RC circuit
With RC circuits is understood in the electrical circuit consisting of an ohmic resistor (R - Resistor eng. ) And a capacitor (C -. Engl capacitor ) are constructed. RC elements are linear, time-invariant systems. In a narrower sense, so that the filter as the low-pass or high-pass meant. If a lowpass, as in the adjacent figure, the capacitor is connected in parallel to the signal output, the high-pass resistor and capacitor are interchanged.
For potential equalization or grounding in the function itself parallel circuits are of capacitor and resistor. In order to limit electromagnetic interference to series circuits are of capacitor and resistor, such as in the snubber.
- 2.1 Low-pass
- 2.2 high pass
Behavior in the time domain
Charging
Exemplary the system response is presented in a step function. The voltage is zero volt to the time zero and then rises immediately. In the capacitor so long as current flows until the plates are electrically charged, and do not accept further charge. This occurs when the capacitor voltage U ( t) is equal to the applied voltage Vmax. One plate is then electrically positive and the other negatively charged. On the negatively charged side there is an excess of electrons.
The charging time of the capacitor is proportional to the magnitude of the resistor R and the capacitance C of the capacitor. The product of resistance and capacitance is called the time constant.
Theoretically, it takes infinitely long time to U ( t) = Umax is. For practical purposes, can be used as the charging time t L, by which the capacitor can be regarded approximately as a fully loaded ( greater than 99 %).
The time constant τ at the same time marks the time at which the voltage applied at the beginning of the curve tangent reaches the final value of the voltage. After this time, the capacitor would be charged to the final value, if you could load it with the constant current. Actually, however, the current at a constant applied voltage decreases exponentially with time.
The maximum current flowing at the time t = 0 This is obtained through resistor R to the Ohm's law, where Umax is the applied voltage of the power source:
The course of the charging voltage U (t) and the respective time value is described by the following equation, where e is Euler's number, t is the time after the start of charge and the time constant are:
It is assumed that the capacitor was uncharged at the beginning. Thus, the voltage is at first zero, and then rises in the form of an exponential function on. After the time the voltage reaches about 63% of the applied voltage Vmax. According to the time of the capacitor more than 99 % charged.
The curve of the current I (t) and the respective time value is described by the following equation:
Here the current is at first and then decreases as an exponential function such as during the unloading process. After the time, the current is only about 37% of its initial value and after the time it is dropped to less than 1%.
Equation charging
Applies to the voltage for charging the capacitor for an ideal voltage source:
This is derived as follows. For the current applies:
For the voltage across the resistor is true:
The voltage across the capacitor is considered:
For a simple circuit consisting of a capacitor and ohmic resistance In accordance mesh sentence:
This differential equation is dissolved by only solves the homogeneous equation by setting the time being:
Since is constant, then:
After the substitution rule:
Is the electric charge of the capacitor at the time, it can not be negative, the time to start of the charging, and has the value of 0 's; it follows:
By exponentiation to the base e is obtained:
In order to now solve the inhomogeneous differential equation, we apply the method of variation of parameters, by considering as a time- dependent, and as it is differentiated and insert it into the original equation.
Use in:
This is according converted and integrated:
As mentioned above, the charging starts when the time. At this time, the charge on the capacitor is:
This must be used in the solution of the ODE:
This is the equation as it stands above. If one chooses as the value of a theoretically fully charged capacitor is calculated from the equation:
Similarly applies to the voltage:
And for the current:
Unloading
The picture shows the discharge, when the capacitor at the beginning to the value Umax is loaded and discharged via the resistor R. Here, both the voltage and the current at the beginning of the largest:
, The voltage over the discharge with time in accordance with
The current (t) is linked via the discharge resistor R to the voltage U, shows the corresponding course
The discharge current is negative at the given reference arrow.
Differential equation of the discharge
For the discharging of the capacitor holds:
This is derived as in the charging process. The dissolved differential equation can be found from there. The initial conditions are only different and the method of variation of parameters is not required:
Is the electric charge of the capacitor at the time, it can not be negative, the time to start of discharge, and has a value of 0 s Here, there is no discharge, but an initial charge; it follows:
By exponentiation to the base e is obtained:
Similarly applies to the voltage:
And for the current:
Impulse response
The impulse response describes the output voltage waveform on a diracimpulsförmige input voltage. The output voltage waveform is described by the time derivative:
It is the instantaneous voltage across the resistor, which causes a charge reversal of the capacitor. The voltage pulse is integrated by the RC network and leaves a capacitor charge, which are then discharged in the form of an e-function.
The rate of voltage rise (volts per second) is an important parameter in the electronics and power electronics.
Periodic signals
The filter effect is particularly evident at wave signals; The filter response is composed of segments of the charging and discharging. The slope is lower, therefore missing in the frequency range of high frequencies. RC elements are accordingly used to reduce interference and a low-pass.
The slope of the voltage across the capacitor at an amplitude U0 of the rectangular voltage source drops from the infinite value of the supplying square wave voltage to a maximum of
From. The maximum charging current ( peak current, pulse current Ip)
This power must be able to withstand, for example, with an RC unit wired switch contacts or semiconductor switches.
Behavior in the frequency range
Low-pass
Resistor and capacitor form a frequency dependent voltage divider, which also causes a phase shift of up to 90 °. The impedances Z are R or. The RC element is for a balanced oscillating voltage of frequency:
And thus the transmission characteristics, which is defined as the ratio of the output to the input voltage:
The normalized frequency Ω = ω/ω0 from the division of angular frequency ω = 2? f and limit angular frequency ( crossover frequency, corner frequency cutoff frequency or English )? c = 1 / τ = 1 / (RC ) results. Hence the cut-off frequency fc, assume the same value in the reactance and resistance, the phase shift of 45 ° and thus the attenuation is about 3 dB:
For low frequencies Ω << 1 H is about 1, the input and output voltage approximately equal, which is why the area also Engl. referred to as the passband. For frequencies Ω >> 1 H falls with 20 dB per decade from = 6 dB per octave. The area is weggefilterte english designated stop band.
At very low frequencies, which are significantly smaller than the cutoff frequency, the charging current of the capacitor is of no consequence and input and output voltage differ only slightly. The phase shift is 0 °.
The frequency increases, it takes - in comparison to the period of oscillation - longer, until the capacitor is charged to the input voltage. Therefore, the phase shift increases.
At very high frequency, this tends to the limit of 90 ° C, but then the voltage across the capacitor is also immeasurably small.
High-pass
The interconnection as a high pass different from that of the low-pass filter by interchanging R and C. Accordingly applies
And
The amplitude response is compared with the low-pass along mirrored, high frequencies pass almost unattenuated.
Description in the spectral range
With an analog derivation obtained for the low-pass
A pole at.
Wherein the high-pass
Also results in a pole at, in addition a zero at the origin. The RC element thus represents a Butterworth filter 1st order dar.