1,2= {- ζ ± √(ζ ) } ω ….(9.9) indicating that the response will actually consist of two components, y1 = A1es1t and y2 = A2es2t . This is not surprising as we now have two energy-storage elements instead of one. If y1 and y2 satisfy Equation (9.3), so does their sum, y = y1 + y2, as you can easily verify by substitution. The most general solution is thus

9.1 Natural Response of Series and Parallel RLC Circuits The Characteristic Equation. (s2 + 2ζω0s + ω20 ) = … (9.8) The most general solution is thus Y(t) = A1es1t + A2es2t … (9.10) where A1 and A2 are suitable constants to be determined on the basis of the initial conditions for y(t) and its derivative. While R, L, and C are always real and positive, s1 and s2 may be real or complex, depending on whether the discriminant (ζ2 - 1) in Equation s1,2= {- ζ ± √(ζ ) } ω ….(9.9), is positive or negative. We have the following important cases: (1) ζ > 1, so that ζ > 0. In this case the roots are real, negative, and distinct. For reasons that will become apparent shortly, the corresponding response is said to be over-damped.

9.1 Natural Response of Series and Parallel RLC Circuits The Characteristic Equation. (s2 + 2ζω0s + ω20 ) = … (9.8) We have the following important cases: … (2) 0 < ζ< 1, so that ζ < 0. In this case the roots are said to be complex conjugate, and the response is said to be under-damped. (3) ζ = 1, so that ζ = 0. The roots are still real and negative, but they are now identical. The response is said to be critically damped. (4) ζ = 0. The roots are said to be purely imaginary, and the response is said to be undamped. Equations (9.4) through (9.6) indicate that ω0 depends only on L and C, while ζ depends also on R.

9.1 Natural Response of Series and Parallel RLC Circuits The Characteristic Equation. (s2 + 2ζω0s + ω20 ) = … (9.8) Varying the Damping Ratio To investigate the various response types we keep L and C fixed and vary R with a potentiometer to achieve different values of ζ . This is shown in Figure 9.2. In either circuit the function of the dc source is to inject energy into the circuit so that the latter can store it in its capacitance and/ or inductance, and reuse it to produce the natural response once the source is excluded from the circuit. Figure 9.2 Investigating the natural response of the series and parallel RLC configurations for different damping conditions.

9.1 Natural Response of Series and Parallel RLC Circuits The Characteristic Equation. (s2 + 2ζω0s + ω20 ) = … (9.8) Varying the Damping Ratio To investigate the various response types we keep L and C fixed and vary R with a potentiometer to achieve different values of ζ . This is shown in Figure 9.2. As usual, we shall assume that the switch has been in the position shown long enough to allow for the circuit to reach its dc steady state, where ic = 0 and vL = 0.

9.1 Natural Response of Series and Parallel RLC Circuits The Characteristic Equation. (s2 + 2ζω0s + ω20 ) = … (9.8) Varying the Damping Ratio To investigate the various response types we keep L and C fixed and vary R with a potentiometer to achieve different values of ζ . This is shown in Figure 9.2. Before considering examples of the various response types, let us find the initial values of y(t) and its derivative because we shall need them later on to calculate the constants A1 and A2.

9.1 Natural Response of Series and Parallel RLC Circuits Example 9.1 In the series RLC circuit of Figure 9.2(a) find the current i(0+) and its derivative di(0+)/ dt just after switch activation. Solution. During the steady state preceding switch activation, the capacitance acts as an open circuit and the inductance as a short circuit. Hence, iL(0+)= 0, and vc(0-)=vS. By the inductance and the capacitance continuity rules, i(t)= iL(0+)= iL(0-)= 0, and vc(0+)= vc(0-)= vS. By the inductance law, di(0+)/dt = diL(0+)/dt = vL(0+)/L. To find vL(0+), apply KVL: vR(0+)+ vC(0+) + vL(0+)= 0, or vL(0+)= -vR(0+) - vc(0+) = -Ri(0+) - vS = - vS. Substituting yields di(0+)/dt = -vS/L. In short, the initial conditions for the given circuit are : Figure 9.2(a) Home Work: Do Exercises 9.1 (page 382 of text book)

9.1 Natural Response of Series and Parallel RLC Circuits Overdamped Response If ζ > 1, the roots are real, negative, and distinct, and they are expressed in nepers/ second (Np/ s), as usual. Y(t) = A1es1t + A2es2t … (9.10) According to Equation (9.10), the response is the sum of two decaying exponentials having, respectively, A1 and A2 as initial values, and Ƭ1 = -1/s1 and Ƭ2 = -1/s2 as time constants, ... (9.13) Where by Equation: we get:

9.1 Natural Response of Series and Parallel RLC Circuits Over-damped Response It is good practice to visualize the roots of the characteristic equation as a pair of points in a plane of roots called the s plane, already introduced in Section 7.3 for first-order circuits. As shown in Figure 9.3(a), a real and negative root pair is represented by a pair of points on the negative portion of the horizontal axis at s1 = - 1/Ƭ1 and s2 = - 1/Ƭ2. As we know, this axis is calibrated in Np/ s.

9.1 Natural Response of Series and Parallel RLC Circuits Over-damped Response We now wish to develop expressions for A1 and A2 in terms of the initial conditions of the circuit. Evaluating the unknown variable and its derivative at t= 0+ we obtain y(0+) = A1e0 + A2e0 = A1 + A2 , and dy(0+)/dt = - (1/Ƭ1)A1e0 – (1/Ƭ2)A2e0 = -(1/Ƭ1)A1 – (1/Ƭ2) A2. Solving for A1 and A2 yields

9.1 Natural Response of Series and Parallel RLC Circuits Over-damped Response… Home Work: Study Examples: 9.2 and 9.3 on pages 383 to 384 of the text book Do Exercises: 9.2 and 9.3 on pages 383 to 384 of the text.

9.1 Natural Response of Series and Parallel RLC Circuits Under-damped Response. If 0< ζ< 1, the discriminant is negative and can be rewritten as √(ζ2-1) = j √(1-ζ2), where j ≡ √ (-1) … (9.17) is a dimensionless quantity known as the imaginary unit. The roots can thus be expressed as: S1,2 = - α ± jωd … (9.18) where α = ζ ω … (9.19), is called the damping coefficient, and ωd = ω0 √ (1- ζ2) … (9.20), called the damped natural frequency For the root s1 = -α + jωd , ωd is also referred to as the imaginary part of s1, while -α is, by contrast, referred to as the real part of s1. For the root s2 = - α - jωd , the real part is still -α; however, the imaginary part is now -ωd.

9.1 Natural Response of Series and Parallel RLC Circuits Under-damped Response… If 0< ζ< 1, the discriminant is negative and can be rewritten as √(ζ2-1) = j √(1-ζ2), where j ≡ √ (-1) … (9.17) is a dimensionless quantity known as the imaginary unit. The roots can thus be expressed as: S1,2 = - α ± jωd … (9.18) where α = ζ ω … (9.19), is called the damping coefficient, and ωd = ω0 √ (1- ζ2) … (9.20), called the damped natural frequency Both s1 and s2 are examples of complex variables. Since their imaginary parts are opposite to each other, s1 and s2 are said to form a complex-conjugate pair.

9.1 Natural Response of Series and Parallel RLC Circuits Under-damped Response… Though both α and ωd have the dimensions of the reciprocal of time, or s-1, we shall continue to express α in nepers/ second (Np/ s), whereas we shall express ωd in radians/ second (rad/ s) to distinguish between the two. To evidence their complex nature, we shall express s1 and s2 in complex Np/ s. In summary, denoting the physical units of a given quantity x as [ x ] , we have : [α] = Np/ s [ωd ] =[ω0] =rad/ s [s1] = [s2] = complex Np/ s To visualize complex roots in the s plane we need two axes: a horizontal axis for plotting real parts, and a vertical axis for imaginary parts. The former is calibrated in Np/ s and is called the real axis; the latter is calibrated in rad/ s and is called the imaginary axis.

9.1 Natural Response of Series and Parallel RLC Circuits Under-damped Response… For instance, the root s1 = -3 + j4 complex Np/ s is the point of the s plane having an abscissa of -3 Np/ s and an ordinate of +4 rad/ s. Its conjugate s2 = j4 complex Np/ s is the point having an abscissa of - 3 Np/ s and an ordinate of -4 rad/ s. As shown in Figure 9.4(a), two conjugate roots are located symmetrically with respect to the real axis.

9.1 Natural Response of Series and Parallel RLC Circuits Under-damped Response… We now wish to find the under-damped response. This is obtained by substituting Equation: S1,2 = - α ± jωd … (9.18) , into Equation : Y(t) = A1es1t + A2es2t … (9.10) However, it is shown in Appendix 3 that y(t) can be put in the more insightful form: Y(t) = Ae-αt Cos (ωd t + θ) … (9.21) where A and θ are the usual initial-condition constants. This function, called a damped sinusoid, has angular frequency ωd, phase angle θ, and an exponentially decaying amplitude, Ae-αt . The rate of decay is governed by the time constant Ƭ = 1/ α = 1/ ζω0 . Moreover. by Equation (9.20), the damped frequency ωd is always less than the undamped frequency ω0.The smaller the value of ζ the slower the decay and the closer ωd is to ω0. We are now better able to appreciate the reason for referring to the roots of the characteristic equation as frequencies.

9.1 Natural Response of Series and Parallel RLC Circuits Under-damped Response… Physically, the oscillations stem from the ability of stored energy to flow back and forth between L and C, whereas the damping stems from energy loss in R. This is similar to a mechanical system consisting of a mass, spring, and damper. If we inject energy into the system by hitting the mass, the system will oscillate at a frequency determined by the mass and spring characteristics. During each cycle, the kinetic energy of the mass is converted to potential energy in the spring and vice versa. At the same time, part of the energy is dissipated into heat because of friction in the damper, thus causing the oscillations to die out. The smaller the friction, the longer the persistence of the oscillations. Clearly, in this analogy the mass and the spring are the energy-storage elements, and the damper simulates dissipation by R.

9.1 Natural Response of Series and Parallel RLC Circuits Critically Damped Response. Suppose we have an underdamped circuit and we vary R to increase its damping ratio. As ζ approaches unity, the discriminant (ζ2 - 1) approaches zero. When ζ crosses unity, the roots change from complex to real because the discriminant crosses zero. Consequently, the response changes from oscillatory to nonoscillatory. Imposing ζ = 1 in Equations (9.5) and (9.6) indicates that this change occurs when R reaches a critical value, Rc such that , for the series circuit: and , for the parallel circuit: Physically, when R= Rc all available energy from the energy-storage elements is dissipated by the resistance within a single cycle, thus precluding any further oscillation. A circuit with ζ = 1 is said to be critically damped because its response represents the borderline between oscillatory and non oscillatory responses.

9.1 Natural Response of Series and Parallel RLC Circuits Critically Damped Response… There is a mathematical peculiarity associated with critical damping because the roots are now equal: s1 = s2 = - ω0. Equation (9.10), whose derivation was based on the assumption of distinct roots, is no longer the correct solution. Mathematically it can be shown that repeated roots give rise to natural components of the form test, along with est. The general form of a critically damped response is then y(t) = A1e-t/Ƭ + A2te-t/Ƭ, or where A1 and A2 are the usual initial-condition constants. It is easy to verify that this is indeed the correct solution by substituting it into Equation (9.3).

9.1 Natural Response of Series and Parallel RLC Circuits Critically Damped Response… When ζ = 1 , the roots are real, negative, and identical. As shown in Figure 9.5(a), their s-plane representation consists of two coincident points located on the real axis at -1/ Ƭ . (a) (b) Figure 9.5 Coincident roots, and example of a critically damped response.

9.1 Natural Response of Series and Parallel RLC Circuits Critically Damped Response… A familiar example of a critically damped system is the automobile suspension system, which is so designed to ensure a smooth ride in spite of possible bumps or potholes in the road. The suspension can be checked by suddenly pushing down either the front or the rear end of the car and then observing how it comes back up. Clearly, this is the natural response of the suspension system. Depending on its damping characteristics, a competent mechanic should be able to diagnose the state of the springs and the shock absorbers. (a) (b) Figure 9.5 Coincident roots, and example of a critically damped response.

9.1 Natural Response of Series and Parallel RLC Circuits Undamped Response. The decay of an under-damped response is caused by energy loss in the resistance. The smaller this loss, the slower the decay. In the limiting case of zero loss, the response would never die out and the outcome would be a sustained oscillation. In our mechanical analogy this situation corresponds to frictionless oscillation. Power loss in the resistance is related to the natural response as PR = Ri2 in the series circuit, and PR = v2/R in the parallel circuit. To achieve lossless operation, we must therefore have: R = 0 … (9.28), for the series circuit and R = ∞ … (9.29), for the parallel circuit. Either condition yields ζ = 0, by Equations (9.5) and (9.6). Letting ζ = 0 in Equations (9.19) through (9.21), we obtain α = 0, ωd = ω0, and y = Ae0cos ( ω0 t + θ), or y(t) = A cos ( ω0 t + θ) … (9.30) where A and θ are the usual initial-condition constants. Since this is an undamped sinusoid, its angular frequency ωo is called the undamped natural frequency. The period of oscillation is T0 = 2π/ω0

9.1 Natural Response of Series and Parallel RLC Circuits Undamped Response… When ζ = 0, the roots are purely imaginary. As shown in Figure 9.6(a), their s-plane representation consists of two points located symmetrically on the imaginary axis at ± ω0.

9.1 Natural Response of Series and Parallel RLC Circuits Undamped Response… When ζ = 0, the roots are purely imaginary. As shown in Figure 9.6(a), their s-plane representation consists of two points located symmetrically on the imaginary axis at ± ω0. Home Work: Study Examples: 9.6 & 9.7 pages 390 to 391 of the text book Do Exercises: 9.6 on page 390 of the text.

9.1 Natural Response of Series and Parallel RLC Circuits Undamped Response… The underdamped natural frequency ωd is always less than the undamped natural frequency ω0, and the under-damped period Td is always longer than the undamped period T0. Increasing ζ decreases ωd and increases Td as per Equations (9.20) and (9.23). As ζ  l, we have ωd  0 and Td  ∞, in agreement with the fact that the response changes from oscillatory to non oscillatory. In a practical LC circuit the undamped condition cannot be achieved because the losses in the stray resistances of its elements and interconnections cause the oscillation to eventually die out. However, using additional circuitry to continuously reinject into the system the exact amount of power that is dissipated in its resistances, it is possible to maintain a sustained oscillation, giving the A appearance of undamped operation.

9.1 Natural Response of Series and Parallel RLC Circuits Undamped Response… This principle forms the basis of a certain class of oscillators. Typically, an oscillator consists of a timing element, such as an LC pair or a quartz crystal to establish the frequency of oscillation, and an active device such as an amplifier to compensate for energy losses. A familiar analog of an electrical oscillator is a mechanical pocket watch, which uses the energy stored in its spring to compensate for the friction losses of its balance wheel and thus maintain a sustained oscillation. The Root Locus. The effect of ζ upon the roots of the characteristic equation can be visualized graphically by means of the root Locus. This is the system of trajectories described by the roots as ζ is varied over its range of possible values. Figure 9.7 shows the locus for a second-order passive RLC circuit as ζ is varied from ζ > 1 all the way down to ζ = 0.

9.1 Natural Response of Series and Parallel RLC Circuits The Root Locus… With ζ > 1, both roots are located on the negative real axis. As ζ decreases, the roots move toward each other on the real axis until they coalesce when the condition ζ = 1 is reached. This corresponds to critical damping. Decreasing ζ below unity causes the roots to split apart and move along symmetric trajectories toward the imaginary axis, which they reach in the limit ζ = 0. Using the Pythagorean Theorem, the radial distance r of an underdamped root from the origin is, by Equations (9.19) and (9.20), r =√ (α2 + ω2d) = ω0 = 1/ √(LC) = constant, indicating that the trajectories are circular arcs with radius ω0. As shown in greater detail in Figure 9.8, given a root pair s1 and s2 on these arcs, we can state the following: (1) The abscissa of s1 (or s2), called the real part of s1 (or s2), represents the negative of the damping coefficient, - α = - ζω0.

9.1 Natural Response of Series and Parallel RLC Circuits The Root Locus… As shown in greater detail in Figure 9.8, given a root pair s1 and s2 on these arcs, we can state the following: (1) The abscissa of s1 (or s2), called the real part of s1 (or s2), represents the negative of the damping coefficient, - α = - ζω0. (2) The ordinate of s1, also called the imaginary part of s1, represents the damped frequency ωd = ω0 √(1- ζ2). (3) The distance of either root from the origin represents the undamped frequency ω0. (4) The damping ratio is readily found as ζ = √ {1-(ωd/ ω0)2}

Having studied the natural or source-free response of RLC circuits, we now investigate the transient response, that is, the response to a dc forcing function x(t) = XS The equation governing the circuit now takes on the general form : The response will generally consist of a transient component and a dc steady state component: y(t) = yxsient + yss … (9.33) The transient component takes on the same functional form as the natural response. Depending on the damping conditions of the circuit, this component will be over-damped if ζ > I, critically damped if ζ = I , and under-damped if 0 < ζ < 1.

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