A resistor–capacitor circuit (RC circuit), or RC filter or RC network, is an electric circuit composed of resistors and capacitors. It may be driven by a voltage or current source and these will produce different responses. A first order RC circuit is composed of one resistor and one capacitor and is the simplest type of RC circuit. RC circuits can be used to filter a signal by blocking certain frequencies and passing others. The two most common RC filters are the high-pass filters and low-pass filters; band-pass filters and band-stop filters usually require RLC filters, though crude ones can be made with RC filters.
Natural response
The simplest RC circuit consists of a resistor with resistance R and a charged capacitor with capacitance C connected to one another in a single loop, without an external voltage source. The capacitor will discharge its stored energy through the resistor. If V(t) is taken to be the voltage of the capacitor's top plate relative to its bottom plate in the figure, then the capacitor current–voltage relation says the current I(t) exiting the capacitor's top plate will equal C multiplied by the negative time derivative of V(t). Kirchhoff's current law says this current is the same current entering the top side of the resistor, which per Ohm's law equals V(t)/R. This yields a linear differential equation
C − d V ( t ) d t ⏞ capacitor current = V ( t ) R ⏞ resistor current , {\displaystyle \overbrace {C{\frac {-\mathrm {d} V(t)}{\mathrm {d} t}}} ^{\text{capacitor current}}=\overbrace {\frac {V(t)}{R}} ^{\text{resistor current}},}
which can be rearranged according to the standard form for exponential decay:
d V ( t ) d t = − 1 R C V ( t ) . {\displaystyle {\frac {\mathrm {d} V(t)}{\mathrm {d} t}}=-{\frac {1}{RC}}V(t).}
This means that the instantaneous rate of voltage decrease at any time is proportional to the voltage at that time. Solving for V(t) yields an exponential decay curve that asymptotically approaches 0:
V ( t ) = V 0 ⋅ e − t R C , {\displaystyle V(t)=V_{0}\cdot e^{-{\frac {t}{RC}}},}
where V0 is the capacitor voltage at time t = 0, and e is Euler's number. The time required for the voltage to fall to V0/e is called the RC time constant and is given by
τ = R C . {\displaystyle \tau =RC.}
When using the International System of Units, R is in ohms, and C is in farads, so τ will be in seconds. At any time N·τ, the capacitor's charge or voltage will be 1/eN of its starting value. So if the capacitor's charge or voltage is said to start at 100%, then 36.8% remains at 1·τ, 13.5% remains at 2·τ, 5% remains at 3·τ, 1.8% remains at 4·τ, and less than 0.7% remains at 5·τ and later. The half-life (t1/2) is the time that it takes for its charge or voltage to be reduced in half:
1 2 = e − t 1 / 2 τ ⇒ t 1 / 2 = ln ( 2 ) τ ≈ 0.693 τ . {\displaystyle {\frac {1}{2}}=e^{-{\tfrac {t_{1/2}}{\tau }}}\quad \Rightarrow \quad t_{1/2}=\ln(2)\,\tau \approx {\text{0.693}}\,\tau .}
For example, 50% of charge or voltage remains at time 1·t1/2, then 25% remains at time 2·t1/2, then 12.5% remains at time 3·t1/2, and 1/2N will remain at time N·t1/2.
RC discharge calculator0.00000110000001111111.36836.810.3681110.1591111 For instance, 1 of resistance with 1 of capacitance produces a time constant of approximately 1 seconds. This τ corresponds to a cutoff frequency of approximately 159 millihertz or 1 radians per second. If the capacitor has an initial voltage V0 of 1 , then after 1 τ (approximately 1 seconds or 1.443 half-lives), the capacitor's voltage will discharge to approximately 368 millivolts:
Complex impedance The RC circuit's behavior is well-suited to be analyzed in the Laplace domain, which the rest of this article requires a basic understanding of. The Laplace domain is a frequency domain representation using complex frequency s, which is (in general) a complex number:
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