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MT-144

MT-144. NETWORK ANALYSIS Mechatronics Engineering (07). TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8). 8.1 Basic RC and RL Circuits 8.2 Transients in First-order Networks 8.3 Step, Pulse and Pulse-Train Responses 8.4 First-order Op-Amp Circuits

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MT-144

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  1. MT-144 NETWORK ANALYSIS Mechatronics Engineering (07)

  2. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 Basic RC and RL Circuits 8.2 Transients in First-order Networks 8.3 Step, Pulse and Pulse-Train Responses 8.4 First-order Op-Amp Circuits 8.5 Transient Analysis Using SPICE

  3. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) Introduction. In Section 7.4 we found that the response to a dc forcing function consists of a transient component and a dc steady-state component. The transient component has the same functional form as the natural response, which is an exponentially decaying function. The dc steady-state component has the same functional form as the forcing function, which is a constant function. We now wish to apply the mathematical concepts of Sections 7.3 and 7.4 to a variety of circuit configurations, but with greater attention to physical behavior. An engineer must always use physical insight to interpret mathematical results. In the present chapter we concentrate on first-order circuits, that is, circuits that contain only one energy-storage element or that contain multiple elements, but in such a way that they can be reduced to a single equivalent element via series / parallel combinations.

  4. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) Introduction… In Chapter 9 we turn our attention to second-order circuits, with special emphasis on circuits that contain one capacitance or one inductance. We begin by examining in great physical detail how energy builds up and decays in simple capacitive and inductive circuits. In so doing, we develop a set of useful rules to facilitate the transient analysis of more complex networks. Next, we study the step and pulse responses of the basic R-C, L-R, C-R, and R-L circuits, and observe how these circuits behave in terms of signal distortion as well as passing or blocking the dc component of an input pulse train. We then turn to capacitive circuits using op amps.

  5. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) Introduction… After examining the classical integrator and differentiator configurations, we use the op amp to illustrate another important systems concept, this time the concept of root location control. In particular, we illustrate the use of an op amp to create a root in the right half of the s plane and thus obtain a diverging response, a situation that cannot be achieved with a purely passive circuit. As usual, we conclude the chapter by illustrating the use of SPICE to perform the transient analysis of circuits. The SPICE facilities introduced in this chapter are the . TRAN statement and the PULSE function.

  6. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS In this section we take a closer look at the physical aspects of the transient behavior of basic RC and RL pairs. However, we must first state two important rules that will greatly facilitate our transient analysis of these circuits. Instantaneous Behavior: The Continuity Conditions Capacitance current and voltage are related as ic = C dvc/dt. The faster the change in vc, the greater ic. If vc were to change instantaneously, dvc/dt would become infinite, and so would ic. An infinite current, however, would require the existence of an infinite amount of power at the capacitor terminals. Since this is physically impossible, we have the following rule, known as the voltage continuity rule for capacitance: Capacitance Rule 1: The voltage across a capacitance cannot change instantaneously, that is, at any instant to we must have vC(t0+) = vc(t0-), where t0+ is the instant right after t0 , and t0- is the instant right before to.

  7. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… Instantaneous Behavior: The Continuity Conditions … Capacitance Rule 1: The voltage across a capacitance cannot change instantaneously, that is, at any instant to we must have vc(to+) = vc(to-), where to+ is the instant right after to, and to- is the instant right before to. Note that the continuity rule applies to vc, not to ic; ic can indeed change instantaneously without violating Rule 1. We can readily visualize this rule using the water tank analogy, where water level is likened to voltage. and water flow to current. To bring about an instantaneous level change would require adding or removing water from the tank in zero time, a physically impossible task. Current flow can undergo brusque changes, however, as when we suddenly open a spigot to let water out of the tank.

  8. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… Instantaneous Behavior: The Continuity Conditions … Turning now to inductance, whose voltage and current are related as vL = L diL/dt, we observe that if iL were to change instantaneously, diL/dt would become infinite and so would vL. This is again physically impossible, as it would require the existence of an infinite amount of power at the inductor terminals. Hence, we have the following current continuity rule for inductance: Inductance Rule 1: The current through an inductance cannot change instantaneously, that is, at any instant to we must have iL(t0+) = iL(t0-). Inductors are also called chokes to reflect their tendency to oppose sudden changes in current. Note that the continuity rule applies to inductive current, not to inductive voltage. The latter i.e. (vL) can indeed undergo instantaneous changes without violating Rule 1.

  9. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior If the voltage vC across a capacitance is known to have stabilized as some constant value Vc, not necessarily zero, the capacitance is said to be in dc steady state. Once in this state, it draws no current from the rest of the circuit because iC = C dvC / dt= C x 0 = 0. This behavior is summarized as follows: Capacitance Rule 2: In dc steady state, a capacitance behaves as an open circuit, that is, ic = 0. Likewise, if the current iL through an inductance is known to have stabilized as some constant value IL , not necessarily zero, the inductance is said to be in dc steady state, and the voltage across its terminals is vL = L diL /dt = L x 0 = 0. Consequently, Inductance Rule 2: In dc steady state, an inductance behaves as a short circuit, that is, vL = 0.

  10. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior… Inductance Rule 2: In dc steady state, an inductance behaves as a short circuit, that is, vL = 0. These rules indicate that if an energy-storage element in a circuit is known to be in dc steady state, we can replace it with an open circuit if it is a capacitance, or a short circuitif it is an inductance, and thus simplify our analysis considerably (assuming ideal capacitors and inductors). Table 8.1 compares capacitance and inductance behavior. Note again the dual behavior of the two elements. (next slide)

  11. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior… Table 8.1 compares capacitance and inductance behavior. Note again the dual behavior of the two elements. (next slide)

  12. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior… The RC Circuit In the circuit of Figure 8.1(a) assume the capacitance is initially discharged so that q= 0. Then, we know q= cv , or we have v = q/C = 0/C = 0. Moreover, i = (0 – v )/ R = (0 – 0) /R = 0.

  13. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… In the circuit of Figure 8.1(a) assume the capacitance is initially discharged so that q= 0. Then, we know q= cv , or we have v = q/C = 0/C = 0. Moreover, i = (0 – v )/ R = (0 – 0) /R = 0. At an instant that we arbitrarily choose as t= 0 we flip the switch to connect R to VS , ending up with the situation of Figure 8.1(b). The resulting current i = (VS - v )/ R will charge C toward VS according to Equation (7.23), which now becomes: RC dv/ dt + v = VS …(8.1)

  14. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) • 8.1 BASIC RC AND RL CIRCUITS… • DC Steady-State Behavior…The RC Circuit… • A word of caution is necessary at this point. The instant t= 0 is exceptional because the switch voltage is in the process of changing from 0 to VS and, as such, is not a single-valued function of time. • RC dv/ dt + v = VS …(8.1) • To avoid any ambiguities, we assume Equation (8.1) to begin at t= 0+, where 0+ denotes the instant just after t = 0, when the switch voltage has fully attained the value VS. • Likewise, t = 0- shall denote the instant just before t = 0, when the switch voltage is still zero. With this in mind, the solution to Equation (8.1) shall also be assumed to begin at t= 0+. Such a solution is provided by Equation (7.41), but with t = 0+ instead of t = 0. • y(t)= y(0) e-t/Ƭ + XS (1- e-t/Ƭ) … (7.41) • Because of the voltage continuity rule, we have v(0+) = v(0-) = 0, indicating that the natural component in Equation (7.41) is zero.

  15. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) • y(t)= y(0) e-t/Ƭ+ XS (1- e-t/Ƭ) … (7.41) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… The solution is thus the forced component. To summarize, just prior to switch activation we have The responses are shown in Figure 8.2. It is interesting to note that v is continuous but i is discontinuous at the origin.

  16. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… The responses are shown in Figure 8.2. It is interesting to note that v is continuous but i is discontinuous at the origin. This does not violate Rule 1, however, as continuity applies to voltage, not to current. Because of its abrupt jump in magnitude, the current waveform is referred to as a spike. • v(t)= vS (1- e-t/RC) • i(t)= (vS/R) e-t/RC)

  17. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… Both transients of Figure 8.2 are governed by the time constant Ƭ = RC. Increasing R reduces the charging current, and increasing C increases the amount of charge that needs to be transferred. In either case the transients will be slowed down. Conversely, decreasing R results in faster transients because the charging current is increased.

  18. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… In the limit R  0, C would try to charge instantaneously. But to do so would require an infinitely large current spike from VS, which we know to be physically impossible. In a practical situation current is limited by the internal resistance of the source and the connecting wires, indicating that charge buildup, however rapid, cannot be instantaneous. The spark that we observe in the laboratory when we connect a capacitor directly across a battery attests to the large current spike involved.

  19. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… Example 8.1 • v(t ≥ 0+)= vS (1- e-t/RC) …(8.2b) Home Work: Do Exercise 8.1 (page 330 of text book)

  20. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… Suppose that after the capacitance of Figure 8.1(a) has fully charged to VS, the switch is flipped back down at an instant that we again choose as t = 0 for convenience. This is shown in Figure 8.3(a). With the source out of the way, we end up with the source-free circuit of Figure 8.3(b). By the voltage continuity rule we must have v(0+) = v(0-) = VS. Consequently, at t = 0+, the top plate holds the charge +CVSand the bottom

  21. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… Consequently, at t = 0+, the top plate holds the charge +CVS and the bottom plate the charge -CVS. Moreover, the initial stored energy in the capacitance is w(0+) = (1/2)CV2S. Thanks to this energy, C will be able to sustain a nonzero voltage even though VS has been switched out of the circuit. This voltage, in turn, will cause R to draw current and gradually discharge C. Though the reference direction is still shown clockwise for consistency, the discharge current actually flows counterclockwise, out of the top plate, through the resistance, and back into the bottom plate.

  22. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… The discharge process is governed by Equation (7.23), which now becomes: RC dv/ dt + v = 0 …(8.4) Rewriting as dv/ dt = -v/ (RC) indicates that v decreases at a rate proportional to v itself. Initially, when v is large, we have a proportionally rapid discharge rate. As the discharge progresses, the rate decreases in proportion, making the discharge slower and slower. This process, similar to emptying a water tank by opening a spigot at the bottom, is an exponential decay.

  23. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… In fact, it is the natural response predicted by Equation (7.33). We thus have: (Fig 8.4) on the next slide y (t) = y(0)e-t/Ƭ... (7.33),

  24. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… Fig 8.3

  25. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… The rate of decay is governed by the time constant RC. The larger R or C, theslower the decay. In the limit R  ∞the decay would become infinitely slow because there would be no current path for C to discharge. Hence, C would retain its initial voltage VS indefinitely, thus providing a memory function.

  26. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… Example 8.2 Fig 8.3 tƐ = - Ƭ lnƐ …(7.34)

  27. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… Example 8.3 y (t) = y(0)e-t/Ƭ... (7.33) or v = Vs e-t/RC v2 = V2S e-2t/RC

  28. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… The CR Circuit. Interchanging the roles of the capacitance and resistance in the R-C circuit of Figure 8.1(a) turns it into the C-R circuit of Figure 8.5 (a). Though the current response is not affected by this interchange, the voltage response will be different because we are now observing it across R rather than C.

  29. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… The CR Circuit… Assume prior to switch activation the circuit is in steady state so that C, by Rule 2, acts as an open circuit. Then, by Ohm's Law, v(0-) = Ri(0-) = R x 0 = 0, indicating that the voltage across C must also be zero.

  30. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… The CR Circuit… As the switch is flipped up, the voltage of the left plate jumps from 0 V to Vs. By Rule 1, the voltage across C right after switch activation must still be 0 V. For this to be possible, the voltage of the right plate must also jump from 0 V to VS,so that v(0+) = VS. With a nonzero voltage drop, R will draw current and cause C to charge exponentially.

  31. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… The CR Circuit… In summary, This response is shown in Figure 8.5(b). We observe that v now exhibits a discontinuity at t = 0. This does not violate Rule 1, however, because this rule applies to the voltage across the capacitance, not to the individual plate voltages.

  32. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… The CR Circuit… In summary, If the switch is left in the up position long enough to allow v to fully decay to 0 V, the capacitance will achieve its dc steady state, in which the voltage at the left plate will be VS and that at the right plate will be 0 V. Consequently, the final stored energy will be wc = (1/2) CVS2.

  33. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… The CR Circuit… Suppose now the switch is flipped back down, at a time that we again choose as t=0, as shown in Figure 8.6(a). This causes the voltage of the left plate to jump from VS to 0 V, and that of the right plate to jump from 0 V to -Vs, by the continuity rule. Consequently, v(0+) = -Vs, causing R to draw current and discharge C, Notethe current will flow from R to C through the closed switch to the lower node.

  34. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The RC Circuit… The CR Circuit… This response is shown in Figure 8.6(b). Voltage is again discontinuous at t = 0, but without violating Rule 1. We also note the creation of a negative voltage transient using a positive voltage source. This feature is exploited in the design of certain types of voltage inverters. Home Work: Do Exercise 8.2 (page 336 of text book)

  35. TRANSIENT RESPONSE OF FIRST ORDER CIRCUITS: (Chapter 8) 8.1 BASIC RC AND RL CIRCUITS… DC Steady-State Behavior…The The RL Circuit In the circuit of Figure 8.7 assume the inductance has zero initial stored energy so that, by Equation (7.19), i(0-) = 0. Flipping up the switch connects the source IS to the RL pair, causing current and, hence, energy to build up in the inductance. For t ≥ 0+, this process is governed by Equation (7.24) By the current continuity rule we must have i(0+)= i(0-)= 0. Consequently, the solution to Equation (8.9) is the forced response Figure 8.7 Flipping a switch to investigate the forced response of the RL Circuit

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