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The 'R+LC' circuit shown in figure 1 is an RLC circuit with a resistor followed by a capacitor and inductor in a parallel configuration. Consider R, L, C arbitrary positive constants. R eeee Figure 1: 'R+LC' circuit (a) Find the differential equation that describes the transient current in the in- ductor. Show all your work (i.e. all the Kirchhoff rules used in the derivation, any relationship among the variables used to get the final result, etc.) (b) What combination of R, L, C will generate i) overdamped, ii) critically damped and iii) underdamped behavior? Let's now consider the case when the transient behavior generates (non-zero) underdamped oscillations. (c) Find the time dependent expression of the current in the inductor. The initial conditions are: IL(t = 0) = 0 and V₁(t = 0) = Vo. Recommendation: check units by using the units of RC, R/L, LC given in class. (d) An LC circuit (i.e. with R= 0) shows SHO at an angular frequency, called the resonant frequency, Res=1/√LC. Find the ratio R+LC/res, where NR+LC is the angular frequency of the underdamped oscillations of the 'R+LC' circuit. Hint: the result must be dimensionless. Use the units of RC, R/L provided in class to check it out. (e) Suppose we are given some values for R, L, C. If we're able to change L, keeping R, C constant, should we increase or decrease L in order to get the underdamped oscillations closer to the LC resonant oscillations? What if we could change the value of C, keeping R, L constant? And what if we could change R, keeping L, C constant?

The 'R+LC' circuit shown in figure 1 is an RLC circuit with a resistor followed by a capacitor and inductor in a parallel configuration. Consider R, L, C arbitrary positive constants. R eeee Figure 1: 'R+LC' circuit (a) Find the differential equation that describes the transient current in the in- ductor. Show all your work (i.e. all the Kirchhoff rules used in the derivation, any relationship among the variables used to get the final result, etc.) (b) What combination of R, L, C will generate i) overdamped, ii) critically damped and iii) underdamped behavior? Let's now consider the case when the transient behavior generates (non-zero) underdamped oscillations. (c) Find the time dependent expression of the current in the inductor. The initial conditions are: IL(t = 0) = 0 and V₁(t = 0) = Vo. Recommendation: check units by using the units of RC, R/L, LC given in class. (d) An LC circuit (i.e. with R= 0) shows SHO at an angular frequency, called the resonant frequency, Res=1/√LC. Find the ratio R+LC/res, where NR+LC is the angular frequency of the underdamped oscillations of the 'R+LC' circuit. Hint: the result must be dimensionless. Use the units of RC, R/L provided in class to check it out. (e) Suppose we are given some values for R, L, C. If we're able to change L, keeping R, C constant, should we increase or decrease L in order to get the underdamped oscillations closer to the LC resonant oscillations? What if we could change the value of C, keeping R, L constant? And what if we could change R, keeping L, C constant?

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
Problem 1P: Visit your local library (at school or home) and describe the extent to which it provides literature...
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Can someone help me with parts D and E please? :)

The 'R+LC' circuit shown in figure 1 is an RLC circuit with a resistor followed by a capacitor and inductor in a parallel configuration. Consider R, L, C arbitrary positive constants. R -0000 L Figure 1: 'R+LC' circuit (a) Find the differential equation that describes the transient current the in- ductor. Show all your work (i.e. all the Kirchhoff rules used in the derivation, any relationship among the variables used to get the final result, etc.) (b) What combination of R, L, C will generate i) overdamped, ii) critically damped and iii) underdamped behavior? Let's now consider the case when the transient behavior generates (non-zero) underdamped oscillations. (c) Find the time dependent expression of the current in the inductor. The initial conditions are: IL(t = 0) = 0 and V₁(t = 0) = Vo. Recommendation: check units by using the units of RC, R/L, LC given in class. (d) An LC circuit (i.e. with R = 0) shows SHO at an angular frequency, called the resonant frequency, Res=1/√LC. Find the ratio R+LC/res, where NR+LC is the angular frequency of the underdamped oscillations of the 'R+LC' circuit. Hint: the result must be dimensionless. Use the units of RC, R/L provided in class to check it out. (e) Suppose we are given some values for R, L, C. If we're able to change L, keeping R, C constant, should we increase or decrease L in order to get the underdamped oscillations closer to the LC resonant oscillations? What if we could change the value of C, keeping R, L constant? And what if we could change R, keeping L, C constant?
Transcribed Image Text:The 'R+LC' circuit shown in figure 1 is an RLC circuit with a resistor followed by a capacitor and inductor in a parallel configuration. Consider R, L, C arbitrary positive constants. R -0000 L Figure 1: 'R+LC' circuit (a) Find the differential equation that describes the transient current the in- ductor. Show all your work (i.e. all the Kirchhoff rules used in the derivation, any relationship among the variables used to get the final result, etc.) (b) What combination of R, L, C will generate i) overdamped, ii) critically damped and iii) underdamped behavior? Let's now consider the case when the transient behavior generates (non-zero) underdamped oscillations. (c) Find the time dependent expression of the current in the inductor. The initial conditions are: IL(t = 0) = 0 and V₁(t = 0) = Vo. Recommendation: check units by using the units of RC, R/L, LC given in class. (d) An LC circuit (i.e. with R = 0) shows SHO at an angular frequency, called the resonant frequency, Res=1/√LC. Find the ratio R+LC/res, where NR+LC is the angular frequency of the underdamped oscillations of the 'R+LC' circuit. Hint: the result must be dimensionless. Use the units of RC, R/L provided in class to check it out. (e) Suppose we are given some values for R, L, C. If we're able to change L, keeping R, C constant, should we increase or decrease L in order to get the underdamped oscillations closer to the LC resonant oscillations? What if we could change the value of C, keeping R, L constant? And what if we could change R, keeping L, C constant?
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