Transient Analysis

Contents

**Describe Transient Analysis and Source Free Response** 1

**Define and differentiate between Natural and Forced response** 2

**Describe Initial and Final conditions in circuit Elements (R, L, and C)** 3

**Describe Particular Integral and Complementary Function of a Differential Equation** 3

**Describe Transient response of R-L circuit having DC Excitation** 4

**Describe Transient response of R-L circuit having AC Excitation** 5

**Recall the concepts of Time constants and plot the Normalised curve** 6

**Determine Transient response of R-C circuit having DC Excitation** 6

**Determine Transient response of R-C circuit having AC Excitation** 7

**Recall the concepts of Time constants and plot the Normalised curve** 8

**Recall Second Order Differential Equation** 9

**Determine Transient response of R-L-C circuit having DC Excitation.** 10

**Determine Transient response of R-L-C circuit having AC Excitation** 11

**Describe Transient Analysis and Source Free Response**

Transient analysis is the study of the behaviour of a circuit in response to a sudden change in the input or in the initial conditions. It is a time-domain analysis technique that is used to determine the voltages and currents in a circuit as a function of time, after a transient event has occurred.

In a circuit, a transient event can be caused by a sudden change in the input, such as a step function, a pulse, or a ramp, or by a change in the initial conditions of the circuit, such as the initial voltage across a capacitor or the initial current through an inductor. The transient response of a circuit is the voltage or current that occurs in response to a transient event.

The source-free response of a circuit is the response that occurs in the absence of any external sources, such as voltage or current sources. The source-free response is caused by the initial conditions of the circuit, and it can be determined by setting all the sources to zero and solving for the voltages and currents in the circuit.

To analyse the transient response of a circuit, the following steps can be followed:

- Determine the circuit equations: Write down the circuit equations that describe the behaviour of the circuit, using the laws of Kirchhoff’s voltage and current, Ohm’s law, and the element equations for capacitors and inductors.
- Determine the initial conditions: Determine the initial voltages and currents in the circuit, which are the values of the voltages and currents at t=0, just before the transient event occurs.
- Determine the response to the transient event: Determine the response of the circuit to the transient event, using the circuit equations and the initial conditions. The response can be determined by solving the equations using mathematical techniques, such as Laplace transforms, differential equations, or numerical methods.
- Determine the steady-state response: Determine the steady-state response of the circuit, which is the response that occurs after the transient response has decayed to zero. The steady-state response can be determined by setting all the initial conditions to zero and solving for the voltages and currents in the circuit.

**Define and differentiate between Natural and Forced response**

In circuit analysis, natural response and forced response refer to the behaviour of a circuit in response to a change in input, such as a sudden change in voltage or current.

The natural response of a circuit is the behaviour of the circuit due to its own inherent characteristics, such as the values of the circuit elements (resistors, capacitors, and inductors) and their initial conditions. The natural response is the transient behaviour of the circuit that occurs without any external input.

The forced response of a circuit is the behaviour of the circuit due to an external input, such as a voltage or current source. The forced response is the steady-state behaviour of the circuit that occurs once the transient behaviour has died out.

To differentiate between the natural and forced response of a circuit, one can use the concept of the complete response, which is the sum of the natural and forced responses. The complete response is the behaviour of the circuit over time that takes into account both the transient and steady-state behaviour.

Overall, the natural response is the initial transient behaviour of a circuit due to its inherent characteristics, while the forced response is the steady-state behaviour of the circuit due to an external input. The complete response takes into account both the natural and forced responses to provide a comprehensive understanding of the circuit’s behaviour over time.

**Describe Initial and Final conditions in circuit Elements (R, L, and C)**

In circuit analysis, initial conditions and final conditions refer to the state of circuit elements, such as resistors (R), inductors (L), and capacitors (C), at specific points in time.

For resistors (R), the initial condition refers to the voltage or current across the resistor at the beginning of a given time interval. The final condition refers to the voltage or current across the resistor at the end of the time interval.

For inductors (L), the initial condition refers to the current through the inductor at the beginning of a given time interval. The final condition refers to the current through the inductor at the end of the time interval.

For capacitors (C), the initial condition refers to the charge on the capacitor at the beginning of a given time interval. The final condition refers to the charge on the capacitor at the end of the time interval.

It is important to consider the initial and final conditions when analysing circuits because they can affect the behaviour of the circuit over time. In some cases, the initial conditions can cause transient behaviour in the circuit, while the final conditions can affect the steady-state behaviour of the circuit.

**Describe Particular Integral and Complementary Function of a Differential Equation**

In differential equations, the particular integral and complementary function refer to the two components that make up the general solution of a differential equation.

The complementary function (CF) is the solution of the homogeneous part of the differential equation, which is obtained by setting the non-homogeneous part of the equation to zero. In other words, the CF satisfies the differential equation with the non-homogeneous part set to zero. The CF is determined by solving the characteristic equation of the homogeneous part of the differential equation.

The particular integral (PI) is a particular solution of the non-homogeneous part of the differential equation. It is obtained by finding a function that satisfies the differential equation with the non-homogeneous part included. The PI can be determined by using methods such as the method of undetermined coefficients or the variation of parameters.

The general solution of the differential equation is the sum of the complementary function and the particular integral. The general solution represents all possible solutions of the differential equation.

To summarise, the complementary function is the solution of the homogeneous part of the differential equation, while the particular integral is a particular solution of the non-homogeneous part of the differential equation. The general solution is the sum of the complementary function and the particular integral, representing all possible solutions of the differential equation.

**Describe Transient response of R-L circuit having DC Excitation**

A DC-excited R-L (resistor-inductor) circuit is a circuit consisting of a resistor (R) and an inductor (L) connected in series to a DC voltage source. When the voltage source is connected to the circuit, the current through the circuit does not change instantaneously due to the inductor’s property of opposing changes in current. This behaviour is known as the transient response of the circuit.

The transient response of an R-L circuit can be described using the time constant of the circuit, which is equal to the ratio of the inductance and the resistance (L/R). The time constant represents the time it takes for the current in the circuit to reach approximately 63.2% of its final value.

Initially, the current in the circuit is zero, and the voltage across the inductor is equal to the voltage source. As time passes, the inductor begins to oppose the increase in current, and the voltage across the inductor decreases. Eventually, the current in the circuit reaches its steady-state value, which is equal to the voltage source divided by the resistance (I = V/R).

The transient response of the R-L circuit is the behaviour of the circuit during the time it takes for the current to reach its steady-state value. During this time, the voltage across the inductor decreases, and the voltage across the resistor increases, as the inductor stores and releases energy. The transient response is a decaying exponential function, with a time constant of L/R, representing the rate at which the current approaches its steady-state value.

Overall, the transient response of an R-L circuit with DC excitation is characterised by the time constant of the circuit and the decaying exponential behaviour of the current during the time it takes to reach its steady-state value.

** ****Describe Transient response of R-L circuit having AC Excitation**

A R-L (resistor-inductor) circuit with AC excitation is a circuit consisting of a resistor (R) and an inductor (L) connected in series to an AC voltage source. When the AC voltage source is connected to the circuit, the current through the circuit changes periodically with the frequency of the AC source. The behaviour of the circuit during the time it takes for the current to reach its steady-state value is called the transient response of the circuit.

The transient response of an R-L circuit with AC excitation can be described using the time constant of the circuit, which is equal to the ratio of the inductance and the resistance (L/R). The time constant represents the time it takes for the current in the circuit to reach approximately 63.2% of its final value.

Initially, the current in the circuit is zero, and the voltage across the inductor is equal to the AC voltage source. As time passes, the inductor begins to oppose the increase in current, and the voltage across the inductor decreases. Eventually, the current in the circuit reaches its steady-state value, which depends on the frequency of the AC source.

The transient response of the R-L circuit is the behaviour of the circuit during the time it takes for the current to reach its steady-state value. During this time, the voltage across the inductor decreases, and the voltage across the resistor increases, as the inductor stores and releases energy. The transient response is a decaying exponential function, with a time constant of L/R, representing the rate at which the current approaches its steady-state value.

Overall, the transient response of an R-L circuit with AC excitation is characterised by the time constant of the circuit and the decaying exponential behaviour of the current during the time it takes to reach its steady-state value. The behaviour of the circuit depends on the frequency of the AC source and the initial conditions of the circuit.

**Recall the concepts of Time constants and plot the Normalised curve**

The time constant of an R-L circuit is a measure of the circuit’s response to changes in current. The time constant, denoted by the symbol τ (tau), is defined as the product of the resistance (R) and the inductance (L), divided by the voltage source frequency (f), or τ = L/Rf.

The time constant represents the time it takes for the current in the circuit to reach approximately 63.2% of its final value during the transient response. The time constant is a critical parameter in the analysis of the transient response of R-L circuits.

The normalised curve of the transient response of an R-L circuit is a graph of the current in the circuit as a function of time, divided by the final steady-state current value. The normalised curve is a dimensionless representation of the transient response, allowing for easy comparison of the behaviour of circuits with different time constants.

The normalised curve is a decaying exponential function, with a time constant equal to one. The equation for the normalised curve is given by i(t)/i(∞) = 1 – e^(-t/τ), where i(t) is the current in the circuit at time t, i(∞) is the steady-state current value, and τ is the time constant of the circuit.

The normalised curve is useful for analysing the behaviour of R-L circuits during the transient response, as it provides insight into the rate at which the current approaches its steady-state value. The normalised curve can also be used to compare the transient response of R-L circuits with different time constants, as the behaviour of the circuits can be easily visualised and compared.

**Determine Transient response of R-C circuit having DC Excitation **

Assume that the capacitor is initially uncharged (i.e., it acts as a short circuit), and that a DC voltage Vdc is suddenly applied to the circuit at time t=0.

To determine the transient response of the circuit, we can use the following steps:

- Determine the time constant of the circuit, which is equal to the product of the resistance and the capacitance: tau = R*C.
- Calculate the initial voltage across the capacitor, which is equal to zero at time t=0.
- Calculate the final voltage across the capacitor, which is equal to the applied DC voltage V
_{dc}. - Write down the differential equation governing the voltage across the capacitor as a function of time:

V_{c}(t) = V_{c}(0) + (V_{dc}– V_{c}(0)) * (1 – exp(-t/tau))

where V_{c}(0) is the initial voltage across the capacitor (which is equal to zero), and exp() is the exponential function.

- Use this equation to plot the time evolution of the voltage across the capacitor. The plot should show that the voltage across the capacitor rises exponentially from zero to the applied DC voltage, with a time constant equal to R*C.

This exponential rise of voltage across the capacitor is the transient response of the circuit. It occurs because the capacitor acts as an open circuit at DC, and initially there is no voltage across it. As the capacitor charges up, the voltage across it increases until it reaches the applied DC voltage.

**Determine Transient response of R-C circuit having AC Excitation**

To determine the transient response of an R-C circuit with AC excitation, you can follow these steps:

- Write the differential equation that describes the circuit. For an R-C circuit with AC excitation, the differential equation is:

RC dV(t)/dt + V(t) = Vm sin(ωt)

where Vm is the amplitude of the AC voltage, ω is the angular frequency, R is the resistance, C is the capacitance, and V(t) is the voltage across the capacitor. - Solve the differential equation to obtain the general solution. To do this, you can use the method of integrating factors. The integrating factor for this equation is e^(t/RC), so multiply both sides by e^(t/RC):

e^(t/RC) RC dV(t)/dt + e^(t/RC) V(t) = e^(t/RC) V_{m}sin(ωt)

The left-hand side can be written as the derivative of the product e^(t/RC) V(t):

d/dt(e^(t/RC) V(t)) = e^(t/RC) RC dV(t)/dt + e^(t/RC) V(t)

Substituting this into the equation above, we get:

d/dt(e^(t/RC) V(t)) = e^(t/RC) V_{m}sin(ωt)

Integrating both sides gives:

e^(t/RC) V(t) = -V_{m}/ω cos(ωt) + A

where A is an integration constant that can be determined from the initial conditions. - Apply the initial conditions to determine the value of A. For example, if the capacitor is initially uncharged (V(0) = 0), then we have:

e^(0/RC) V(0) = -V_{m}/ω cos(0) + A

A = V_{m}/ω

So the general solution becomes:

V(t) = V_{m}e^(-t/RC) + Vm/ω cos(ωt)

This is the transient response of the R-C circuit with AC excitation.

Note that this solution has two terms: the first term decays exponentially with a time constant of RC, while the second term oscillates at the angular frequency ω. The first term represents the initial discharge of the capacitor, while the second term represents the AC voltage that is being applied to the circuit. The transient response dies out as time goes on, and the circuit reaches a steady-state response where the capacitor is fully charged and the voltage across it oscillates at the same frequency as the applied AC voltage.

**Recall the concepts of Time constants and plot the Normalised curve**

The time constant of an RC circuit is the time it takes for the capacitor to charge or discharge to 63.2% of its final voltage in response to a step input. Mathematically, the time constant τ is given by:

τ = RC

where R is the resistance of the circuit and C is the capacitance.

The time constant is an important parameter of an RC circuit because it determines how quickly the circuit responds to changes in the input voltage. A smaller time constant means that the circuit responds more quickly, while a larger time constant means that the circuit responds more slowly.

The normalised curve of an RC circuit is a plot of the voltage across the capacitor as a function of time, normalised by the final voltage of the capacitor. The normalised voltage V/V_{0} is given by:

V/V_{0} = 1 – e^(-t/τ)

where V is the voltage across the capacitor at time t, V_{0} is the final voltage across the capacitor, and τ is the time constant of the circuit.

The normalised curve starts at zero and rises exponentially towards unity, with a time constant of τ. After a time of approximately 5τ, the voltage across the capacitor reaches 99.3% of its final value. The normalised curve can be used to compare the response of different RC circuits with different time constants.

**Recall Second Order Differential Equation**

A second-order differential equation is an equation that describes the behaviour of a system that depends on two variables and their derivatives with respect to a single independent variable. The general form of a second-order differential equation is:

a(d^2y/dx^2) + b(dy/dx) + cy = f(x)

where y is the dependent variable, x is the independent variable, a, b, and c are constants, and f(x) is a function of x.

The term “second order” refers to the highest derivative in the equation, which is the second derivative d^2y/dx^2. The equation can be classified based on the values of the constants a, b, and c. Depending on the values of these constants, the equation can be classified as:

- Homogeneous or non-homogeneous: If f(x) = 0, the equation is homogeneous, otherwise, it is non-homogeneous.
- Linear or nonlinear: If the equation can be written in the form shown above, it is linear, otherwise, it is nonlinear.
- Constant coefficient or variable coefficient: If a, b, and c are constants, the equation has constant coefficients, otherwise, it has variable coefficients.

Solving a second-order differential equation involves finding the general solution that satisfies the equation for all values of the independent variable. The solution typically involves finding the roots of the characteristic equation, which is obtained by setting the homogeneous equation (i.e., f(x) = 0) equal to zero. The roots of the characteristic equation determine the form of the general solution, which may involve exponential functions, trigonometric functions, or a combination of both. The general solution can be further modified to include any particular solution that satisfies the non-homogeneous equation (i.e., f(x) ≠ 0).

Second-order differential equations are commonly used to model a wide range of physical phenomena, such as the motion of objects under the influence of forces, the behaviour of electrical circuits, and the vibrations of mechanical systems.

**Determine Transient response of R-L-C circuit having DC Excitation.**

Assume that the inductor is initially uncharged (i.e., it acts as a short circuit), and that a DC voltage Vdc is suddenly applied to the circuit at time t=0.

To determine the transient response of the circuit, we can use the following steps:

- Determine the natural frequency of the circuit, which is given by:

w_{n}= 1 / sqrt(L*C)

where sqrt() is the square root function.

- Determine the damping ratio of the circuit, which is given by:

zeta = R / (2 * sqrt(L*C)) - Determine the type of response, which depends on the value of the damping ratio zeta:
- If zeta < 1 (underdamped case), the response is oscillatory, and the voltage across the capacitor will oscillate back and forth before settling to its steady-state value.
- If zeta = 1 (critically damped case), the response is non-oscillatory, and the voltage across the capacitor will approach its steady-state value as quickly as possible without overshooting.
- If zeta > 1 (overdamped case), the response is non-oscillatory, and the voltage across the capacitor will approach its steady-state value more slowly than in the critically damped case.

- Calculate the initial voltage across the capacitor and the initial current through the inductor, which are equal to zero at time t=0.
- Calculate the final voltage across the capacitor, which is equal to the applied DC voltage Vdc.
- Write down the differential equation governing the voltage across the capacitor as a function of time:

V_{c}(t) = V_{dc}+ (0 – V_{dc}) * exp(-t / (zeta * tau)) * cos(omega_d * t)

**Determine Transient response of R-L-C circuit having AC Excitation**

An RLC circuit is a circuit that contains a resistor (R), an inductor (L), and a capacitor (C) connected in series or in parallel. When an AC voltage source is applied to an RLC circuit, the circuit responds with a transient response and a steady-state response.

The transient response is the response of the circuit to the initial conditions, such as the initial charge on the capacitor and the initial current through the inductor. The steady-state response is the response of the circuit after the transient response has died out, and the circuit has reached a steady-state condition.

To determine the transient response of an RLC circuit with AC excitation, we can write the second-order differential equation that describes the behavior of the circuit. The differential equation for an RLC circuit with AC excitation can be written as:

L(d^2i/dt^2) + R(di/dt) + (1/C)i = (1/C)Vsin(ωt)

where i is the current through the circuit, V is the amplitude of the AC voltage source, ω is the frequency of the AC voltage source, and the other variables have their usual meanings.

The transient response of the circuit depends on the initial conditions, which can be determined by solving the differential equation for i(t) using the appropriate boundary conditions. The solution to the differential equation is a sum of the homogeneous and particular solutions, which can be written as:

i(t) = i_{h}(t) + i_{p}(t)

where i_{h}(t) is the homogeneous solution and i_{p}(t) is the particular solution.

The homogeneous solution describes the natural response of the circuit, which depends on the values of L, R, and C. The homogeneous solution can be written as:

i_{h}(t) = A_{1}e^(r_{1}t) + A_{2}e^(r_{2}t)