Circuit RC

Circuit series

Tension

One expresses the complex impedance condenser in the form:

Z_C (S) = \ frac {1} {Cs} = \ frac {1} {jC \ Omega}

By regarding the circuit as a Tension divider , it is possible to write:

V_C (S) = \ frac {Z_C} {R + Z_C} V_ {in} (S) = \ frac {1} {1 + RCs} V_ {in} (S) = \ frac {1} {1+jRC \ Omega} V_ {in} (S)

V_R (S) = \ frac {R} {R + Z_C} V_ {in} (S) = \ frac {RCs} {1 + RCs} V_ {in} (S) = \ frac {jRC \ Omega} {1+jRC \ Omega} V_ {in} (S)

Transfer transfer function

The Transfer function transfer of the condenser is equal to:

H_C (S) = {V_C (S) \ over V_ {in} (S)} = {1 \ over 1 + RCs} = {1 \ over 1 + jRC \ Omega}

In a similar way, the transfer transfer function of resistance is equal to:

H_R (S) = {V_R (S) \ over V_ {in} (S)} = {RCs \ over 1 + RCs} = {jRC \ Omega \ over 1 + jRC \ Omega}

Poles and zeros

The two transfer transfer functions have only one pole with:

S = - {1 \ over RC}

Moreover, the transfer transfer function of resistance has a zero in the beginning.

Profit and phase

For a dipole, one can write the transfer transfer function in the form

H (S) = G e^ {J \ phi} \,

, where

G \,

is the profit dipole and

\ phi \,

his phase.

By posing $s= \ sigma + J \ Omega \,$ , the profit for each of the two circuit components RC is:

G_C = \ frac {1} {\ sqrt {1 + \ left (\ Omega RC \ right) ^2}}

and

G_R = \ frac {\ Omega RC} {\ sqrt {1 + \ left (\ Omega RC \ right) ^2}}

One can also write the profit in decibels, defined by: $G_ {dB} = 20 \ times \ log {G} = 20 \ times \ log {\ left| H_C (S) \ right|}$ , from where:

G_ {dB, C} = 20 \ times \ log {\ frac {1} {\ sqrt {1 + \ left (\ Omega RC \ right) ^2}}}

and

G_ {dB, R} = 20 \ times \ log {\ frac {\ Omega RC} {\ sqrt {1 + \ left (\ Omega RC \ right) ^2}}}

Similarly, their phase is:

\ phi_C = \ arctan \ left (- \ Omega RC \ right)

and

\ phi_R = \ arctan \ left (\ frac {1} {\ Omega RC} \ right)

Intensity

The intensity of the current is the same one in all the circuit, since it is about a circuit series:

I (S) = \ frac {V_ {in} (S)} {R+Z_C} = {Cs \ over 1 + RCs} V_ {in} (S)

Impulse response

The Impulse response is the Transformée of opposite Laplace of the corresponding transfer transfer function and represents the response of the circuit to a impulse.

For the condenser:

h_C (T) = {1 \ over RC} e^ {- T/RC} U (T) = {1 \ over \ tau} e^ {- T/\ tau} U (T)

where $u (T) \,$ is the Fonction of Heaviside and $\ tau \ = \ RC$ is the Time-constant .

For resistance:

h_R (T) = - {1 \ over RC} e^ {- T/RC} U (T) = - {1 \ over \ tau} e^ {- T/\ tau} U (T)

Field of the frequencies

The analysis of the circuit in the Domaine of the frequencies makes it possible to determine which Fréquence S the filter rejects or accepts.

When $\ Omega \ to \ infty$ :

G_C \to 0

G_R \to 1

When $\ Omega \ to 0$ :

G_C \to 1

G_R \to 0

Thus, when the exit of the filter is taken on the condenser, the high frequencies are attenuated and the low frequencies passed, a standard behavior of a low-pass Filtre. If the exit is taken on resistance, the reverse occurs and the circuit behaves like a high-pass Filtre.

The Frequency cut-off of the circuit is equal to:

f_c = \ frac {1} {2 \ pi RC}

(in Hz)

The phase also depends on the frequency:

When $\ Omega \ to 0$ :

\ phi_C \ to 0

\ phi_R \ to 90^ {\ circ} = \ pi/2

When $\ Omega \ to \ infty$ :

\ phi_C \ to -90^ {\ circ} = - \ pi/2

\ phi_R \ to 0

At the low frequencies, the terminal voltage of the condenser is in phase with that of the entry signal, while the terminal voltage of resistance is in phase lead. At the raised frequencies, the terminal voltage of resistance is in phase with the entry signal, while the terminal voltage of the condenser is late of phase.

Temporal field

By supposing that the circuit is subjected to a level of tension in entry (

V_ {in} = 0 \,

for

t = 0 \,

and

V_ {in} = V \,

if not):

V_ {in} (S) = \ frac {V} {S}

V_C (S) = V \ frac {1} {1 + sRC} \ frac {1} {S}

V_R (S) = V \ frac {sRC} {1 + sRC} \ frac {1} {S}

The Transformée of opposite Laplace of these expressions gives:

V_C (T) = V \ left (1 - e^ {- t/RC} \ right)

V_R (T) = Ve^ {- t/RC} \,

In this case, the condenser takes care and the tension on its terminals tends towards V, while that at the boundaries of resistance tends towards 0.

Circuit RC has a Time-constant , generally noted $\ tau = RC \,$ , representing the time which the tension takes to approach its end value with better than $1/e \,$ .

It is also possible to derive these expressions from the differential equations describing the circuit:

\ frac {V_ {in} - V_C} {R} = C \ frac {dV_C} {dt}

V_R = V_ {in} - V_C \,

The solutions are exactly the same ones as those obtained by the transform of Laplace.

Integrator

High frequency, i.e. if

\ Omega >> \ frac {1} {RC}

, the condenser does not have time to take care and the tension on its terminals remains weak.

As follows:

V_R \ approx V_ {in}

and the intensity in the circuit is thus worth:

I \ approx \ frac {V_ {in}} {R}

Like,

V_C = \ frac {1} {C} \ int_ {0} ^ {T} Idt

one obtains:

V_C \ approx \ frac {1} {RC} \ int_ {0} ^ {T} V_ {in} dt

The terminal voltage of the condenser thus integrates the tension of entry and the circuit behaves like a integrating Montage.

Shunting device

Low frequency, i.e. if

\ Omega << \ frac {1} {RC}

, the condenser has time to take care almost completely.

Then,

I \ approx \ frac {V_ {in}} {1/j \ Omega C}

V_ {in} \ approx \ frac {I} {J \ Omega C} \ approx V_C

Now,

V_R = IR = C \ frac {dV_C} {dt} R

V_R \ approx RC \ frac {dV_ {in}} {dt}

The terminal voltage of resistance drift thus the tension of entry and the circuit behaves like a Montage shunting device.

Parallel circuit

Parallel circuit RC is generally of an interest less than circuit RC series: the output voltage being equal to the tension of entry, it can be used like only filters supplied with a Power source.

The intensities in the two dipoles are:

I_R = \ frac {V_ {in}} {R}

I_C = J \ Omega CV_ {in} \,

The current in the condenser is out of phase of 90° compared to the current of entry (and of resistance).

Subjected to a level of tension, the condenser takes care quickly and can be regarded as an open circuit, the circuit behaving consequently like a simple resistance.

See too

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