Practical Electronics/Parallel RC

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Parallel RC[edit]

A parallel RC Circuit

Circuit Impedance[edit]

Circuit Response[edit]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle V = (IR - RC \frac {dV}{dt}) }

Parallel RL[edit]

An RL parallel circuit\

Circuit Impedance[edit]

Circuit Response[edit]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle I = I_R + I_L}

Parallel LC[edit]

LC circuit diagram

Circuit Impedance[edit]

Circuit response[edit]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle I = \frac{1}{L} \int V dt + C \frac{dV}{dt}}

Parallel RLC[edit]

RLC parallel circuit.png

Circuit Impedance[edit]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \frac{1}{Z} = \frac{1}{R} + \frac{1}{j\omega L} + j\omega C }

Circuit response[edit]

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle I = I_R + I_L + I_C}
Failed to parse (Conversion error. Server ("https://wikimedia.org/api/rest_") reported: "Cannot get mml. Server problem."): {\displaystyle I={\frac {V}{R}}+{\frac {1}{L}}\int Vdt+C{\frac {dV}{dt}}}

Natural Respond[edit]

Forced Respond[edit]

Second ordered equation that has two roots

ω = -α ±

Where

Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \beta = \frac{1}{\sqrt{LC}}}

The current of the network is given by

A eω1 t + B eω2 t

From above

When , there is only one real root
ω = -α
When , there are two real roots
ω = -α ±
When Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle {\alpha^2 < \beta^2}} , there are two complex roots
ω = -α ± j

Resonance Response[edit]

At resonance, the impedance of the frequency dependent components cancel out . Therefore the net voltage of the circui is zero

and

At Resonance Frequency

.
. Current is at its maximum value

Further analyse the circuit

At ω = 0, Capacitor Opened circuit . Therefore, I = 0 .
At ω = 00, Inductor Opened circuit . Therefore, I = 0 .


With the values of Current at three ω = 0 , , 00 we have the plot of I versus ω . From the plot If current is reduced to halved of the value of peak current Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle I = \frac{V}{2R}} , this current value is stable over a Frequency Band ω1 - ω2 where ω1 = ωo - Δω, ω2 = ωo + Δω


  • In RLC series, it is possible to have a band of frequencies where current is stable, ie. current does not change with frequency . For a wide band of frequencies respond, current must be reduced from it's peak value . The more current is reduced, the wider the bandwidth . Therefore, this network can be used as Tuned Selected Band Pass Filter . If tune either L or C to the resonance frequency . Current is at its maximum value . Then, adjust the value of R to have a value less than the peak current by increasing R to have a desired frequency band .


  • If R is increased from R to 2R then the current now is which is stable over a band of frequency
ω1 - ω2 where
ω1 = ωo - Δω
ω2 = ωo + Δω

For value of I < . The circuit respond to Wide Band of frequencies . For value of < I > . The circuit respond to Narrow Band of frequencies

Summary[edit]

Circuit Symbol Series Parallel
RC
RLC series circuit.png
A parallel RC Circuit
Impedance Z
Frequency



Voltage V Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle I = \frac {V}{R} + C\frac{dV}{dt}}
Current I Failed to parse (MathML with SVG or PNG fallback (recommended for modern browsers and accessibility tools): Invalid response ("Math extension cannot connect to Restbase.") from server "/mathoid/local/v1/":): {\displaystyle \frac {dV}{dt} = \frac{1}{C}(I - \frac{V}{R})}
Phase Angle Tan θ = 1/2πf RC
f = 1/2π Tan CR
t = 2π Tan CR
Tan θ = 1/2πf RC
f = 1/2π Tan CR
t = 2π Tan CR