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Understanding Voltage Dividers and Complex Impedance in Electrical Circuits

This lecture covers the fundamentals of voltage dividers, impedance, and the application of complex numbers in electrical engineering. Key concepts include algebraic operations with complex numbers, graphical representation, and polar form conversion. The course also highlights how to set up differential equations for LCR circuits under sinusoidal voltage and solve these equations using complex exponentials. Additionally, practical measurements of current and voltage using ammeters and voltmeters are discussed, emphasizing ideal conditions for measurement accuracy.

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Understanding Voltage Dividers and Complex Impedance in Electrical Circuits

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  1. Lecture B Voltage dividers Impedance Complex numbers

  2. Lecture B Voltage dividers Impedance Complex numbers

  3. B1a 5 W I3 What is I3? 2 A 4 A 5 A 10 A 14.5 A 4 W 1 W + 10 V

  4. B1b 5 W I3 = 2 A What is IBatt? 2 A 4 A 5 A 10 A 14.5 A 4 W 1 W + IBatt = ? 10 V

  5. What is Vout/Vin? B2 C) A) B) D) E)

  6. Things you need to know about complex numbers for Phys3330 1.Perform algebraic operations on complex numbers and represent a given complex number graphically and express it in polar form. 2. Represent a sinusoidal function as the real and imaginary part of an exponential and use this representation for adding trigonometric functions. 3. Set up a linear differential equation to describe the behavior of LCR circuit that is subject to an applied sinusoidal voltage. 4. Use complex exponentials to solve homogenous and inhomogeneous linear differential equations with constant coefficients.

  7. 1. Complex number - The imaginary unit: and , where x & y are REAL numbers General complex number real part of z, x= Re z imaginary part of z, y= Im z Arithmetic with complex numbers z1,2: + − × 1 ÷

  8. Complex conjugate (z → z*) : Replace i with -i → → the modulus of z

  9. 2. Power Series for exponential and trigonometric functions: Euler’s Formula Now compare trigonometric and hyperbolic function in complex number:

  10. 3. Polar representation of a complex number z=x+iy y (imaginary) Representing z=x+iyby the point (x,y) z=x+iy Then, r ϕ where, x (real) y x So we can always write ✵Polar representation is advantageous for multiplication and division! Let then

  11. B3 L R What is Zeq? A) B) C) D) E)

  12. Aliasing Red = real signal Blue = scope display Black points = sampling times

  13. B4 Is it possible to measure the current generated by the power supply using the scope? No, it only measures voltage Yes, just turn the knob on the scope from “V” to “I” Yes, put a resistor in the circuit and measure the voltage across it.

  14. B5 To measure the current thru resistor 3, how should the ammeter be attached? e) MORE than one of these choices is ok.

  15. B6 • An ideal ammeter should have • Zero resistance • Infinite resistance • A well defined resistance > • 0 (e.g. 1 or 1k) • Shiny red color

  16. B7 To measure the voltage across resistor 3, how should the voltmeter be attached (assume you only attach one at a time)? e) MORE than one of these choices is ok.

  17. B8 • An ideal voltmeter should have • Zero resistance • Infinite resistance • A well defined resistance >0 (e.g. 1 or 1k) • Shiny red color

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