When the output of an electric circuit for the given input varies with respect to time, then it is called a time response. The time response will consist of Below two parts.
- Transient Response
- Steady-state Response
1) Transient Response: When an input is applied to an electric circuit, the output takes a time to reach a steady-state. Hence the output will be considered in a transient state till it goes to a steady-state. Therefore, the response of the electric circuit during the transient state is considered a transient response.
Presence or Absence of Transients: Transients occur in the response due to instant change in the sources that are applied to the electric circuit and or due to switching action in the circuit.
The two possible switching actions are:- Opening switch and Closing switch.
There are 3 basic components in Linear Electrical Circuits:
- Resistance (R)
- Capacitance (C)
- Inductance (L)
The transient part will absent in the response of an electrical circuit or network if the circuit contains only resistances. This is because the resistor is having the ability to adjust any amount of voltage and current in it.
The transient part present in the response of an electrical circuit or network due to the presence of energy storing elements like inductor and or capacitor. This is Because they can’t change the energy stored in those elements instantly.
Let us Assume the switching action takes place at time t = 0. The Inductor current does not change instantaneously (due to inductor property) when the switching action takes place. That means, the value of inductor current just after the switching action as well as just before the switching action will same.
Mathematically, Inductor current can be represented as IL(0-) = IL(0+)
The capacitor voltage does not change instantaneously similar to that of the inductor current when the switching takes place. That means, the value of capacitor voltage just after the switching action as well as just before the switching action will same.
Mathematically, Capacitor Voltage can be represented as VC(0-) = VC(0+)
Note: VR, VL, and VC are the voltages across R, L, and C elements respectively while iR, iL, and iC Represent the current through R, L, and C elements respectively.
Now, Steps for Finding Transient Response :
Step 1: Identify the variable of interest (Inductor current for R-L circuit, Capacitor voltage for R-C circuit).
Step 2: Determine the initial value of the variable according to the circuit.
Step 3: Calculate the final value of the variable according to the circuit.
Step 4: Calculate the time constant for the circuit (Circuit after switching).
Resistive circuits will have NO Transient:
Consider the resistive circuit as shown below
When the switch is ON, the voltage across R becomes V volts immediately (in zero time).
Instantaneous Voltage v(t) = V = iR for t ≥ 0
Instantaneous Voltage v(t) = 0 for t <0
First Order RC Circuit:
Using a loop, the sum of the voltage will be zero.
VTh = RTh · i (t) + v(t)
... Eq. (1)
Substitute in the capacitor current.
... Eq. (2)
This simplifies into the differential equation,
... Eq. (3)
Move the second term to the right-hand side and then divide by the numerator
... Eq. (4)
The indefinite integral resolves to the following form
... Eq. (5)
D is a constant of integration. Removing the natural log and solving for v(t) shows
... Eq. (6)
The constant eD represented by A, can be found at time t = 0
eDA=v(0)- VTH ... Eq. (7)
We can also solve for the final steady state.
... Eq. (8)
Substitute eq. (8) and (7) into eq. (6).
... Eq. (9)
the time constant from the product in the exponential term.
... Eq. (10)
Therefore, the final form of the complete response is :-
Inductor and Thevenin’s Equivalent Circuit
Below is an inductor element connected to a circuit that has been reduced to its Thevenin’s equivalent.
Applying KVL to the loop of this circuit
VTh = RTh · i (t) + v(t) ... Eq. (11)
The voltage across an inductor is given by:-
... Eq. (12)
Use this in eq. (11).
... Eq. (13)
On Rearrange the equation into a form that is easier to integrate.
... Eq. (14)
on Divide by the term in brackets, and integrate.
... Eq. (15)
The integral becomes,
... Eq. (16)
... Eq. (17)
At time t = 0, the constant eD = A is revealed.
... Eq. (25)
As time moves to infinity, the steady-state or forced response is found.
... Eq. (18)
The time constant is,
... Eq. (19)
Therefore the complete response of the current through an inductor element connected to a Thevenin equivalent circuit is given by
The derived equation of voltage and current can be used for both RL & RC circuits.
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