Maximum and Asymptotic Behaviour of Parallel Impedance
About points...
We associate a certain number of points with each exercise.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
When you click an exercise into a collection, this number will be taken as points for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit the number of points for the exercise in the collection independently, without any effect on "points by default" as represented by the number here.
That being said... How many "default points" should you associate with an exercise upon creation?
As with difficulty, there is no straight forward and generally accepted way.
But as a guideline, we tend to give as many points by default as there are mathematical steps to do in the exercise.
Again, very vague... But the number should kind of represent the "work" required.
About difficulty...
We associate a certain difficulty with each exercise.
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
When you click an exercise into a collection, this number will be taken as difficulty for the exercise, kind of "by default".
But once the exercise is on the collection, you can edit its difficulty in the collection independently, without any effect on the "difficulty by default" here.
Why we use chess pieces? Well... we like chess, we like playing around with \(\LaTeX\)-fonts, we wanted symbols that need less space than six stars in a table-column... But in your layouts, you are of course free to indicate the difficulty of the exercise the way you want.
That being said... How "difficult" is an exercise? It depends on many factors, like what was being taught etc.
In physics exercises, we try to follow this pattern:
Level 1 - One formula (one you would find in a reference book) is enough to solve the exercise. Example exercise
Level 2 - Two formulas are needed, it's possible to compute an "in-between" solution, i.e. no algebraic equation needed. Example exercise
Level 3 - "Chain-computations" like on level 2, but 3+ calculations. Still, no equations, i.e. you are not forced to solve it in an algebraic manner. Example exercise
Level 4 - Exercise needs to be solved by algebraic equations, not possible to calculate numerical "in-between" results. Example exercise
Level 5 -
Level 6 -
Question
Solution
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Exercise:
Qualitatively graph the impedance for an ac circuit with a resistor resistance R a capacitor capacitance C and a coil inductance L in parallel. Discuss the maximum and the asymptotic behaviour.
Solution:
The impedance is given by Zomega fracsqrtfracR^+leftfracomega L-omega Cright^ The expression reaches the maximum value when the expression in parenthesis is equal to zero i.e. for an angular frequency omega_ with fracomega_ L omega_ C omega_ fracsqrtL C The impedance at omega_ is sscZmax Zomega_ fracsqrtfracR^ R For omegato the term with the inductance ist the dominant term: Zomega &propto fracsqrtfracleftomega Lright^ omega L quad textrmfor omegato For omegato infty the term with the capacitance ist the dominant term: Zomega &propto fracsqrtleftomega Cright^ fracomega C quad textrmfor omegato infty A typical graph for two different values of R is displayed in the figure below. center includegraphicswidthtextwidth#image_path:parallel-impedance# center
Qualitatively graph the impedance for an ac circuit with a resistor resistance R a capacitor capacitance C and a coil inductance L in parallel. Discuss the maximum and the asymptotic behaviour.
Solution:
The impedance is given by Zomega fracsqrtfracR^+leftfracomega L-omega Cright^ The expression reaches the maximum value when the expression in parenthesis is equal to zero i.e. for an angular frequency omega_ with fracomega_ L omega_ C omega_ fracsqrtL C The impedance at omega_ is sscZmax Zomega_ fracsqrtfracR^ R For omegato the term with the inductance ist the dominant term: Zomega &propto fracsqrtfracleftomega Lright^ omega L quad textrmfor omegato For omegato infty the term with the capacitance ist the dominant term: Zomega &propto fracsqrtleftomega Cright^ fracomega C quad textrmfor omegato infty A typical graph for two different values of R is displayed in the figure below. center includegraphicswidthtextwidth#image_path:parallel-impedance# center
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Exercise:
Qualitatively graph the impedance for an ac circuit with a resistor resistance R a capacitor capacitance C and a coil inductance L in parallel. Discuss the maximum and the asymptotic behaviour.
Solution:
The impedance is given by Zomega fracsqrtfracR^+leftfracomega L-omega Cright^ The expression reaches the maximum value when the expression in parenthesis is equal to zero i.e. for an angular frequency omega_ with fracomega_ L omega_ C omega_ fracsqrtL C The impedance at omega_ is sscZmax Zomega_ fracsqrtfracR^ R For omegato the term with the inductance ist the dominant term: Zomega &propto fracsqrtfracleftomega Lright^ omega L quad textrmfor omegato For omegato infty the term with the capacitance ist the dominant term: Zomega &propto fracsqrtleftomega Cright^ fracomega C quad textrmfor omegato infty A typical graph for two different values of R is displayed in the figure below. center includegraphicswidthtextwidth#image_path:parallel-impedance# center
Qualitatively graph the impedance for an ac circuit with a resistor resistance R a capacitor capacitance C and a coil inductance L in parallel. Discuss the maximum and the asymptotic behaviour.
Solution:
The impedance is given by Zomega fracsqrtfracR^+leftfracomega L-omega Cright^ The expression reaches the maximum value when the expression in parenthesis is equal to zero i.e. for an angular frequency omega_ with fracomega_ L omega_ C omega_ fracsqrtL C The impedance at omega_ is sscZmax Zomega_ fracsqrtfracR^ R For omegato the term with the inductance ist the dominant term: Zomega &propto fracsqrtfracleftomega Lright^ omega L quad textrmfor omegato For omegato infty the term with the capacitance ist the dominant term: Zomega &propto fracsqrtleftomega Cright^ fracomega C quad textrmfor omegato infty A typical graph for two different values of R is displayed in the figure below. center includegraphicswidthtextwidth#image_path:parallel-impedance# center
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