Mechatronics Engineering MT-144 NETWORK ANALYSIS Mechatronics Engineering (06)

ENERGY STORAGE ELEMENTS (Chapter 7) Capacitance Inductance Natural Response of RC and RL Circuits Response to DC and AC Forcing Functions

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to an AC Forcing Function As we shall see in greater detail in Chapter 10, an ac forcing function is a sinusoidal function of the type: X(t) = Xm cos ωt ...(7.47) where Xm is called the amplitude and ω the angular frequency of the ac signal, whose units are radians/ s (rad/s). This means that in the capacitive circuit of Figure 7.14 we have vs = Vm cosωt, and in the inductive circuit we have is = Im cosωt. Substituting Equation (7.47) into Equation (7.38) yields:

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to an AC Forcing Function or

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to an AC Forcing Function ...(7.36) … (7.37) … (7.38) Substituting from (7.50) into Equation (7.36), using also Equation (7.37), and then regrouping, the complete response is obtained: as Eqn. (7.51)

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to an AC Forcing Function The complete response is obtained: as Eqn. (7.51); see previous slide An example of a complete response is shown below in Figure 7.19. Back To Example 7.10

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to an AC Forcing Function The Transient and AC Steady-State Components Equation (7.51) reveals that the response to an ac forcing function consists of two components, namely, an exponentially decaying component, called the transient component, and an ac component, called the ac steady-state component because this is the value to which the complete response will settle once the transient component has died out. We emphasize this by writing

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to an AC Forcing Function The Transient and AC Steady-State Components… We emphasize this by writing: y(t)= yxsient + yss … (7.52) where, and,

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … Response to an AC Forcing Function The Transient and AC Steady-State Components… The quantities ym and φ are called, respectively, the amplitude and the phase angle of the steady-state component. Again, we observe that yxient has the same functional form as ynatural and that it consists of two terms: the natural response itself, y(0)e-t/Ƭ, and the term Xme-t/Ƭ / (1+ω2Ƭ2), brought about by the forcing function. We also note that yss has the same functional form as the forcing function: they are both sinusoidal, and they have the same frequency ω. Moreover, both the amplitude and phase angle of yss are frequency dependent parameters. It is common practice to refer to the ac steady-state component as simply the ac response. Like the transient response, the ac response is of fundamental importance because a number of important circuit features can be deduced by observing the ac response. The design specifications for circuit systems are often given in terms of ac-response parameters. The ac response may be studied in detail in Chapters 10 through 12.

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation …Response to an AC Forcing Function…The Transient and AC Steady-State Components… To Fig. 7.19

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … CONCLUDING REMARKS We conclude with the following observations, which hold regardless of whether the forcing function is of the dc or ac type: (1) The transient component is functionally similar to the natural response, that is, it is an exponentially decaying function. (2) The steady-state component is functionally similar to the forcing function, that is, it is of the dc type in the case of a dc forcing function, and of the ac type in the case of an ac forcing function. These observations, to which we shall return time and again, provide an important perspective for the material of the subsequent study. Keep them in mind. Home Work: Do Exercise 7.13 (page 320)

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … SUMMARY The capacitor and the inductor are energy-storage elements. The expression for capacitive energy is w= (1/2)Cv2, and that for inductive energy is w = (1/2)Li2 . The capacitance law is i= C dv/dt , and the inductance law is v = L di/dt. These laws are dual of each other. Since these laws are time-dependent, the capacitance and the inductance are also referred to as dynamic elements. . For a capacitance to carry current, its voltage must change. If the voltage is kept constant, the current is zero. We also say that the capacitance provides a memory function. For an inductance to develop voltage, its current must change. If the current is kept constant, the voltage is zero. . Home Work: Students are to study the contents of “Summary” --- this and next three slides; this is good information.

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … SUMMARY … The analysis of circuits with energy-storage elements is still based on the combined use of Kirchhoff s laws and the element laws. However, the circuit equations are now differential equations as opposed to the algebraic equations of purely resistive circuits. The manner in which a circuit reacts to the application of a forcing signal is called the response. . A circuit with energy-storage elements is capable of producing a response even in the absence of any forcing function. This response is called the source-free response or the natural response because it is produced by the circuit spontaneously, using the energy available from its energy-storage elements.

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … SUMMARY … The natural response of a first-order circuit is a decaying exponential. The rate of decay is governed by the time constant Ƭ. Quantitatively, Ƭ is the amount of time it takes for the natural response to decay to (1/e) or 37% of its initial value. A source-free circuit can be described in terms of its characteristic equation. The root s of this equation is related to the time constant Ƭ as s= - 1/ Ƭ , and it lies on the negative portion of the horizontal axis of the s plane. The further away the root is from the origin, the smaller the time constant, and the faster the rate of decay of the natural response. Conversely, the closer the root to the origin, the longer the rate of decay. A root at the origin corresponds to the memory function.

ENERGY STORAGE ELEMENTS 7.4 RESPONSE TO DC AND AC FORCING FUNCTIONS General Solution to the Differential Equation … SUMMARY … When a forcing function is present, the response consists of two components, the natural component and the forced component. The natural component is an exponential decay, but the forced component depends on the functional form of the forcing function. The two cases of greatest. practical interest are the response to a dc forcing function, called the transient response, and to an ac forcing function, called the ac response. In both cases the complete response consists of a transient component having the same functional form as the natural response, and a steady state component having the same functional form as the forcing function. The designation steady state refers to the condition achieved once the transient part has died out. The ac response has the same frequency as the ac forcing function, differing only in amplitude and phase angle. Moreover, amplitude and phase angle are ,frequency dependent.