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Lecture #17 EGR 260 – Circuit Analysis

Lecture #17 EGR 260 – Circuit Analysis. Reading Assignment: Chapter 7 in Electric Circuits, 9 th Ed. by Nilsson . Example: Capacitor charging circuit. Find v(t) for t > 0. Example: Capacitor discharging circuit. Find v(t) for t > 0.

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Lecture #17 EGR 260 – Circuit Analysis

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  1. Lecture #17 EGR 260 – Circuit Analysis Reading Assignment:Chapter 7 in Electric Circuits, 9th Ed. by Nilsson Example: Capacitor charging circuit. Find v(t) for t > 0. Example: Capacitor discharging circuit. Find v(t) for t > 0.

  2. Lecture #17 EGR 260 – Circuit Analysis 1st-order Circuits with Dependent Sources Dependent sources affect the resistance seen by the inductor or capacitor and therefore affect the value of  for the circuit. Two approaches can be used to find : 1) When REQ seen by an inductor or a capacitor, remove the L or C, kill any independent sources, and place any value independent source at the terminals. Then 2) Write a DE for the circuit (any variable) and  can be easily determined from the DE since it has the form:

  3. Lecture #17 EGR 260 – Circuit Analysis Example: Find v(t) for t > 0 if v(0) = 2 V. A) Use method 1: Find t = REQC using

  4. Lecture #17 EGR 260 – Circuit Analysis Example: (continued) Find v(t) for t > 0 if v(0) = 2 V. B) Use method 2: Write a DE for v(t).

  5. Lecture #17 EGR 260 – Circuit Analysis • Unit Step Functions • Unit step functions have several important uses in electrical engineering, including: • representing piecewise-continuous signals • representing switches • defining functions for use with one-sided Laplace transforms (in EGR 261) Definition: u(t) = unit step function where and u(t) is represented by the graph shown below.

  6. Lecture #17 EGR 260 – Circuit Analysis • A good way to think of a unit step function is as follows: • u(argument) = 1 for argument > 0 • u(argument) = 0 for argument < 0 • the transition in u(argument) occurs where argument = 0 Example: Graph 4u(t - 2) u(argument) = 0 for argument < 0. Note that for t < 2, u(t - 2) has a negative argument. For example, when t = 1, u(t - 2) = u(1 - 2) = u(-1) = 0. u(argument) = 1 for argument > 0. Note that for t > 2, u(t - 2) has a positive argument. For example, when t = 3, u(t - 2) = u(3 - 2) = u(+1) = 1. The transition occurs when the argument = 0. Note that when t = 2, u(t - 2) = u(2 - 2) = u(0).

  7. Lecture #17 EGR 260 – Circuit Analysis • Example: Graph the following functions: • 1) -2u(t - 10) • 2) 3u(t + 2) • 4u(-t) • 4) 4u(2-t)

  8. Lecture #17 EGR 260 – Circuit Analysis Example: Graph the following functions (continued): 5) 6u(-4-t) 6) u(-t) 7) Show that 1 - u(t) = u(-t) 8) sin(t) and sin(t)u(t)

  9. Lecture #17 EGR 260 – Circuit Analysis Example: Graph the following functions (continued): 9) u(t) - u(t - 2) 10) f(t) = 2t 11) 2t[u(t) - u(t - 2)] - discuss the concept of a “window”

  10. Lecture #17 EGR 260 – Circuit Analysis Example: Graph the following functions (continued): 12) f(t)[u(t - 2) - u(t - 4)] for any f(t) 13) (2t + 6)[u(t + 2) - u(t - 2)]

  11. Lecture #17 EGR 260 – Circuit Analysis • There are two common types of problems in representing functions using unit step functions: • 1) Determining the function that represents a given graph • Approach: Represent each unique portion of the function using unit step “windows” • 2) Graphing a function specified by unit steps • Approach: As each unit step function “turns on”, graph the cumulative function. • Examples of determining the function that represents a given graph: • Approach: Represent each unique portion of the function using unit step “windows”

  12. Lecture #17 EGR 260 – Circuit Analysis • Examples of graphing a function specified using unit steps functions: • Approach: As each unit step function “turns on”, graph the cumulative function. • 1) Graph f(t) = 2u(t) + 4u(t – 2) – 8u(t – 4)

  13. Lecture #17 EGR 260 – Circuit Analysis 2) Graph f(t) = (t + 2)u(t + 1) + (3 – t)u(t – 2) - (2t-5)u(t – 4) 3) Graph f(t) = 4sin(4t)[u(t) – u(t – 1)]

  14. Lecture #17 EGR 260 – Circuit Analysis Using unit step functions to replace switches in circuits Unit step functions are commonly used to represent switches in circuits. Consider the following examples. • Example: A unit step function is used below to replace a switch connecting a voltage source. • Discuss the value of VX. • Discuss the forms of the solution.

  15. Lecture #17 EGR 260 – Circuit Analysis • Example: A unit step function can be used to replace a switch disconnecting a current source. • Discuss the concept of a “make before break” switch • Discuss the value of IX. • Draw two possible circuits using unit step functions that are equivalent to the • circuit shown below.

  16. Lecture #17 EGR 260 – Circuit Analysis Example: Determine an expression for v(t) in the circuit below. Use the “shortcut method.”

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