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2. Consider the LRC circuit below. Vin Vo sin wt i lo sin(wt + o) C www R L (a) Find an analytic expression for the equivalent impedance of the R, L, and C combination. Remember that Zc = 1/(jwC) and ZL = jwL and that you can combine impedances in series and parallel the same way that you would combine resistances. Give your answer in the form Z = X+jY (i.e. find X and Y). (b) The current in the resistor is given by i = Io sin(wt + o). Find Io and tano in terms of Vo, w, R, L, and C. (c) Plot VR/Vo as a function of frequency. VR represents the amplitude of the voltage across the resistor. Use component values L = 300 μH, C = 15 μF, and R = 1000 2 and plot over a frequency range of f= 2200-2500 Hz. Don't forget that w 2πf. Don't draw your graph by hand, use plotting software. (d) What is the impedance of the parallel LC part of the circuit when w=1/√LC? Compare this to the impedance of a series combination of L and C when w=1/√LC.

2. Consider the LRC circuit below. Vin Vo sin wt i lo sin(wt + o) C www R L (a) Find an analytic expression for the equivalent impedance of the R, L, and C combination. Remember that Zc = 1/(jwC) and ZL = jwL and that you can combine impedances in series and parallel the same way that you would combine resistances. Give your answer in the form Z = X+jY (i.e. find X and Y). (b) The current in the resistor is given by i = Io sin(wt + o). Find Io and tano in terms of Vo, w, R, L, and C. (c) Plot VR/Vo as a function of frequency. VR represents the amplitude of the voltage across the resistor. Use component values L = 300 μH, C = 15 μF, and R = 1000 2 and plot over a frequency range of f= 2200-2500 Hz. Don't forget that w 2πf. Don't draw your graph by hand, use plotting software. (d) What is the impedance of the parallel LC part of the circuit when w=1/√LC? Compare this to the impedance of a series combination of L and C when w=1/√LC.

Introductory Circuit Analysis (13th Edition)
Introductory Circuit Analysis (13th Edition)
13th Edition
ISBN:9780133923605
Author:Robert L. Boylestad
Publisher:Robert L. Boylestad
Chapter1: Introduction
Section: Chapter Questions
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In class we showed that any complex number Z can be written in either of the following two forms: Z = X+jY Z = |Z| ej To go between these two forms, you can use the following relationships: |Z| = √X² +Y² Y X X = |Z| cos o Y = |Z| sino We also introduced one more quantity called the complex conjugate of Z, which is denoted Z*. To find the complex conjugate of Z, you simply reverse the sign in front of all the j's. So, for example, if: Z = X+jY= |Z| e¹ then: tan o = Z* = X-jY= |Z| e-j The complex conjugate is useful because it can be used to calculate |Z|, the magnitude of Z. Consider, for example, the following product: ZZ* = (x+jY) (X -jY) = x² + y² = |Z|². Alternatively, using the exponential form of Z: ZZ = (|Z|e¹) (|Z| e-jº) = |Z|². In both cases we get that ZZ* = |Z|².
Transcribed Image Text:In class we showed that any complex number Z can be written in either of the following two forms: Z = X+jY Z = |Z| ej To go between these two forms, you can use the following relationships: |Z| = √X² +Y² Y X X = |Z| cos o Y = |Z| sino We also introduced one more quantity called the complex conjugate of Z, which is denoted Z*. To find the complex conjugate of Z, you simply reverse the sign in front of all the j's. So, for example, if: Z = X+jY= |Z| e¹ then: tan o = Z* = X-jY= |Z| e-j The complex conjugate is useful because it can be used to calculate |Z|, the magnitude of Z. Consider, for example, the following product: ZZ* = (x+jY) (X -jY) = x² + y² = |Z|². Alternatively, using the exponential form of Z: ZZ = (|Z|e¹) (|Z| e-jº) = |Z|². In both cases we get that ZZ* = |Z|².
2. Consider the LRC circuit below. Vin = Vo sin wt i = lo sin(wt + o) C R L (a) Find an analytic expression for the equivalent impedance of the R, L, and C combination. Remember that Zc = 1/(jwC) and Zz = jwL and that you can combine impedances in series and parallel the same way that you would combine resistances. Give your answer in the form Z = X+jY (i.e. find X and Y). (b) The current in the resistor is given by i = Io sin(wt + o). Find Io and tan o in terms of Vo, w, R, L, and C. (c) Plot VR/Vo as a function of frequency. VR represents the amplitude of the voltage across the resistor. Use component values L = 300 µH, C = 15 µF, and R = 1000 2 and plot over a frequency range of f = 2200-2500 Hz. Don't forget that w 2πf. Don't draw your graph by hand, use plotting software. = (d) What is the impedance of the parallel LC part of the circuit when w=1/√LC? Compare this to the impedance of a series combination of L and C when w=1/√LC.
Transcribed Image Text:2. Consider the LRC circuit below. Vin = Vo sin wt i = lo sin(wt + o) C R L (a) Find an analytic expression for the equivalent impedance of the R, L, and C combination. Remember that Zc = 1/(jwC) and Zz = jwL and that you can combine impedances in series and parallel the same way that you would combine resistances. Give your answer in the form Z = X+jY (i.e. find X and Y). (b) The current in the resistor is given by i = Io sin(wt + o). Find Io and tan o in terms of Vo, w, R, L, and C. (c) Plot VR/Vo as a function of frequency. VR represents the amplitude of the voltage across the resistor. Use component values L = 300 µH, C = 15 µF, and R = 1000 2 and plot over a frequency range of f = 2200-2500 Hz. Don't forget that w 2πf. Don't draw your graph by hand, use plotting software. = (d) What is the impedance of the parallel LC part of the circuit when w=1/√LC? Compare this to the impedance of a series combination of L and C when w=1/√LC.
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