TAILIEUCHUNG - Lecture Electric circuits analysis - Lecture 27: Unit step function u(t)
Lecture Electric circuits analysis - Lecture 27: Unit step function u(t). In this chapter, the following content will be discussed: Unit step function, step response of RC circuit, step response of RL circuit, source free RL and RC circuits. | Previous Lecture 26 Source free RL and RC Circuits. Today's Lecture 27 Unit Step Function Step Response of RC circuit Step Response of RL Circuit Unit step function u(t) The unit step function u(t) is 0 for negative values of t and 1 for positive values of t. The unit step function. In mathematical terms, Undefined at t = 0, where it changes abruptly from 0 to 1 Lecture 27 If the abrupt change occurs at t = to (where to > 0) instead of t = 0, the unit step function becomes The unit step function delayed is by to. If the change is at t = −to, the unit step function becomes The unit step function is advanced by to. We use the step function to represent an abrupt change in voltage or current, like the changes that occur in the circuits of control systems and digital computers. For example, the voltage (a) Voltage source of Vou(t), (b) its equivalent circuit. (a) Current source of Iou(t), (b) its equivalent circuit Step Response of an RC Circuit When the dc source of an RC circuit is suddenly applied, the voltage or current source can be modeled as a step function, and the response is known as a step response. The step response of a circuit is its behavior when the excitation is the step function, which may be a voltage or a current source. Our goal is to find capacitor voltage as the circuit response. An RC circuit with voltage step input. This is known as the complete response of the RC circuit to a sudden application of a dc voltage source, assuming the capacitor is initially charged. Response of an RC circuit with initially charged capacitor. If we assume that the capacitor is uncharged initially, we set V0= 0 Figure shows the plots of capacitor voltage v(t) and capacitor current i(t). Step response of circuit with initially uncharged capacitor: (a) voltage response, (b) current response. Systematic approach or, a short-cut method for finding the step response of an RC or RL circuit The natural response or transient response is the circuit’s temporary response that will die out with time. The forced response or steady-state response is the behavior of the circuit a long time after an external excitation is applied. To find the step response of an RC circuit requires three things: 1. The initial capacitor voltage v(0). 2. The final capacitor voltage v(∞). 3. The time constant τ . If the switch changes position at time t = to instead of at t = 0, there is a time delay in the response so that above Eq. becomes where v(to) is the initial value at t = t+ Example 1 The switch in Fig. has been in position A for a long time. At t = 0, the switch moves to B. Determine v(t) for t > 0 and calculate its value at t = 1 s and 4 s. Example 2 In the following Fig., the switch has been closed for a long time and is opened at t = 0. Find i and v for all time. Step Response of an RL Circuit Our goal is to find the inductor current i as the circuit response. An RL circuit with a step input voltage. This is the complete response of the RL circuit Total response of the RL circuit with initial inductor current Io. Thus, to find the step response of an RL circuit requires three things: 1. The initial inductor current i(0) at t = 0+. 2. The final inductor current i(∞). 3. The time constant τ . if the switching takes place at time t = t0 instead of t = 0,then above Eq. becomes Step responses of an RL circuit with no initial inductor current: (a) current response, (b) voltage response Example 3 Find i(t) in the circuit in following Fig. for t > 0. Assume that the switch has been closed for a long time. Example 4 At t = 0, switch 1 in the following circuit is closed, and switch 2 is closed 4 s later. Find i(t) for t > 0. Calculate i for t = 2 s and t = 5 s.
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