Electronics/RC transient

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For a series RC consist of one resistor connected with one capacitor in a closed loop.

Circuit Impedance[edit]

In Polar Form Z/_θ

 Z = R \angle 0 + \frac{1}{\omega C} \angle - 90
 Z = |Z| \angle \theta = \sqrt{R^2 + (\frac{1}{\omega C})^2}\angle Tan^ -1 \frac{1}{\omega RC}


In Complex Form Z(jω)

Z = Z_R + Z_C = R + \frac{1}{j\omega C} = \frac{1}{j\omega C} (1 + j\omega RC)
T = RC

Differential Equation of circuit at equilibrium[edit]

C \frac{dV}{dt} + \frac{V}{R} = 0
\frac{dV}{dt} = - \frac{1}{RC} V
\frac{1}{V} dV  = - \frac{1}{RC} dt
\int \frac{1}{V} dV  = - \int \frac{1}{RC} dt
\ln V =  - \frac{1}{RC}t + C
 V  =  e^(\frac{-t}{RC} + C) = e^C \cdot e^\frac{-t}{RC}
 V  = A e^{\left(\frac{-t}{RC}\right)} = A e^{\left(\frac{-t}{T}\right)}

Time Constant[edit]

T = R C
t V(t)  % Vo
0 A = eC = Vo 100%
1/RC .63 Vo 60% Vo
2/RC Vo
3/RC Vo
4/RC Vo
5/RC .01 Vo 10% Vo


Capacitor Charge-Discarge.svg

Angle Difference Between Voltage and Current[edit]

Current leads Voltage on an angle; Does this mean in measuring a angle of DC current causes a decrease in current? Let us take some time to understand the previous math in determing a factor of sinuous solution.

Tan\theta = \frac{1}{\omega RC} = \frac{1}{f} \frac{1}{2\pi RC} = t \frac{1}{2\pi RC}

Change the value of R and C will change the value of Angle Difference, Angular Frequency, Frequency and Time

\omega= \frac{1}{RC} \frac{1}{Tan\theta}
f = \frac{1}{2\pi} \frac{1}{RC} \frac{1}{Tan\theta}
t = 2\pi RC Tan\theta

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Application[edit]

RC switch.PNG

When the switch is open, the initial voltage across the capacitor is zero. When the switch closes (which we will refer to as time zero) the capacitor charges via the resistor to V_0.

When the switch is closed, the circuit must follow the relationship:

V_0=v_c(t)+i_c(t)R
V_0=v_c(t)+\frac{dv_c(t)}{dt}RC

which is derived by analysing the circuit using Kirchoff's Voltage Law.

By letting \tau=RC and rearranging the equation:

\frac{dv_c(t)}{dt}+\frac{1}{\tau}v_c(t)=\frac{1}{\tau}V_0

This is a first order linear differential equation with integrating factor:

e^{\int \frac{1}{\tau}dt}
e^{\frac{t}{\tau}}


Multiplying both sides by the integrating factor:

\frac{dv_c(t)}{dt}e^{\frac{t}{\tau}}+\frac{1}{\tau}v_c(t)e^{\frac{t}{\tau}}=\frac{1}{\tau}V_0e^{\frac{t}{\tau}}

Note that:

\frac{d}{dt}[e^{t/\tau}v_c(t)]=\frac{dv_c(t)}{dt}e^{\frac{t}{\tau}}+\frac{1}{\tau}v_c(t)e^{\frac{t}{\tau}}

Substituting and integrating both sides:

e^{\frac{t}{\tau}}v_c(t)=V_0e^{\frac{t}{\tau}}+K

where K is the integration constant.

When t=0

v_c(0)=0

Therefore:

K=-V_0

When t>0 this gives:

e^{\frac{t}{\tau}}v_c(t)=V_0e^{\frac{t}{\tau}}-V_0
v_c(t)=V_0-V_0e^{-\frac{t}{\tau}}
v_c(t)=V_0(1-e^{-\frac{t}{\tau}})

when t<0:

v_c(t)=0
Rc circuit plot.png