## What is transient analysis?

The changes in driving voltage and current are determined by using transient analysis. Due to the capacitance and inductance present in the circuit, the driving voltage can be hard to predict. The capacitance and inductance of some circuits may be large enough so that the response deviates from its original value that was needed to design.

We can examine the following behavior using transient analysis:

- When the circuit is driven with a constant direct current or DC voltage, the approach to the steady-state over time can be examined.
- When a DC voltage/current source changes in magnitude then how the circuit current and voltage changes can also be determined.
- How the phase and magnitude of the current and voltage are different from those of the driver in alternating current or AC circuit is also determined.
- In the time domain how the circuit responds to different arbitrary driving waveforms can be also determined.

Transient analysis is also a time-dependent simulation and it helps to measure the time-domain circuit response. The waveform is exactly similar to the results viewed in Probe when we probe the circuit with a logic analyzer or an oscilloscope. We can perform the transient analysis on both digital and analog circuits. Voltage is the variable on the y-axis and time on the x-axis. In the case of AC analysis, it gives the voltage/current Vs frequency in a linear version of the circuit whereas, in the case of DC analysis, it gives DC voltage/current component Vs a stepped voltage/current component. In the case of transient analysis in the transmission line, we only need to understand the traveling waves on a transmission line.

## Steady state and transient response

Most of us are familiar with transient analysis in an RC (resistor-capacitor) series circuit with a DC source. The potential is dropped entirely across the capacitor and the charges get accumulation on it when we switch on the DC source. Eventually, the current in the circuit falls to zero similar to the potential that drops to zero across the resistor. This behavior is only because the circuit approaches a steady state. In an RL (resistor-inductor) series circuit, once the DC source is switched ON the inductor induces back EMF (electromotive force) causing a transient response in the current. While the voltage drops across the inductor slowly to zero, the current slowly rises to its steady state value according to Ohm's law. All the potential is dropped across the resistor.

Once the DC source is switched off, the current in this type of circuit slowly reduces to zero. The current and voltage in this type of circuit are exponentially rising/falling functions with time. The amount of start time required for the current/voltage to rise to the maximum/minimum of its value is generally infinite. One of the most important things in the transient analysis is to evaluate the time constant of the circuit. The value of the time constant tells us how fast the exponential curves rise/fall with time. The transient response of a system is the response to the change of the system from an equilibrium state to a steady state.

## Transient analysis simulation

Complex solution techniques and power simulation tools are developed for the analysis of modern systems. Transient analysis is a type of time-domain simulation so we can use it to examine the phase and magnitude of current in any circuit which is driven with AC voltage or current source with a specific frequency. We can place probes at specific levels in the circuit which will provide us the current at that location using a "simulation program with integrated circuit emphasis" or SPICE simulator that includes a "graphical user interface" or GUI. We can also measure the voltage drop across specific elements in the circuit by getting a time-domain graph.

Those circuits which have natural resonance can use this analysis to determine the level of damping in the system as well as the natural resonance frequency. For instance, it can be used in an RLC (resistor-inductor-capacitor) series circuit that is driven with a DC source. As the current approaches the steady state it will exhibit overdamped decay, perfectly damped decay, or underdamped oscillation. This solution analysis is used to take out the time constant and the natural resonance frequency from a graph of voltage/current. A netlist is a circuit description in SPICE that consists of a statement that defines each circuit element.

## Numeric integration

This is the era of COVID-19 but we have learned in school that how to formulate differential equations and find the solution to describe an RC circuit. Then the input is given and we try to find a nice analytical function for the output using calculus techniques. These analytical abilities are not present in SPICE. But instead, it tries to find an approximate analytical solution using numeric integration at discrete time points. If we plot all the discrete points it should match will the graph of analytical functions. There is a formula named forward-Euler (FE) formula which is intuitive but is not considered the best method for transient simulations. Instead of the FE formula, the backward-Euler (BE) formula is used by the SPICE simulator to give approximate simulation results. This method is the less sensitive and more accurate one-to-step size of timestep as compared to the FE formula and this method is used for transient simulations. In non-linear circuit analysis, we can use the nodal analysis to find our solution by first converting the energy storage elements into their linear companion models.

## Context and Applications

This topic is significant in the professional exams for undergraduate, graduate, and postgraduate courses.

- Bachelors in Electrical Engineering
- Masters in Electrical Engineering

## Practice Problems

1. Why transient analysis is used?

- To find the phase and magnitude of current/voltage
- To find only the phase of a voltage
- To find the phase and magnitude of current
- None of these

**Answer:** Option a

**Explanation:** Transient analysis is a type of time-varying simulation. It can be used to examine the phase and magnitude of current in any circuit or network which is operated with AC voltage or current source with a specific frequency.

2. What is the response type when the switch S is closed in an RC circuit?

- Decays with time
- Rise with time when a power supply is given
- Do not vary with time
- First increases and then decreases

**Answer: **Option a

**Explanation:** When the switch S is closed in an RC circuit the response decays with time and is given by the equation ${V}_{C}={V}_{S}(1-{e}^{\frac{-t}{RC}})$

3. What is meant by zero initial condition of the system?

- Zero input reference signal
- Non-zero stored energy
- No initial movement of moving parts
- The system is at rest and no energy is stored in its elements.

**Answer: **Option d

**Explanation:** When all the initial conditions of a system are equal to zero then the system is said to be relaxed and no energy is stored at any of its elements.

4. What is the value of the time constant in the RC series circuit?

- L/R
- RC
- R
- L

**Answer:** Option b

**Explanation**: The voltage ${V}_{C}$ across a capacitor is given as ${V}_{t}={V}_{O}(1-{e}^{\frac{-t}{RC}})$. Here RC is the time constant.

5. How many time constants do the transient part take to reach more than ninety-nine percent of its final value?

- 2
- 3
- 4
- 5

**Answer: **Option d

**Explanation:** After a number of five-time constants the transient part of the response will reach more than ninety-nine percent of its final value.

## Related Concepts

- Steady state analysis
- AC transients
- Application of Laplace transform in transients

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