EECE 311 Lecture Notes #3 CHAPTER 7: RESPONSE OF FIRST- ORDER RL AND RC CIRCUITS ASSIGNMENTS Read the textbook, Electric Circuits, 9th Ed., Chapter 7, pp.228-259. Do HW#4: Chapter 7: 4, 5, 7, 12, 14, 16, 23, 26, 30, 35, 36, 37, 51, 53, 55, 56, 85. OBJECTIVES Be able to determine the natural response of both RL and RC circuits. Be able to determine the step response of both RL and RC circuits. Know how to analyze circuits with sequential switching. Be able to analyze op amp circuits containing resistors and a single capacitor CONTENTS 7.1 The Natural Response of an RL Circuit 7.2 The Natural Response of an RC Circuit 7.3 The Step Response of RL and RC Circuits 7.4 A General Solution for Step and Natural Responses 7.5 Sequential Switching 7.6 Unbounded Response 7.7 The Integrating Amplifier TYPES OF FIRST ORDER CIRCUITS 7.1 THE NATURAL RESPONSE OF AN RL CIRCUIT The natural response of an RL circuit can best be described in terms of the circuit shown in Fig. 7.3. Finding the natural response requires finding the voltage and current at the terminals of the resistor after the switch has been opened, that is, after the source has been disconnected and the inductor begins releasing energy If we let t = 0 denote the instant when the switch is opened, the problem becomes one of finding v(t) and i(t) for t ≥0. For t ≥ 0, the circuit shown in Fig. 7.3 reduces to the one shown in Fig. 7.4 EXPRESSIONS FOR THE CURRENT FOR RC CIRCUIT Kirchhoff's voltage law to obtain an expression involving i, R, and L. Summing the voltages around the closed loop gives Where use the passive sign convention. The highest order derivative appearing in the equation is 1; hence the term first-order. Initial inductor current: The first instant after the switch has been opened, the current in the inductor remains unchanged. If we use 0- to denote the time just prior to switching, and 0+ for the time immediately following switching, then Natural response of an RL circuit: The initial current in the inductor is oriented in the same direction as the reference direction of i. fig 7.5 shows this response SIGNIFICANCE OF TIME CONSTANT FOR RC CIRCUIT The expressions for i(t) (Eq. 7.7) and v(t) (Eq. 7.8) include a term of the form e-(R/L)t The coefficient of t—namely, R/L— determines the rate at which the current or voltage approaches zero. The reciprocal of this ratio is the time constant of the circuit, denoted SIGNIFICANCE OF TIME CONSTANT FOR RC CIRCUIT Table 7.1 gives the value of e- t/ for integral multiples of from 1 to 10. The existence of current in the RL circuit shown in Fig. 7.1(a) is a momentary event and is referred to as the transient response of the circuit. The response that exists a long time after the switching has taken place is called the steady-state response. The phrase a long time then also means the time it takes the circuit to reach its steady-state value. TYPICAL STEPS INVOLVED IN CALCULATION FOR NATURAL RESPONSE OF AN RL CIRCUIT Calculating the natural response of an RL circuit can be summarized as follows: 1. Find the initial current, I0, through the inductor. 2. Find the time constant of the circuit, = L/R. 3. Use Eq. 7.15, Ioe- t/ , to generate i(t) from Io and . All other calculations of interest follow from knowing i(t). Text Examples 7.1 and 7.2 illustrate the numerical calculations associated with the natural response of an RL circuit. 7.2 THE NATURAL RESPONSE OF AN RC CIRCUIT We begin by assuming that the switch has been in position a for a long time, allowing the loop made up of the dc voltage source Vg, the resistor R1, and the capacitor C to reach a steady-state condition. Thus the voltage source cannot sustain a current, and so the source voltage appears across the capacitor terminals. When the switch is moved from position a to position b (at t = 0), the voltage on the capacitor is Vg. Because there can be no instan-taneous change in the voltage at the terminals of a capacitor, the problem reduces to solving the circuit shown in Fig. 7.11 EXPRESSION FOR VOLTAGE OF AN RC CIRCUIT Using the lower junction between R and C as the reference node and summing the currents away from the upper junction between R and C gives Initial capacitor voltage: The initial voltage on the capacitor equals the voltage source voltage Vg, or Time constant for RC circuit: The time constant for the RC circuit equals the product of the resistance and capacitance, namely, Natural Response of an RC circuit: Fig 7.12 shows the graphical representation of the response and the equation is given as TYPICAL STEPS INVOLVED IN CALCULATION FOR NATURAL RESPONSE OF AN RC CIRCUIT Calculating the natural response of an RC circuit can be summarized as follows: 1. Find the initial voltage, v0, through the capacitor. 2. Find the time constant of the circuit, = RC. 3. Use Eq. 7.25, v(t)= Voe- t/ , to generate v(t) from vo and . All other calculations of interest follow from knowing v(t). Text Examples 7.3 and 7.4 illustrate the numerical calculations associated with the natural response of an RL circuit. 7.3 THE STEP RESPONSE OF RL CIRCUIT After the switch in Fig. 7.16 has been closed, Kirchhoff's voltage law requires that After deriving equation for natural response of RL circuit from above equation we get, Fig 7.17 shows response for the RL circuit when Io = 0 Also Fig 7.18 shows Inductor voltage vs. time 7.3 THE STEP RESPONSE OF RC CIRCUIT We can find the step response of a first-order RC circuit by analyzing the circuit shown in Fig. 7.21. For mathematical convenience, we choose the Norton equivalent of the network connected to the equivalent capacitor. From the above equation we derive equation for step response of RC circuit. 7.4 A GENERAL SOLUTION FOR STEP RESPONSE AND NATURAL RESPONSES To generalize the solution of these four possible circuits, we let x(t) represent the unknown quantity, giving x(t) four possible values. It can represent the current or voltage at the terminals of an inductor or the current or voltage at the terminals of a capacitor. We know that the differential equation describing any one of the four circuits in Fig. 7.24 takes the form Where the value of K can be zero, the final value of x will be constant; that is, the final value must satisfy Eq. 7.54, and, when x reaches its final value, the derivative dx/dt must be zero. After deriving general solution for natural and step responses of RL and RC circuits. Time t0 = time of change or switching DETAILED STEPS INVOLVED TO COMPUTE STEP AND NATURAL RESPONSES When computing the step and natural responses of circuits, it may help to follow these steps:…. Identify the variable of interest for the circuit. For RC circuits, it is most convenient to choose the capacitive voltage; for RL circuits, it is best to choose the inductive current. Determine the initial value of the variable, which is its value at to. Note that if you choose capacitive voltage or inductive current as your variable of interest, it is not necessary to distinguish between t = t0- and t = t0.+ This is because they both are continuous variables abler. If you choose another variable, you need to remember that its initial value is defined at t = to+. Calculate the final value of the variable, which is its value as t —> ∞. Calculate the time constant for the circuit. Note: The expressions t0- and t0+ are analogous to 0- and 0+. Thus x(t0-) is the limit of x(t) as t —> t0 to from the left, and x(t0+) is the limit of x(t) as t —> to from the right. 7.5 SEQUENTIAL SWITCHING Whenever switching occurs more than once in a circuit, we have sequential switching. For example, a single, two-position switch may be switched back and forth, or multiple switches may be opened or closed in sequence. The time reference for all switching's cannot be t = 0. We determine the voltages and currents generated by a switching sequence by using the tech-niques described previously in this chapter. We derive the expressions for v(t) and i(t) for a given position of the switch or switches and then use these solutions to determine the initial conditions for the next position of the switch or switches. With sequential switching problems, a premium is placed on obtaining the initial value x(to). Recall that anything but inductive currents and capacitive voltages can change instantaneously at the time of switching. Thus solving first for inductive currents and capacitive voltages is even more pertinent in sequential switching problems. Drawing the circuit that pertains to each time interval in such a problem is often helpful in the solution process. Refer Text examples 7.11, 7.12 7.6 UNBOUNDED RESPONSE A circuit response may grow, rather than decay, exponentially with time. This type of response, called an unbounded response, is possible if the circuit contains dependent sources. In that case, the Thévenin equivalent resistance with respect to the terminals of either an inductor or a capacitor may be negative. This negative resistance generates a negative time con-stant, and the resulting currents and voltages increase without limit. In an actual circuit, the response eventually reaches a limiting value when a component breaks down or goes into a saturation state, prohibiting fur-ther increases in voltage or current. When we consider unbounded responses, the concept of a final value is confusing. Hence, rather than using the step response solution given in Eq. 7.59, we derive the differential equation that describes the circuit containing the negative resistance and then solve it using the separation of variables technique. Refer Text example 7.13 7.7 INTEGRATING AMPLIFIER We assume that the operational amplifier is ideal. Thus we take advantage of the constraints 7.7 INTEGRATING AMPLIFIER Vo(to) is the initial o/p voltage which is 0 , Further deriving we get, 7.7 INTEGRATING AMPLIFIER The output voltage is proportional to the integral of the input voltage only if the op amp operates within its linear range, that is, if it doesn't saturate. Refer Text Examples 7.14 and 7.15 further illustrate the analysis of the integrating amplifier SUMMARY A first-order circuit may be reduced to a Thévenin (or Norton) equivalent connected to either a single equiva-lent inductor or capacitor. The natural response is the currents and voltages that exist when stored energy is released to a circuit that contains no independent sources. The time constant of an RL circuit equals the equiva-lent inductance divided by the Thévenin resistance as viewed from the terminals of the equivalent inductor. The time constant of an RC circuit equals the equiva-lent capacitance times the Thévenin resistance as viewed from the terminals of the equivalent capacitor. The step response is the currents and voltages that result from abrupt changes in dc sources connected to a circuit. Stored energy may or may not be present at the time the abrupt changes take place. The solution for either the natural or step response of I both RL and RC circuits involves finding the initial and final value of the current or voltage of interest and the time constant of the circuit. Equations 7.59 and 7.60 summarize this approach. Sequential switching in first-order circuits is analyzed by dividing the analysis into time intervals correspon-ding to specific switch positions. Initial values for a particular interval are determined from the solution corresponding to the immediately preceding interval. An unbounded response occurs when the Thévenin resistance is negative, which is possible when the first-order circuit contains dependent sources. An integrating amplifier consists of an ideal op amp, a capacitor in the negative feedback branch, and a resis-tor in series with the signal source. It outputs the inte-gral of the signal source, within specified limits that avoid saturating the op amp.
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