Electronic components R,L,C can be connected in parallel to form RL, RC, LC, RLC series circuit
- RC Parallel
- RL Parallel
- LC Parallel
- RLC Parallel
![](//upload.wikimedia.org/wikipedia/commons/thumb/9/90/RC_switch.svg/200px-RC_switch.svg.png)
The total Impedance of the circuit
![{\displaystyle Z=Z_{R}+Z_{C}=R+{\frac {1}{j\omega C}}={\frac {1+j\omega RC}{j\omega C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/000ce729085225cfe4e44c9246382dd33c3d8ed8)
)
- T = RC
At Equilibrium sum of all voltages equal zero
![{\displaystyle C{\frac {dV}{dt}}+{\frac {V}{R}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/96a94c702185bbaee3ee9d2da9b6a21b7c1cb29b)
![{\displaystyle {\frac {dV}{dt}}=-{\frac {1}{RC}}V}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a755ccfbbdab10d3b0a1fb3df251e91d4b1bbf00)
![{\displaystyle {\frac {1}{V}}dV=-{\frac {1}{RC}}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9b26936d7200e6fa59c58e223084f706f4d3952a)
![{\displaystyle \int {\frac {1}{V}}dV=-\int {\frac {1}{RC}}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/01619c70798213ecc28308a35f9145a706956798)
- ln V =
![{\displaystyle -{\frac {1}{RC}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8115d5a07714c73707b5a2f1d6749ecad1a7bf6e)
![{\displaystyle V=e^{-}({\frac {1}{RC}})t+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5a1d04d0d1e6f73bea933301fddf7388c5f7abb9)
![{\displaystyle V=Ae^{-}({\frac {1}{T}})t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a33007bb390c0f3c7b3f3f3d82df4d59f6b49d0)
![{\displaystyle A=e^{C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/faf7b0cb669e664691530e2f84e867d7e1851ce4)
- T = RC
Circuit's Impedance in Polar coordinate
![{\displaystyle Z=Z_{R}+Z_{C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cc21deea946526b602a102dc02e2a1202095fd0)
![{\displaystyle Z=R\angle 0+{\frac {1}{\omega C}}\angle -90}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2b1add29c9683be215bcb9ad6fcc52e82a394182)
![{\displaystyle {\sqrt {R^{2}+({\frac {1}{\omega C}})^{2}}}\angle Tan^{-}1{\frac {1}{\omega RC}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6d71ab7a86d03c4b3fa577a6ff4a93f5f94b248a)
Phase Angle Difference Between Voltage and Current
There is a difference in angle Between Voltage and Current . Current leads Voltage by an angle θ
![{\displaystyle Tan\theta ={\frac {1}{\omega RC}}={\frac {1}{2\pi fRC}}={\frac {1}{2\pi }}{\frac {t}{T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2716471f8c011ee9132c4ad99e06323685bbe7)
RL series circuit has a first order differential equation of voltage
![{\displaystyle {\frac {d}{dt}}f(t)+{\frac {t}{T}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fef221b15787c20eb1e50cef3baab4f2132c2aca)
Which has one real root
![{\displaystyle V(t)=Ae^{\frac {-t}{T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e478842f7a6d07f5c6f21d3652cb4f268cb4bfe7)
![{\displaystyle A=e^{c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e64c15dbde84a803cd6955d0e28aacccb97cefd6)
The Natural Response of the circuit at equilibrium is a Exponential Decrease function
Phase Angle Difference Between Voltage and Current
![{\displaystyle Tan\theta ={\frac {1}{\omega RC}}={\frac {1}{2\pi fRC}}={\frac {1}{2\pi }}{\frac {t}{T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ee2716471f8c011ee9132c4ad99e06323685bbe7)
The total Circuit's Impedance In Rectangular Coordinate
![{\displaystyle Z=Z_{R}+Z_{L}=R+j\omega L}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3feb7598c4ed1c78469f834da3859d6ca5ed7244)
![{\displaystyle Z={\frac {1}{R}}(1+j\omega T)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/822c34bb73bf59cb2d81cb13aab5727b32a27e76)
![{\displaystyle T={\frac {L}{R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8f2d42d4886f4d6812d2b29c8376f1071de1e085)
At Equilibrium sum of all voltages equal zero
![{\displaystyle L{\frac {dI}{dt}}+IR=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2588f7e7b12ed734ef23fde1c40a2e0ef23c60c7)
![{\displaystyle {\frac {dI}{dt}}=-I{\frac {R}{L}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9711a9fb278cdadb5ba7ccb83f5e3f024061b576)
![{\displaystyle \int {\frac {1}{I}}dI=-\int {\frac {L}{R}}dt}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c86f726fc80894a70f0416560cc74e5f904efe6b)
- ln I =
![{\displaystyle (-{\frac {L}{R}}+c)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/359cb8b4f430cc6c9b83b8c4d4b14f3b63707c3b)
- I =
![{\displaystyle e^{(}-{\frac {L}{R}}+c)t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b8c1de91a822e78cb26c123017622e41c90963b9)
- I =
![{\displaystyle e^{c}e^{(}-{\frac {L}{R}}t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/51544320380aafab2b52918d543b7ca1b86e4604)
- I =
![{\displaystyle Ae^{(}-{\frac {L}{R}}t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0375d6bd95ccfd40a04cc320dbba9aebd21dae1e)
Circuit's Impedance In Polar Coordinate
![{\displaystyle Z=Z_{R}+Z_{L}=R\angle 0+\omega L\angle 90}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9a8f4c7955f352ce6afaf49ea424a308399f10da)
![{\displaystyle {\sqrt {R^{2}+(\omega L)^{2}}}\angle Tan^{-}1\omega {\frac {L}{R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/aa5659350216245f178fd596ff270c5f68b304a5)
Phase Angle of Difference Between Voltage and Current
![{\displaystyle Tan\theta =\omega {\frac {L}{R}}=2\pi f{\frac {L}{R}}=2\pi {\frac {T}{t}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f67a4ae80edaef762d568e44b84cbbe1d7388998)
In summary RL series circuit has a first order differential equation of current
![{\displaystyle {\frac {d}{dt}}f(t)+{\frac {1}{T}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/333cb14e9846b9d413e0278a616ebb3bff6557d6)
Which has one real root
![{\displaystyle I(t)=Ae^{\frac {t}{T}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f98d0641ce14bfa30abccef6c413cdf8fa5f18b7)
![{\displaystyle A=e^{c}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e64c15dbde84a803cd6955d0e28aacccb97cefd6)
The Natural Response of the circuit at equilibrium is a Exponential Decrease function
Phase Angle of Difference Between Voltage and Current
![{\displaystyle Tan\theta =\omega {\frac {L}{R}}=2\pi f{\frac {L}{R}}=2\pi {\frac {T}{t}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f67a4ae80edaef762d568e44b84cbbe1d7388998)
The Total Circuit's Impedance in Rectangular Form
. ZL = ZC
. ZL = ZC
Circuit's Natural Response at equilibrium
![{\displaystyle L{\frac {dI}{dt}}+{\frac {1}{C}}\int Idt=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e6154437f80ef11d397a657bf703da6722b4fce)
![{\displaystyle {\frac {d^{2}I}{dt^{2}}}+{\frac {1}{LC}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7aa5d77828f7fa7ab349066267b570a2f5d51621)
![{\displaystyle s^{2}+{\frac {1}{LC}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/247529843c7dca164500d0cf8d82c2782088920c)
![{\displaystyle s=\pm {\sqrt {\frac {1}{LC}}}t=\pm \omega t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/cea40109b2b178979cc2b20c06671566a9205078)
![{\displaystyle I=e^{(}st)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c0232ff9300c6e6a239dc5f0f2dda7f66e11ff58)
![{\displaystyle I=e^{j}\omega t+e^{-}j\omega t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/825de5cc1cf7a617b98c8495a90affc12dcc435c)
![{\displaystyle I=ASin\omega t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0e9ac2e623accb2105fb1ff129416b6465f2c23b)
The Natural Response at equilibrium of the circuit is a Sinusoidal Wave
At Resonance, The total Circuit's impedance is zero and the total volages are zero
![{\displaystyle \omega L={\frac {1}{\omega C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/dcb11634ca16211fd8ab0612e5120df68ef9d89c)
![{\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a101aa792a66998c31e8b1a3b81061dee244c95c)
![{\displaystyle V_{L}=-V_{C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/30aae9006a33fd17921d2f41ff4ffe3e6f1ea4c8)
The Resonance Reponse of the circuit at resonance is a Standing (Sinusoidal) Wave
![](//upload.wikimedia.org/wikipedia/commons/thumb/4/4e/RLC_series_circuit.png/100px-RLC_series_circuit.png)
At Equilibrium, the sum of all voltages equal to zero
![{\displaystyle L{\frac {dI}{dt}}+IR+{\frac {1}{C}}\int Idt=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4e8735ed0a77e79c08a1b613d8eb1d9365c94bc6)
![{\displaystyle {\frac {dI}{dt}}+I{\frac {R}{L}}+{\frac {1}{LC}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1561db08862170dc2d939a67e004519ffcbf45ac)
![{\displaystyle {\frac {d^{2}I}{dt^{2}}}+{\frac {R}{L}}{\frac {dI}{dt}}+{\frac {1}{LC}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/754431b408f76d7b36a3cb0144d26a3dd901edac)
![{\displaystyle s^{2}+{\frac {R}{L}}s+{\frac {1}{LC}}=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d886feb5100a04906eb8248b717bb523a9f7f8a)
![{\displaystyle s=(-\alpha \pm \lambda )t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e775fbd0e0dd8732e942177a4f392f6ecd4d9b)
Với
và
![{\displaystyle \beta ={\frac {1}{LC}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/06d8917f5f97df3550c9c9f875ade761dfed144a)
![{\displaystyle \lambda ={\sqrt {\alpha ^{2}-\beta ^{2}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/56ab74e57be6d68bc13c3439cc5b01418d349482)
Khi
![{\displaystyle s=-\alpha t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8a37499ebc06fe0365db95df6d176e6adcebfcdf)
![{\displaystyle I=e^{(}-\alpha )t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/59fba1d0fb710a89505b519ac764bffe1343e2e5)
- The response of the circuit is an Exponential Deacy
Khi
![{\displaystyle s=(-\alpha \pm \lambda )t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e775fbd0e0dd8732e942177a4f392f6ecd4d9b)
![{\displaystyle I=e^{-}\alpha t\pm (e^{\lambda }t+e^{-}\lambda t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/18b8156e99a469c3d775d150cdc2d5dd37743ea7)
- The response of the circuit is an Exponential Deacy
Khi
![{\displaystyle s=(-\alpha \pm \lambda )t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b4e775fbd0e0dd8732e942177a4f392f6ecd4d9b)
![{\displaystyle I=e^{-}\alpha t\pm (e^{j}\lambda t+e^{-}j\lambda t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/afd7e330c9042aa450d3db271655d7f5d9a2b346)
- The response of the circuit is an Exponential decay sinusoidal wave
Điện Kháng Tổng Mạch Điện
![{\displaystyle Z=Z_{R}+Z_{L}+Z_{C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5e3e1cc50641d9e3565916b66cd8bf60ca409938)
![{\displaystyle Z=R+j\omega L+{\frac {1}{j\omega C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5671544c54db4f6cd557a6ae8db39179cb493473)
![{\displaystyle Z={\frac {1}{j\omega C}}(j\omega ^{2}+j\omega {\frac {R}{L}}+{\frac {1}{LC}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/65fa68fd7d7f7f5daf826945abf280b695e9f940)
The total impedance of the circuit
![{\displaystyle Z=Z_{R}+Z_{L}+Z_{C}=R+0=R}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78acec3edc67492eef537e01da96cecf8f06b2ef)
![{\displaystyle I={\frac {V}{R}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a3f79c0e6370e9516dd8670460b5b119f526e159)
![{\displaystyle Z_{L}=Z_{C}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/fdca5f8ea2f39956afc4eb1c755ea137f756a661)
![{\displaystyle j\omega L={\frac {1}{j\omega C}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/78b53a582b173962b28de6e165142f615e46627a)
![{\displaystyle \omega ={\sqrt {\frac {1}{LC}}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a101aa792a66998c31e8b1a3b81061dee244c95c)
At resonance frequency
the total impedance of the circuit is Z = R ; at its minimum value and current will be at its maximum value :
Look at the circuit, at
, Capacitor opens circuit . Therefore, current is equal to zero . At
, Inductor opens circuit . Therefore, current is equal to zero
Series RC and RL has a Character first order differential equation of the form
![{\displaystyle {\frac {df(t)}{dt}}+\omega t=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/21ac34a3c7ff2886554ef3f4fe7c98e9a5e583e6)
that has Decay exponential function as Natural Response
![{\displaystyle f(x)=Ae^{(}-{\frac {t}{T}})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/646ef0e35768c669273f660a7e1f221bd93412e0)
- f(t) = i(t) for series RL
- f(t) = v(t) for series RC
Series LC and RLC has a Characteristic Second order differential equation of the form
![{\displaystyle {\frac {d^{2}f(t)}{dt}}+\omega t=0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e3758fa68d1e79ac24f76248615a0226d4dff7a)
![{\displaystyle f(x)=e^{(}\pm \omega t)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/74f6a88dfb9a72826dfade9299bfae7bde521443)
![{\displaystyle f(x)=e^{(}\omega t)+e^{(}-\omega t)=ASin\omega t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/98d4080689d52bd1aea962f84332f65ad5db39bd)
At equilibrium , the Natural Response of the circuit is Sinusoidal Wave
![{\displaystyle f(x)=ASin\omega t}](https://wikimedia.org/api/rest_v1/media/math/render/svg/32426d3db7e14f8a53135c4fa20627b6bb8e2476)
At Equilibrum , the Resonance Response is Standing Wave Reponse