Introduction
As we all know, the most basic passive linear components are resistors (R), capacitors (C) and inductive components (L). These components can be used to form 4 different circuits: RC circuit, RL circuit, LC circuit and RLC circuit. They have some important properties for analog electronics, and can be used as passive filters. In practice, capacitors (and RC circuits) are usually used instead of inductors to form filter circuits. This is because capacitors are easier to manufacture with smaller size. This article mainly introduces the RC Circuit in series and parallel state.
RC circuit (resistor–capacitor circuit), also called RC filter or RC network, has a resistor and a capacitor in series connection. When connected to a DC voltage source, the capacitor charges exponentially in time. That is, a capacitor can store energy, and when a resistor placed in series with it will control the rate at which it charges or discharges. This produces a characteristic time dependence that turns out to be exponential.
RC Circuits Basic Explained
Catalog
Ⅰ RC Circuit Basics
1.1 What is RC Circuit?
For a RC circuit (resistorcapacitor circuit), the primary composes of a resistor and a capacitor. According to the arrangement of resistors and capacitors, it can be divided into a RC series circuit and a RC parallel circuit. In addition, simple RC parallel circuits cannot resonate, because resistor does not store energy. However, LC parallel circuits can resonate. RC circuits are widely used in analog circuits and pulse digital circuits. If a RC parallel circuit connected in series in the circuit, it can attenuate lowfrequency signals, and if it connected in parallel in the circuit, it can attenuate highfrequency signals. That is filtering.
RC circuit is common element in electronic devices. It also play an important role in the transmission of electrical signals in nerve cells. A capacitor can store energy and a resistor placed in series with it will control the rate at which it charges or discharges.
Figure 1. Passive Lowpass RC Circuit
1.2 RC Circuit Characteristics
In the analog circuit, the passive RC filter circuit can be divided into a lowpass filter circuit and a highpass filter circuit according to the connection and size of the capacitor.
The lowpass filter circuit is somewhat equal to the integrator circuit (capacitor C is in parallel at the output.), but both circuits are applied to different requirements. The integrator circuit mainly uses the integration effect of the capacitor C when it is charged. In the case of square wave input, periodic sawtooth wave (triangular wave) will generate, so the capacitor C and resistor R are selected according to the square wave. While the lowpass filter circuit bypasses the higher frequency signal (because XC=1/( 2πfC), when f is larger, XC is smaller, which is equivalent to a short circuit), so the value of capacitor C is determined by referring to the value of the low frequency. For the filter circuit of the power supply, theoretically the larger the value of C, the better.
Figure 2. Low Pass Filter Circuit
The highpass filter circuit has the same form as the differential circuit or the coupling circuit. In the pulse digital circuit, due to the different relationship between RC and pulse width, it is divided into a differential circuit and a coupling circuit. In an analog circuit, choosing an appropriate capacitance C value can pass higher frequency signals selectively, even block DC and lowfrequency signals. For example, a capacitor connected in series with a tweeter, is to prevent the low pitch from entering the tweeter to avoid burnout. What’s more, in the multistage AC amplifier circuit, the highpass filter circuit is also a coupling circuit.
Figure 3. High Pass Filter Circuit
Ⅱ How to Calculate RC Circuit?
From a mathematical point of view, suppose that the RC circuit has been connected to a DC power supply with a voltage value of U0. The voltage on the capacitor is equal to the power supply’s, and at a certain moment t0 the left end S of the resistor is grounded, then the capacitor discharges. In the theoretical analysis, the time t0 is taken as the zero point of time.
According to KVL's law, establish the circuit equation:
The initial condition is .
This is a firstorder homogeneous differential equation, and its general solution is .
After substituting into the original equation:
The characteristic equation is .
The characteristic root is .
According to , get .
Therefore, the required initial value of the differential equation is
It can be seen that the voltage attenuation speed on the capacitor depends on the , and its size only depends on the circuit structure and component parameters.
When the unit of resistance is Ω and the unit of capacitance is F, the unit of product RC is seconds (s), which is represented by τ, then the capacitor voltage can be written as .
t 
τ 
2τ 
3τ 
4τ 
5τ 
... 
∞ 
u_{c}(t) 
Uo 
0.368Uo 
0.135Uo 
0.05Uo 
0.018Uo 
0.0067Uo 
... 
∞ 
0 
The τ time constant is the time it takes for the capacitor voltage to drop to 1/e=36.8% of the initial value. Specifically, it is the time required to charge the capacitor, through the resistor, from an initial charge voltage of zero to approximately 63.2% of the value of an applied DC voltage, or to discharge the capacitor through the same resistor to approximately 36.8% of its initial charge voltage.
When t=4t, the capacitor voltage is very small, and it is generally considered that the circuit enters a steady state, which is also called the zero input response of the RC firstorder circuit.
Ⅲ RC Circuits Classification
3.1 Series and Parallel Circuits
 RC Series Circuit
In circuit, the capacitor cannot flow DC current, and R & C have an obstructive effect on the current. So the total impedance is determined by the resistance and capacitive reactance, and it changes with frequency. RC series circuit has a turning frequency: f0=1/2πR1C1. When the input signal frequency is greater than f0, the total impedance is basically unchanged, and it is equal to R1.
 RC Parallel Circuit
The RC parallel circuit can pass both DC and AC signals. It has the same turning frequency as the RC series circuit: f0=1/2πR1C1. On the one hand, when the input signal frequency is less than f0, the total impedance of the circuit is equal to R1, on the other hand, when the input signal frequency is greater than f0, the capacitive reactance of C1 is relatively small, and the total impedance is the sum of resistance and capacitance. In addition, when the frequency is high to a certain level, the total impedance is zero.
What’s more, as frequency increases, the capacitor will act like a short circuit to high frequency current in its path. At low frequencies, the capacitor tends to block current flow.
3.2 Example: RC Low Pass Filter
 Circuit Analysis
To create a passive lowpass filter, we need to combine the resistor elements with the reactance elements. That is a circuit consisting of a resistor and a capacitor or an inductor. Theoretically speaking, the RL lowpass topology is equivalent to the RC lowpass topology in terms of filtering ability. However, in practice, RC circuits are more common.
Figure 4. RC Lowpass Filter
As shown in the figure, connecting a resistor in series with the signal path and a capacitor in parallel with the load, an RC lowpass response can be generated. In the figure, the load is a single part, but in actual circuits, it may be more complicated, such as the input stage of an analogtodigital converter, amplifier, or oscilloscope to measure the response of the filter.
If a resistor and a capacitor form a frequencydependent voltage divider circuit, we can intuitively analyze the filtering function of the RC lowpass circuit.
Figure 5. Change RC Lowpass Filter into a Voltage Divider
When the frequency of the input signal is low, the impedance of the capacitor is high than the resistor. Therefore, most of the input voltage will drop on the capacitor (and both ends of the load, which is in parallel with the capacitor). When the input frequency is higher, the impedance of the capacitor is lower than the impedance of the resistor, which means that the resistor voltage decreases and less voltage is transferred to the load. Therefore, low frequencies pass and high frequencies are blocked.
 Cutoff Frequency
Where the filter does not cause significant attenuation for a frequency range is called the passband, and the opposite is called the stopband. Analog filters, such as RC lowpass filters, always gradually transit from the passband to the stopband. This means that it cannot be recognized that the filter stops passing the signal and starts blocking one frequency of the signal. This is why the cutoff frequency concept introduced.
When checking the frequency response graph of the RC filter, the signal spectrum is "cut" into two halves of the image, one of which is retained and one is discarded. Because as the frequency moves from below the cutoff point to above the cutoff value, the attenuation gradually increases.
The cutoff frequency of the RC lowpass filter is actually the frequency at which the input signal amplitude is reduced by 3dB (this value is chosen because a 3dB reduction is equal to a 50% reduction in power). Therefore, the cutoff frequency is also called 3dB frequency. The term bandwidth refers to the width of the passband of the filter. For a lowpass filter, its bandwidth is equal to the 3dB frequency (as shown in the figure below).
Figure 6. Cutoff Frequency 3dB
 Filter Response Calculation
We can discuss the theoretical behavior of the lowpass filter by a typical voltage divider. The output of the resistor divider is expressed as following:
The RC filter uses an equivalent structure, using a capacitor XC replace R2. Then we need to calculate the total impedance and place it in the denominator, so there is
The reactance of a capacitor represents the opposite amount of current, but unlike resistance, the opposite amount depends on the frequency of the signal passing through the capacitor. Therefore, we must calculate the reactance at a specific frequency. The equation we use for this as follows:
In the above design example: R≈160Ω and C=10nF. We assume that the magnitude of VIN is 1V, so we can simply remove VIN from the calculation. First, let's calculate the amplitude of V_{OUT} with a sine wave frequency:
While suppressing noise, the amplitude of the sine wave is basically unchanged. Because the cutoff frequency (100kHz) we chose is much higher than the sine wave frequency (5kHz).
Let’s see how the filter successfully attenuates the noise component.
The noise amplitude is only about 20% of its original value.
Ⅳ Visualizing Filter Response
4.1 Frequency Response
The most convenient way to assess the effect of a filter on a signal is to examine the frequency response graph. That is Bode plot, which has amplitude (in decibels) on the vertical axis and frequency on the horizontal axis; the horizontal axis usually has a logarithmic scale so that the physical distance between 1Hz and 10Hz is the same as 10Hz to 100Hz and 100Hz to 1kHz. This configuration allows us to quickly and accurately evaluate the behavior of the filter over a large frequency range.
Figure 7. Bode Plot
Each point on the curve represents the amplitude that the output signal is 1V and the frequency is equal to the corresponding value on the horizontal axis. For example, when the input frequency is 1MHz, the output amplitude (assuming the input amplitude is 1V) will be 0.1V (because 20dB corresponds to a tenfold reduction factor).
The curve in the passband is almost completely flat, and then as the input frequency approaches the cutoff frequency, it starts to drop faster. Finally, the rate of change of attenuation becomes stable, that is, for every ten times the input frequency increases, the amplitude of the output signal decreases by 20dB.
4.2 Low Pass Filter Phase Shift
The way in which the filter modifies the amplitude of various frequency components in the signal has been discussed above. However, in addition to amplitude effects, reactive circuit elements always involve phase shifts.
The concept of phase refers to the value of the periodic signal at a specific moment in the cycle. Therefore, when we say that a circuit causes a phase shift, we mean that it creates a misalignment between the input signal and the output signal. That is the input and output signals no longer start and end their periods at the same time. The phase shift value, such as 45° or 90°, indicates how much misalignment has been created.
Each reactance element in the circuit introduces a 90° phase shift, but this phase shift does not occur at the same time. The phase of the output signal is the same as the amplitude of the output signal, and it changes gradually as the input frequency increases. In the RC lowpass filter, we have a reactive element (capacitor), so the circuit will eventually introduce a 90° phase shift.
As with the amplitude response, the phase response can be most easily evaluated by examining the graph on the horizontal axis which represents the logarithmic frequency. The following description is the general pattern.
The phase shift is initially 0°, and it gradually increases until it reaches 45° at the cutoff frequency. During this part of the response, the rate of change is increasing. With time, the phase shift continues to increase, but the rate of change is decreasing. As the phase shift approaches 90°, the change of rate becomes very small.
Figure 8. Phase Shift
4.3 Secondorder Lowpass Filter
As above mentioned, we have assumed that the RC lowpass filter consists of a resistor and a capacitor. This configuration is a firstorder filter. The "order" of passive filters is determined by the number of reactive components (ie capacitors or inductors) in the circuit. Higherorder filters have more reactive components, which lead to more phase shift and steeper rolloff.
Secondorder filters are usually built a resonant circuit consisting of inductors and capacitors (this topology is called "RLC", or resistorinductorcapacitor circuit). However, it is also possible to create a secondorder RC filter. As shown in the figure below, all we need to do is to cascade two firstorder RC filters.
Figure 9. Secondorder Filter Circuit
Although this topology can produce a secondorder response, it is not widely used. Because its frequency response is usually not as good as a secondorder active filter or a secondorder RLC filter.
 Frequency Response
We can try to create a secondorder RC lowpass filter by designing a firstorder filter based on the required cutoff frequency, that is connecting two firstorder stages in series. This set has a similar overall frequency response, with a maximum rolloff of 40dB/decade instead of 20dB/decade.
However, we cannot simply connect these two stages together and analyze the circuit as a secondorder lowpass filter. In addition, even if we insert a buffer between the two stages so that the first RC stage and the second RC stage can be used as independent filters, the attenuation at the original cutoff frequency will be 6dB instead of 3dB. Because the two stages work independently.
Figure 10. Frequency Response of RCRC Filter
A limitation of the secondorder RC lowpass filter is that the designer cannot tune the conversion from passband to stopband by adjusting the Q factor (this parameter indicates the degree of damping of the frequency response.) of the filter. If two identical RC lowpass filters are cascaded, the overall transfer function corresponds to the secondorder response, but the Q factor is always 0.5. When Q = 0.5, the filter is at the boundary of overdamping, which results in a "sag" frequency response in the transition region. While secondorder active filters and secondorder resonant filters do not have this limitation, designers can control the frequency response of the transition region.
Ⅴ Conclusion
All electrical signals contain a mixture of requiring frequency and unwanted ones. Undesirable frequency components are usually caused by noise and interference, and in some cases they have a negative impact on the performance of the system.
Filters are circuits that react to different parts of the signal spectrum in different ways. The lowpass filter is designed to pass low frequency components and block high frequency components. The output voltage of an RC lowpass filter can be calculated by considering the circuit as a voltage divider (frequencyindependent) composed of resistance and reactance.
The graph of amplitude (in dB, on the vertical axis) vs. log frequency (in Hz, on the horizontal axis) is a convenient and effective way to check the theoretical behavior of the filter. You can also use phase and log frequency graph determines the amount of phase shift that will be applied to the input signal.
The secondorder filter provides a steeper rolloff, and its response is useful when the signal cannot provide broadband separation between the desired frequency and the unwanted one. You can make a secondorder RC lowpass filter by connecting two identical firstorder RC lowpass filters, but the overall 3 dB frequency will be lower than expected.
In RC filtering circuit, the capacitor can store energy, and the resistor placed in series with it can control the chargedischarge rate. And this produces a characteristic time dependence that turns out to be exponential.
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2 comments
According to your view, filtering is one part of rc circuits, too much words for the theory, but you don't tell why are rc circuits used?
author
Re:
The RC circuit is used in camera flashes, pacemaker, timing circuit etc. ... The RC signal filters the signals by blocking some frequencies and allowing others to pass through it. It is also called firstorder RC circuit and is used to filter the signals bypassing some frequencies and blocking others.
Hey, I am a student, can you tell me how does RC circuit work? be specific, thanks~
author
Re:
RC Circuits is a circuit with both a resistor (R) and a capacitor (C). ... A capacitor can store energy and a resistor placed in series with it will control the rate at which it charges or discharges. This produces a characteristic time dependence that turns out to be exponential.