The above equation indicates the solution of a first-order differential equation of a series R-C circuit. Substituting value of K’ in equation (3) we get, substituting equation (1) into equation (3), we get, Substitute this into equation (2), we get, Now i(t) is the current through the capacitor and it can be expressed in terms of voltage across capacitor as Hence applying KVL to the circuit, we get, Now the voltage source is introduced in the circuit. This is an initial condition hence we can write,īecause the voltage across the capacitor cannot change instantaneously. Let us assume that the capacitor is initially fully discharged and switch (K) is kept open for a very long time and it is closed at. i.e.,įorced Response of Driven Series RC Circuit That means the voltage across the capacitor at instant will remain the same just after transition at instant. In the case of a capacitor, the voltage across the capacitor cannot be changed instantaneously. That means the current through inductor at instant will remain same just after transition at instant. In the case of an inductor, the current through it cannot be changed instantaneously. Therefore, total response = forced response + natural response What is an Initial Condition? The natural response is also called the zero input response because the source of supply is turned off. The natural response is one in which source of supply is turned off but the circuit does including the initial conditions (initial voltage on capacitors and current in inductors). The forced response is one in which source of supply is turned on but with the initial conditions (internally stored energy) assumed to be zero. These responses can be combined using the principle of superposition.
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The total response of a circuit is equal to the forced response plus natural response. If the change is an abrupt step the response is called the step response. When something changes in a circuit, as a switch closes, the voltage and current also change and adjust to the new conditions. Similarly, the transfer function from the input voltage to the voltage across the resistor is The transfer function from the input voltage to the voltage across the capacitor is The current is the same everywhere in the series R-C circuit. Voltage: By applying the voltage divider rule, the voltage across the capacitor is: Represents sinusoidal angular frequency (radians per second)Ĭurrent: The current is same everywhere in series R-C circuit. Impedance: The complex impedance, of a capacitor C is Transfer Function of the Parallel RC Circuit:Ĭapacitor C behaves as a in the frequency domain with a voltage source of in series with it where is the initial voltage across the capacitor. Above equation is the first-order differential equation of an R-C circuit.
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