We are now in the First- Order Circuits. The lessons that we have studied before are very applicable in solving for the first-order circuits. Before we are only dealing with resistors and power source in our circuit but now we will not only analyze the resistors also capacitors and inductors .
In this lesson we will examine the "simple circuit": a circuit comprising a resistor and capacitor and a circuit comprising a resistor and an inductor. These are called RC and RL circuits.
A first-order circuit is characterized by a first-order differential equation.
The Source-Free RC Circuit
A source-free RC circuit occurs when its dc source is suddenly disconnected.
The energy already stored in the capacitor is released to the
resistors.
A source-free RC circuit occurs when its dc source is suddenly disconnected.
The energy already stored in the capacitor is released to the
resistors.
Our objective is to determine the circuit response, which, for pedagogic reasons, we assume to be the voltage, v(t) across the capacitor. Since the capacitor is initially charged, we can assume that at time the initial voltage is
This shows that the voltage response of the RC circuit is an exponential decay of the initial voltage. Since the response is due to the initial energy stored and the physical characteristics of the circuit and not due to some external voltage or current source, it is called the natural response of the circuit.
The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation.
The natural response of a circuit refers to the behavior (in terms of voltages and currents) of the circuit itself, with no external sources of excitation.
The natural response is illustrated graphically in the graph above. Note that at t=0 we have the correct initial condition as in Eq. (7.1). As t increases, the voltage decreases toward zero. The rapidity with which the voltage decreases is expressed in terms of the time constant, denoted by t, the lowercase Greek letter tau.
The time constant of a circuit is the time required for the response to decay to a factor of
1/e or 36.8 percent of its initial value.
The time constant of a circuit is the time required for the response to decay to a factor of
1/e or 36.8 percent of its initial value.
The Key to Working with a Source-free RC Circuit Is Finding:
1. The initial voltage v(0) = Vo across the capacitor.
2. The time constant t.
1. The initial voltage v(0) = Vo across the capacitor.
2. The time constant t.