RC Circuit Analysis: Formula, Classification, and Application Circuit

This post was most recently updated on May 24th, 2022

RC circuit (resistor-capacitor circuit), also known as RC filter circuit, is composed of a resistor and a capacitor. According to the arrangement of resistor and capacitor, it can be divided into RC series circuit and RC parallel circuit; pure RC parallel circuit cannot resonate, because the resistor does not store energy. RC circuits are widely used in analog circuits and pulse digital circuits. If an RC parallel circuit is connected in series in the circuit, it will attenuate low-frequency signals, and if it is connected in parallel, it will attenuate high-frequency signals, that is, filtering. In this article, Easybom will conduct a series of RC circuit analysis for electronic engineers or enthusiasts.

RC circuit classification

In the following part, we will discuss the classification of RC circuits in detail: series RC circuitparallel RC circuit, and series-parallel RC circuit.

(1) RC series circuit

The characteristics of the RC series circuit: due to the existence of the capacitor, the DC current cannot flow, and both the resistor and the capacitor have a blocking effect on the current. The total impedance is determined by the resistance and the capacitive reactance, and the total impedance changes with the frequency. There is a corner frequency in the RC series connection: f0=1/2πR1C1. When the input signal frequency is greater than f0, the total impedance of the RC circuit is basically unchanged, and its magnitude is equal to R1.

Screenshot 130

(2) RC parallel circuit

An RC parallel circuit can pass both DC and AC signals. It has the same corner frequency as the RC series circuit: f0=1/2πR1C1. When the input signal frequency is less than f0, the signal is DC relative to the circuit, and the total impedance of the circuit is equal to R1; when the input signal frequency is greater than f0, the capacitive reactance of C1 is relatively small, and the total impedance is the resistance value plus the capacitive reactance. When the frequency reaches a certain level, the total impedance becomes 0.

Screenshot 131

(3) RC series-parallel circuit

There are two corner frequencies f01 and f02 in the RC series-parallel circuit: f01=1/2πR2C1, f02=1/2πC1*[R1*R2/(R1+R2)]. When the signal frequency is lower than f01, C1 is equivalent to an open circuit, and the total impedance of the circuit is R1+R2.

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When the signal frequency is higher than f02, C1 is equivalent to a short circuit, and the total impedance of the circuit is R1.

When the signal frequency is higher than f01 and lower than f02, the total impedance of the circuit varies from R1+R2 to R1.

Screenshot 132

Transient response

According to the external excitation in the circuit, the transient response of the RC circuit is divided into three types;

1. Zero state response: The energy storage element in the circuit has no initial energy storage after switching, and is only maintained by the excitation power supply.

2. Zero input response: There is no independent power supply in the circuit after switching, and the response is only maintained by the initial energy storage of the energy storage element.

3. Full response: After switching, there are independent excitation power supply, energy storage element, and initial energy storage in the circuit, and they maintain the response together.

Source of time constant in an RC circuit

There is a time constant τ=RC in the RC circuit, which is the time constant of the reaction circuit decaying with time and having a transition period. Then why is it called the time constant, and what does it have to do with time?

When the DC power supply at both ends of AB is suddenly disconnected, the current through AB is interrupted instantaneously, but the voltage across the capacitor will not change suddenly (the moment before and after the interruption), so Uc=UR. The energy stored in C at this moment is released to the resistor R, thus forming a series combination.

Screenshot 133

According to the current at the node (KCL), we need to use calculus:

Screenshot 134

The initial condition u(0)=V0, we get

Screenshot 135

The relationship of the capacitor voltage decay with time is obtained, and it decays exponentially.

Then the rate at which the capacitor voltage decays with time is:

Screenshot 136

Therefore, when t=0, the decay rate of the initial voltage V0 can be obtained:

Screenshot 137

The time required for the initial voltage V0 to decay to 0 is:

Screenshot 138

From this, RC is obtained as the time required for the initial voltage V0 to decay to 0 at a constant rate. That’s why it’s called the time constant of the RC circuit, the minus sign in the formula means that the voltage is decaying over time.

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RC Circuit Applications

RC circuits are widely used in analog circuits and pulse digital circuits. Due to the differences in the form of the circuit, the signal source, and the R and C components parameters, various RC circuit applications are formed: differential circuit, integral circuit, coupling circuit, filter circuit, and phase shift circuit.

1. RC differential circuit

Take the two ends of the resistor in the RC series circuit as the output terminal, and select the appropriate circuit parameters to make the time constant τ﹤tp (the pulse width of the rectangular pulse). Since the charging and discharging of the capacitor proceeds quickly, the voltage uc(t) on the capacitor C is close to the input voltage ui(t), and the output voltage is:

Screenshot 139

The above formula shows that the output voltage uo(t) approximately has a differential relationship with the input voltage ui(t), so this circuit is called a differential circuit.

Screenshot 140

2. RC integral circuit

If the two ends of the capacitor of the RC circuit are used as the output end, and the circuit parameters satisfy the condition of τ>tp (the pulse width of the rectangular pulse), it becomes an integrating circuit. Since this circuit capacitor charges and discharges very slowly, the voltage ur(t) across the resistor R is approximately equal to the input voltage ui(t), and its output voltage uo(t) is:

Screenshot 141

The above formula shows that the output voltage uo(t) is approximately integral with the input voltage ui(t).

The integrator is essentially a low-pass filter, and the longer the integration time, the lower its cutoff frequency.

Screenshot 142

3. RC Filter circuit

The filter circuit is a kind of electrical circuit that can make the useful frequency signal pass smoothly and suppress and attenuate the useless frequency signal. Due to the basic nature of capacitors resisting low frequencies and passing high frequencies, the basic component of the filter circuit is still an RC circuit. When the output voltage is taken from the resistor, it is a high-pass filter; when the output voltage is taken from the capacitor, it is a low-pass filter.

Screenshot 143

In order to cut off the influence of the load on the RC circuit, the RC circuit and the integrated operational amplifier are often combined to form an active filter. The figure below shows the first-order active low-pass filter circuit. Swap the positions of R and C in the figure to obtain a first-order active high-pass filter. In order to make the suppressed frequency components attenuate faster beyond the cut-off frequency, several RC circuits can be used in series to obtain a high-order active filter, or RC circuits of different properties can be used in series and parallel with each other to obtain the so-called bandpass filters and band-stop filters, etc.

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4. RC coupling circuit

The RC coupling circuit is the resistor-capacitor coupling circuit, which is the basic form of the multi-stage amplifier interstage coupling method. The figure below shows a two-stage amplifier. The output voltage of the first stage is added to the second stage through the RC resistor-capacitor coupling circuit shown in the figure below, where C = C2, R is R5 and R6 connected in parallel, Ui is the no-load output voltage of the first stage, and Uo is the input voltage of the second stage. In fact, the input coupling circuit and the output coupling circuit of the whole amplifier are RC coupling circuits whose output voltage is taken from the resistor. The output voltage for this coupling circuit can be expressed as:

Screenshot 144
Screenshot 145
Screenshot 146

5. RC Phase Shift Circuit

The RC circuit is used as a two-terminal transmission network. If the output voltage is taken from the resistor, the phase of the output voltage will advance; if the output voltage is taken from the capacitor, the phase of the output voltage will lag. This advance or lag can be up to 90 degrees, but the amplitude of the output voltage is also close to 0 at this time. Generally in the circuit, the signal is passed through the RC circuit.

There is both a certain phase shift and a certain voltage amplitude, so the RC circuit becomes a phase shift circuit. In the circuit, according to different needs, several RC circuits are connected in series to realize a certain angle phase shift for the signal of a certain frequency. The figure below is an RC phase-shifted sine wave oscillator circuit.

Screenshot 147

The three-section RC phase shift circuit is not only a positive feedback network but also a frequency selection network in the oscillation circuit. The circuit parameters are reasonably selected, and a signal of a certain frequency is passed through the RC phase shift circuit. The average phase shift of each section is 60 degrees, and the total phase shift is 180 degrees so that the oscillation balance condition is satisfied, and the signal of this frequency oscillates.