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Q&A Series RLC circuit

It all depends on the values of the components:If the system will very slowly decay until the energy of the system reaches 0. If the system undergoes something which will look like a part of...

posted 3y ago by MissMulan‭  ·  edited 3y ago by MissMulan‭

Answer
#3: Post edited by user avatar MissMulan‭ · 2021-09-08T16:07:51Z (about 3 years ago)
  • It all depends on the values of the components:If
  • ![](https://electrical.codidact.com/uploads/8ym42jU8HUER4jr6uEFSL2QW)
  • the system will very slowly decay until the energy of the system reaches 0.
  • If
  • ![](https://electrical.codidact.com/uploads/mL9YkvUSVf8BH1MW96VuTu2V)
  • the system undergoes something which will look like a part of a oscillation and loses its energy very quickly
  • If
  • ![](https://electrical.codidact.com/uploads/dBSd36PuPH6yJ84shk7qdTZW)
  • it oscillates with decreasing amplitude until its energy reaches 0.
  • In our case:
  • ![](https://electrical.codidact.com/uploads/xqBmzivFipgBcRrLYvTGdEMr)
  • ![](https://electrical.codidact.com/uploads/fwzhAAuRZbiuXVPvAq3HP5Ae)
  • it will very slowly decay without any oscillation.In order to find the equation of current of this RLC circuit we must be introduced to 2 things:
  • Neper angular frequency -> a feature of damped systems
  • In the case of the series RLC circuit it is:
  • ![](https://electrical.codidact.com/uploads/D2iMorYukryhtQqg3bXhY92V)
  • The natural angular frequency -> the frequency response of the system under no damping
  • In a RLC circuit it is equal to:
  • ![Image alt text](https://electrical.codidact.com/uploads/RhZNfiqZx9ZGmHSnftjnLHU3)
  • The equation of current of this overdamped system takes this form:
  • ![Image alt text](https://electrical.codidact.com/uploads/u2cWGVD5oMCCR6gkzbvwqrc6)
  • where :
  • ![](https://electrical.codidact.com/uploads/ZRDLjwdRtkrFGKfr9bTVtt6G)
  • (1)
  • and:
  • ![](https://electrical.codidact.com/uploads/imNfVLKa1zjacV1MTP8iWjUh)
  • Now lets go calculate s1 and s2:
  • ![](https://electrical.codidact.com/uploads/qBbadovLpxGormHJeBW61wWb)
  • After the switch is closed:
  • ![Image alt text](https://electrical.codidact.com/uploads/wNSsrYNSB1hu6UgorJnL7Z8i)
  • and due to L1:
  • ![](https://electrical.codidact.com/uploads/oCDeSgowdkLP5a5xyeveXy3s)
  • so the voltage between L1 is:
  • ![](https://electrical.codidact.com/uploads/1YMPV5DcPvqiHdw8FoB8mDQU)
  • By substituting the values of s1,s2 VL1(0+),L1 and IL1(0+) to (1) and solving the system we get:
  • ![](https://electrical.codidact.com/uploads/rDVwMFUgTWfrNYEAW3hoabfv)
  • This is the equation of the current of the loop consisting of C1,L1,R2 after the switch is closed.
  • It all depends on the values of the components:If
  • ![](https://electrical.codidact.com/uploads/8ym42jU8HUER4jr6uEFSL2QW)
  • the system will very slowly decay until the energy of the system reaches 0.
  • If
  • ![](https://electrical.codidact.com/uploads/mL9YkvUSVf8BH1MW96VuTu2V)
  • the system undergoes something which will look like a part of a oscillation and loses its energy very quickly
  • If
  • ![](https://electrical.codidact.com/uploads/dBSd36PuPH6yJ84shk7qdTZW)
  • it oscillates with decreasing amplitude until its energy reaches 0.
  • In our case:
  • ![](https://electrical.codidact.com/uploads/xqBmzivFipgBcRrLYvTGdEMr)
  • ![](https://electrical.codidact.com/uploads/fwzhAAuRZbiuXVPvAq3HP5Ae)
  • it will very slowly decay without any oscillation.In order to find the equation of current of this RLC circuit we must be introduced to 2 things:
  • Neper angular frequency -> a feature of damped systems
  • In the case of the series RLC circuit it is:
  • ![](https://electrical.codidact.com/uploads/D2iMorYukryhtQqg3bXhY92V)
  • The natural angular frequency -> the frequency response of the system under no damping
  • In a RLC circuit it is equal to:
  • ![Image alt text](https://electrical.codidact.com/uploads/RhZNfiqZx9ZGmHSnftjnLHU3)
  • The equation of current of this overdamped system takes this form:
  • ![Image alt text](https://electrical.codidact.com/uploads/u2cWGVD5oMCCR6gkzbvwqrc6)
  • where :
  • ![](https://electrical.codidact.com/uploads/ZRDLjwdRtkrFGKfr9bTVtt6G)
  • (1)
  • and:
  • ![](https://electrical.codidact.com/uploads/imNfVLKa1zjacV1MTP8iWjUh)
  • Now lets go calculate s1 and s2:
  • ![](https://electrical.codidact.com/uploads/qBbadovLpxGormHJeBW61wWb)
  • After the switch is closed:
  • ![Image alt text](https://electrical.codidact.com/uploads/wNSsrYNSB1hu6UgorJnL7Z8i)
  • and due to L1:
  • ![](https://electrical.codidact.com/uploads/oCDeSgowdkLP5a5xyeveXy3s)
  • so the voltage between L1 is:
  • ![](https://electrical.codidact.com/uploads/1YMPV5DcPvqiHdw8FoB8mDQU)
  • By substituting the values of s1,s2 VL1(0+),L1 and IL1(0+) to (1) and solving the system we get:
  • ![](https://electrical.codidact.com/uploads/rDVwMFUgTWfrNYEAW3hoabfv)
  • This is the equation of the current of the loop consisting of C1,L1,R1 after the switch is closed.
#2: Post edited by user avatar MissMulan‭ · 2021-09-06T19:16:59Z (about 3 years ago)
  • It all depends on the values of the components:If
  • ![](https://electrical.codidact.com/uploads/8ym42jU8HUER4jr6uEFSL2QW)
  • the system will very slowly decay until the energy of the system reaches 0.
  • If
  • ![](https://electrical.codidact.com/uploads/mL9YkvUSVf8BH1MW96VuTu2V)
  • the system undergoes something which will look like a part of a oscillation and loses its energy very quickly
  • If
  • ![](https://electrical.codidact.com/uploads/dBSd36PuPH6yJ84shk7qdTZW)
  • it oscillates with decreasing amplitude until its energy reaches 0.
  • In our case:
  • ![](https://electrical.codidact.com/uploads/6dxozgLnSfUfFf7antzszExj)
  • since:
  • ![](https://electrical.codidact.com/uploads/fwzhAAuRZbiuXVPvAq3HP5Ae)
  • it will very slowly decay without any oscillation.In order to find the equation of current of this RLC circuit we must be introduced to 2 things:
  • Neper angular frequency -> a feature of damped systems
  • In the case of the series RLC circuit it is:
  • ![](https://electrical.codidact.com/uploads/pnofzBNJ1DicX31dBBRPRvvh)
  • The natural angular frequency -> the frequency response of the system under no damping
  • In a RLC circuit it is equal to:
  • ![Image alt text](https://electrical.codidact.com/uploads/RhZNfiqZx9ZGmHSnftjnLHU3)
  • The equation of current of this overdamped system takes this form:
  • ![Image alt text](https://electrical.codidact.com/uploads/u2cWGVD5oMCCR6gkzbvwqrc6)
  • where :
  • ![](https://electrical.codidact.com/uploads/ZRDLjwdRtkrFGKfr9bTVtt6G)
  • (1)
  • and:
  • ![](https://electrical.codidact.com/uploads/imNfVLKa1zjacV1MTP8iWjUh)
  • Now lets go calculate s1 and s2:
  • ![](https://electrical.codidact.com/uploads/qBbadovLpxGormHJeBW61wWb)
  • After the switch is closed:
  • ![Image alt text](https://electrical.codidact.com/uploads/wNSsrYNSB1hu6UgorJnL7Z8i)
  • and due to L1:
  • ![](https://electrical.codidact.com/uploads/oCDeSgowdkLP5a5xyeveXy3s)
  • so the voltage between L1 is:
  • ![](https://electrical.codidact.com/uploads/1YMPV5DcPvqiHdw8FoB8mDQU)
  • By substituting the values of s1,s2 VL1(0+),L1 to (1) and solving the system we get:
  • ![](https://electrical.codidact.com/uploads/rDVwMFUgTWfrNYEAW3hoabfv)
  • This is the equation of the current of the loop consisting of C1,L1,R2 after the switch is closed.
  • It all depends on the values of the components:If
  • ![](https://electrical.codidact.com/uploads/8ym42jU8HUER4jr6uEFSL2QW)
  • the system will very slowly decay until the energy of the system reaches 0.
  • If
  • ![](https://electrical.codidact.com/uploads/mL9YkvUSVf8BH1MW96VuTu2V)
  • the system undergoes something which will look like a part of a oscillation and loses its energy very quickly
  • If
  • ![](https://electrical.codidact.com/uploads/dBSd36PuPH6yJ84shk7qdTZW)
  • it oscillates with decreasing amplitude until its energy reaches 0.
  • In our case:
  • ![](https://electrical.codidact.com/uploads/xqBmzivFipgBcRrLYvTGdEMr)
  • ![](https://electrical.codidact.com/uploads/fwzhAAuRZbiuXVPvAq3HP5Ae)
  • it will very slowly decay without any oscillation.In order to find the equation of current of this RLC circuit we must be introduced to 2 things:
  • Neper angular frequency -> a feature of damped systems
  • In the case of the series RLC circuit it is:
  • ![](https://electrical.codidact.com/uploads/D2iMorYukryhtQqg3bXhY92V)
  • The natural angular frequency -> the frequency response of the system under no damping
  • In a RLC circuit it is equal to:
  • ![Image alt text](https://electrical.codidact.com/uploads/RhZNfiqZx9ZGmHSnftjnLHU3)
  • The equation of current of this overdamped system takes this form:
  • ![Image alt text](https://electrical.codidact.com/uploads/u2cWGVD5oMCCR6gkzbvwqrc6)
  • where :
  • ![](https://electrical.codidact.com/uploads/ZRDLjwdRtkrFGKfr9bTVtt6G)
  • (1)
  • and:
  • ![](https://electrical.codidact.com/uploads/imNfVLKa1zjacV1MTP8iWjUh)
  • Now lets go calculate s1 and s2:
  • ![](https://electrical.codidact.com/uploads/qBbadovLpxGormHJeBW61wWb)
  • After the switch is closed:
  • ![Image alt text](https://electrical.codidact.com/uploads/wNSsrYNSB1hu6UgorJnL7Z8i)
  • and due to L1:
  • ![](https://electrical.codidact.com/uploads/oCDeSgowdkLP5a5xyeveXy3s)
  • so the voltage between L1 is:
  • ![](https://electrical.codidact.com/uploads/1YMPV5DcPvqiHdw8FoB8mDQU)
  • By substituting the values of s1,s2 VL1(0+),L1 and IL1(0+) to (1) and solving the system we get:
  • ![](https://electrical.codidact.com/uploads/rDVwMFUgTWfrNYEAW3hoabfv)
  • This is the equation of the current of the loop consisting of C1,L1,R2 after the switch is closed.
#1: Initial revision by user avatar MissMulan‭ · 2021-09-06T17:03:40Z (about 3 years ago)
It all depends on the values of the components:If


![](https://electrical.codidact.com/uploads/8ym42jU8HUER4jr6uEFSL2QW)

the system will very slowly decay until the energy of the system reaches 0.

If 

![](https://electrical.codidact.com/uploads/mL9YkvUSVf8BH1MW96VuTu2V)

the system undergoes something which will look like a part of a oscillation and loses its energy very quickly

If

![](https://electrical.codidact.com/uploads/dBSd36PuPH6yJ84shk7qdTZW)

it oscillates with decreasing amplitude until its energy reaches 0.

In our case:

![](https://electrical.codidact.com/uploads/6dxozgLnSfUfFf7antzszExj)
since:

![](https://electrical.codidact.com/uploads/fwzhAAuRZbiuXVPvAq3HP5Ae)

it will very slowly decay without any oscillation.In order to find the equation of current of this RLC circuit we must be introduced to 2 things:

Neper angular frequency ->  a feature of damped systems


In the case of the series RLC circuit it is:

![](https://electrical.codidact.com/uploads/pnofzBNJ1DicX31dBBRPRvvh)

The natural angular frequency -> the frequency response of the system under no damping 

In a RLC circuit it is equal to:

![Image alt text](https://electrical.codidact.com/uploads/RhZNfiqZx9ZGmHSnftjnLHU3)

The equation of current of this overdamped system takes this form:

![Image alt text](https://electrical.codidact.com/uploads/u2cWGVD5oMCCR6gkzbvwqrc6)

where :

![](https://electrical.codidact.com/uploads/ZRDLjwdRtkrFGKfr9bTVtt6G)
(1)

and:

![](https://electrical.codidact.com/uploads/imNfVLKa1zjacV1MTP8iWjUh)

Now lets go calculate s1 and s2:

![](https://electrical.codidact.com/uploads/qBbadovLpxGormHJeBW61wWb)

After the switch is closed:

![Image alt text](https://electrical.codidact.com/uploads/wNSsrYNSB1hu6UgorJnL7Z8i)

and due to L1:

![](https://electrical.codidact.com/uploads/oCDeSgowdkLP5a5xyeveXy3s)

so the voltage between L1 is:

![](https://electrical.codidact.com/uploads/1YMPV5DcPvqiHdw8FoB8mDQU)

By substituting the values of s1,s2 VL1(0+),L1 to (1) and solving the system we get:

![](https://electrical.codidact.com/uploads/rDVwMFUgTWfrNYEAW3hoabfv)

This is the equation of the current of the loop consisting of C1,L1,R2 after the switch is closed.