Fundamentals of calculating switching operations

Switching operations are calculated on the basis of differential equations (DGL).

Applying Kirchhoff’s laws to circuits with energy storage devices leads to equations for voltages and currents in the respective components. This means that the respective instantaneous values of voltages and currents are not sufficient to describe the state of the system.

The time courses of voltages and currents during switching operations can be determined by solving their respective differential equations.

1 Definitions

Table 1 provides an overview of important terms with abbreviated definitions used here. Certain terms are explained and discussed in more detail in the following sections. In particular, the definitions used here for transients, compensation, settling and switching processes

are explained in section 1.2 in contrast to the definitions found in other sources and the reasons for the choice of definitions used here are given.

The table serves as a reference for the use of terms in this module and can be used as a glossary.

1.1 Explanations

CONDITIONS:

Balanced: Describes a steady state after a balancing process.

Settled in: Describes a steady state immediately after a transient process. The term is also generally used for steady states after compensation processes in which no oscillation occurs and which are also independent of the direction of energy flow.

Stationary: State description for time-constant variables (e.g. with DC voltage supply) or time-periodic variables (e.g. with AC voltage) [?, p. 362]

SIZES:

The quantities decay constant, damping constant, natural frequency, resonance frequency and time constant are important parameters for describing the oscillatory behaviour of systems. They are explained in more detail in Chapter ?? using the example of an RLC series resonant circuit.

CASES:

The distinction between aperiodic case, aperiodic limiting case, periodic case in linear, time-invariant, oscillatory systems is explained in Chapter ?? using the example of an RLC series resonant circuit.

1.2 Discussion and delimitation

This section is not essential for understanding the module content and should be regarded as supplementary information. As some of the terms listed in Table 1 are used differently in the literature, the choice of definitions is justified and defined here.

PROCEDURES:

Compensation process, m.

• Weißgerber: Process from steady state after intervention to steady state
• Hagmann: Process from change to steady state
• here, based on Hagemann: Process when a specific steady state is desired

Like Hagmann, except that quasi „interrupted“ compensation processes are also referred to as compensation processes. This definition includes applications in power electronics such as step-up and step-down converters, assuming real components (\(R>0\)).

Switching operation, m.

• Weißgerber: Compensation process after switching[?, p. 1]
• Hagmann: Synonym: compensation process, immediately after circuit change [sic!], transitions to steady state[?, p. 362]
• here: Compensation process triggered by switching (based on Hagmann)

In this module, the term ‘switching operation’ refers only to the process after (!) switching, which is triggered by this. The definition used here allows compensation processes that are independent of switching to be excluded.

Oscillation process, m.

Refers here to exergonic compensation processes, i.e. compensation processes that take place with positive net -energy dissipation. According to the literal meaning of the word, a settling process is the counterpart to a settling-in process.

Settling-in process, m.

Here, in the narrower sense, refers to endosmic compensation processes, i.e. compensation processes that take place with a positive net energy supply. Usually used here in the broader sense as a synonym for all compensation processes. Refers here, as usual, to both oscillatory and non-oscillatory processes. [Cf. the words oscillate, swing] The definition has been chosen to facilitate a more detailed classification of possible compensation processes. Transient, w.

Here considered synonymous with the process of compensation, freely translatable as „transition“from Latin. Comparison transient (adj.) from Latin transire English (to) pass (through), opposite of persistent (adj.) from Latin persistere English to persist.

2 Calculation methods for switching operations

There are basically two methods available for calculating switching operations. Both methods are based on solving the differential equation (DE) of a circuit.

The DE is solved as follows:

• in the time domain (by decomposing into transient and steady states) or
• in the image domain (using Laplace transformation).

Figure 1 compares the calculation process in the time and image domains as a flowchart.

The naming and presentation of both methods are based on [?, p. 51], but can also be found in a similar form in other textbooks on electrical engineering, control engineering and systems theory.

Solving the ODE in the time domain is the generally applicable method and is covered in detail in this module. The computational effort can vary greatly depending on the complexity of the ODE. The method is based on decomposing the system into a transient and a steady state.

Solving the ODE in the frequency domain using the Laplace transform is a special method that can be applied to linear, time-invariant systems with energy storage. It is suitable for vanishing initial conditions (see chapter 4). In some cases, the method can reduce the computational effort, as the transformed ODE can be solved algebraically. The method is not discussed further in this module.

The idea of switching to the image domain can be found again in Section 3 when setting up a differential equation and in Section 4.3 when applying complex AC calculation (see Module 7) to determine steady-state conditions.

3 Differential equations

The calculation of switching operations begins with the formulation of differential equations (ODG). The ODG for voltages and currents are based in principle on Kirchhoff’s rules: \begin {align} \text {Knot rule:} & \qquad \sum _{j=1}^{n} i_j = 0 \qquad \text {mit $n$ Stromzweigen im Knoten}\label {eq:grundlagen:kirchhoff:knotenregel} \\ \text {Mesh rule:} & \qquad \sum _{k=1}^{m} u_k = 0 \qquad \text {mit $m$ Teilspannungen in Masche}\label {eq:grundlagen:kirchhoff:maschenregel} \end {align}

and the following current-voltage relationships (component equations) for \(R\), \(L\) and \(C\): \begin {align} &\text {Resistance:} & u_R(t) &= R \cdot i_R(t) \vphantom {\mathrm{d}t } &&\label {eq:grundlagen:bauteilgleichung:widerstand}\\ &\text {inductance:}& u_L(t) &= L \cdot \mathrm{d}t \, i_L(t) &&\label {eq:grundlagen:bauteilgleichung:induktivitaet}\\ &\text {Capacity:} & i_C(t) &= C \cdot \mathrm{d}t \, u_C(t) &&\label {eq:grundlagen:bauteilgleichung:kapazitaet} \end {align}

The most effective order in which to set up, transform and combine the equations depends on the respective circuit topology and the target variable.

Similar to solving differential equations (see section 2 and figure 1 on calculation methods), differential equations can be set up in both the time domain and the image domain.

The method in the time domain is the method discussed in more detail in this module. Figure 2 shows a schematic comparison of both approaches.

For simple circuits with a single mesh (pure series connection) or two individual nodes (pure parallel connection), the differential equations can be derived directly from Kirchhoff’s rules and the component equations (time domain). If necessary, the equation must be differentiated to obtain the required derivatives (or to remove integrals). For more complex networks, the computational effort is significantly greater.

One way to minimise the amount of calculation required is to use complex AC calculation methods (image area). This approach also works for DC circuits, as frequencies are not actually calculated. The aim is to set up the DGL by algebraically transforming the complex Kirchhoff’s rules. Here, the circuit elements are replaced by impedances or admittances, and the quantity to be investigated is determined, for example, by a complex voltage divider. The transformation to the differential equation is performed by back-transforming the complex quantities into the time domain. \(\mathrm {j}\omega \) is replaced by \(\frac {\mathrm {d}}{\mathrm {d} t}\).

The advantage of the approach in the image area is that complex voltage and current dividers can be used. This makes the approach particularly suitable for mixed series and parallel connections.

A similar approach would be the mesh flow method in matrix form with complex quantities. [See Module 7] However, this will not be considered further at this point.

3.1 Properties and general form

Only the following components are used in all of the circuits examined here: Capacitors, inductors, resistors, ideal voltage sources, ideal current sources, ideal electrical conductors and ideal switches. No other components such as transistors, diodes or operational amplifiers are used in the circuits.

To calculate switching operations, this module always assumes ideal components in isothermal, homogeneous, isotropic media. Other influences such as temperature dependencies, non-linear effects or similar factors are not taken into account.

The components listed are all linear and time-invariant. The same properties also apply to circuits that are composed solely of these components. This means that they are linear, time-invariant systems (LZI systems). Time invariance for switches applies with restrictions (in sections) for all time periods outside of switching times and without restrictions for all other components.

The following properties of the differential equations can be derived from the properties of the circuits:

• ordinarily (nur abhängig von einer Variable, hier der Zeit)
• linear (Linearität der Schaltungselemente)
• const. coefficients \(a_i=konst.\) (Zeitinvarianz der Schaltungselemente)
• inhomogeneous \(b(t)\neq 0\) (bei Anregung durch Spannungs- oder Stromquellen)
• homogeneous \(b(t)= 0\) (ohne Anregung durch Spannungs- oder Stromquellen)

For a considered circuit variable \(y\) as a function of time \(t\), this results in the following form for ordinary, inhomogeneous, linear ODEs with constant coefficients:

\begin {align} \label {eq:grundlagen:dgl:inhomogen} \sum _{i=0}^{n} a_i \cdot \frac {\mathrm{d} ^i y(t)}{\mathrm {d}t^i} = b(t)\\[2pt]\nonumber \text {mit} a_i, b(t) \in \mathbb {R}; i, n \in \mathbb {N} \end {align}

With the order \(n\), the disturbance function \(b(t)\) and the constant coefficients \(a_i\).

3.2 Order of differential equations

The structure of a circuit determines the form of the differential equations (DGLs) used to describe its system variables (currents, voltages). The order of the DGL corresponds directly to the number of energy storage devices that cannot be combined into a single energy storage device.[?, p. 4] Figure 3 shows several circuits and the order of the corresponding differential equations as examples.

The order of the differential equation determines the number of constants in the general solution of the differential equation.

4 Calculation method in the time domain

The calculation of switching operations in the time domain is based on the solution of differential equations (ODEs) for the circuit variables. In general, there are an infinite number of solutions for ODEs.

Knowledge of the initial conditions (IC) is important for limiting the general solution. The IC are values of circuit variables at a specific point in time \(t_0\) during the observation period (switching process). These are, for example, voltages or currents of individual components or their derivatives.

For ODEs of order n, n ABs are necessary to obtain a unique solution. If the ABs are not explicitly given, they can be derived from the circuit topology and the state immediately before switching, if necessary. A distinction must be made between which circuit variables can jump and which are continuous.

Since inductors and capacitors are energy storage devices, their behaviour at a point in time \(t\) depends on the energy supplied/removed up to that point.

In a brief comparison, with \(E = \int p(t) \,\mathrm {d}t = \int u(t) \cdot i(t) \,\mathrm {d}t\) and Eqs. ?? and ??: \begin {align} \textbf {Capacity:} && E_{el} &= \frac {1}{2} C \cdot u^2 &&\Rightarrow \qquad u = steadily && \text {(Electric en. storage)} \\[+4pt] \textbf {Inductance:} && E_{mag} &= \frac {1}{2} L \cdot i^2 &&\Rightarrow \qquad i = steadily && \text {(Magnetic en. storage)} \end {align}

Since energy can only move at a finite speed, continuity applies to \(u_C\) and \(i_L\): \begin {align} u_{C,t-} &= u_{C,t+} & i_{L,t-} &= i_{L,t+} \\ \text {mit} \lim _{\pm \infty \rightarrow t} u_C(t) &= u_{C,t\pm } & \lim _{\pm \infty \rightarrow t} i_L(t) &= i_{L,t\pm } \end {align}

This means that the voltage of a capacitance, just like the current of an inductance, is the same immediately before switching as it is at the beginning of the following switching process.

4.1 Decomposition of compensation processes and solution of inhomogeneous differential equations

The resulting time course of a variable \(y(t)\) can be broken down during a compensation process. in a volatile state \(y_f\) und in a steady state \(y_e\). \begin {equation} \label {eq:grundlagen:zerlegung} y(t) := y_{\mathrm {f}}(t) + y_{\mathrm {e}}(t) \end {equation} The steady state \(y_{\mathrm {e}}\) describes the stationary state that the variable \(y\) strives to reach. The transient state \(y_{\mathrm {f}}\), also known as the free state, represents the decaying part of \(y\):

\begin {align} \lim _{t \to \infty } y_{\mathrm {e}}(t) &= \lim _{t \to \infty } y(t) & y_{\mathrm {e}} &= \text {stationary} \label {eq:grundlagen:eingeschwungen}\\ \lim _{t \to \infty } y_{\mathrm {f}}(t) &= 0 & y_{\mathrm {f}} &= y - y_{\mathrm {e}} \label {eq:grundlagen:fluechtig} \end {align}

Figure 4 shows an example of a compensation process with decomposition of a variable \(y(t)\) into a transient and steady state. The graph corresponds to the voltage curve of a capacitor when charged with DC voltage across an ohmic resistor, as in Example ??.

Since the quantity \(y(t)\) according to equation ?? transitions to \(y_e(t)\) for \(t \to \infty \), \(y_e(t)\) must also satisfy the differential equation from equation ?? including the disturbance term \(b\): \begin {align} \label {eq:grundlagen:dgl:eingeschwungen} \sum _{i=0}^{n} a_i \cdot \frac {\mathrm{d} ^i(t)}{\mathrm {d}t^i}\,y_{\mathrm {e}} &= b(t) \end {align}

If we subtract the ODE for \(y_e\) from Eq. ?? from the general inhomogeneous ODE from Eq. ??, we obtain the homogeneous ODE for the transient state using the decomposition according to Eq. 1: \begin {align} \label {eq:grundlagen:dgl:fluechtig} \sum _{i=0}^{n} a_i \cdot \frac {\mathrm{d} ^i(t)}{\mathrm {d}t^i}\,y_{\mathrm {f}} = 0 \end {align}

Mathematically, the decomposition of the time course \(y(t)\) into a transient and steady state corresponds to the decomposition of the solution of the inhomogeneous ODE \(y\) into the solution of the homogeneous ODE \(y_{\mathrm {h}}\) (without disturbance term) and into a particular solution \(y_{\mathrm {p}}\) of the inhomogeneous ODE with:

\begin {align} \text {homogeneous solution}\qquad y_{\mathrm {h}} &= y_{\mathrm {f}} \qquad \text {volatile state}\\ \text {particular solution}\qquad y_{\mathrm {p}} &= y_{\mathrm {e}} \qquad \text {settled state} \end {align}

The breakdown of compensation processes into a steady state and a transient state can be found in a similar form in the textbooks by Albach[?], Hagmann[?] and Weißgerber[?].

4.2 Calculation of the homogeneous solution

For solving ordinary, linear, homogeneous differential equations with constant coefficients of the form:

\begin {equation} \label {eq:grundlagen:dgl:homogen} \sum _{i=0}^{n} a_i \cdot \mathrm{d}t [i]\, y_{\mathrm {h}} = 0 \end {equation}

the approach of an exponential function is generally used:

\begin {equation} \label {eq:grundlagen:dglexponentialansatz} y_{\mathrm {h}}(t) = K \cdot \mathrm {e}^{\lambda t} \end {equation}

Here, \(K\) is a constant and \(\lambda \) is the solution of the characteristic polynomial and is also referred to as the eigenvalue. By inserting the exponential approach in 2, differentiating and factoring out: \begin {align} \sum _{i=0}^{n} a_i \cdot \mathrm{d}t [i]\, K \cdot \mathrm {e}^{\lambda t} &= 0 \nonumber \\[2pt] \left (\sum _{i=0}^{n} a_i \cdot \lambda ^i \right ) K \cdot \mathrm {e}^{\lambda t} &= 0 \nonumber \\[4pt] \end {align}

the characteristic polynomial can be established: \begin {align} \sum _{i=0}^{n} a_i \cdot \lambda ^i &= 0 \end {align}

The maximum number of eigenvalues is given by the order of a differential equation. Since the coefficients \(a_i\) are real, the eigenvalues \(\lambda _i\) must also be real, or complex conjugate in pairs.

Condition: In order to satisfy the homogeneous differential equation in Equation 2, the real part of all eigenvalues must be negative \(\Re \{\underline {\lambda }_i\}<0\). Only then will the exponential terms from the approach in Equation 3 converge for \(t \to \infty \).

The general homogeneous solution \(y_{\mathrm {h}}\) of an nth-order ODE is the linear combination of n linearly independent solutions of the homogeneous ODE.

The form of the solution depends on whether there are distinct eigenvalues, identical eigenvalues (multiple zeroes of the characteristic polynomial) or complex conjugate eigenvalues in pairs.

If all eigenvalues \(\lambda _i\) are different, the homogeneous solution is: \begin {equation} \label {eq:grundlagen:homogeneloesung:lambdaungleich} y_{\mathrm {h}}(t) = \sum _{i=1}^{n} K_i \cdot \mathrm {e}^{\lambda _i t} \qquad \text {für} \lambda _i \neq \lambda _j \ \forall \ i,j \end {equation} For complex conjugate eigenvalues in pairs, this solution can be reformulated. [Derivation in section ??.] For \(\lambda _1 = \lambda _2^* = a + \mathrm {j} b \in \mathbb {C}\), the homogeneous solution is:

\begin {equation} \label {eq:grundlagen:homogeneloesung:lambdakomplex} \begin {aligned} y_{\mathrm {h}}(t) &= K_1 \cdot \mathrm {e}^{a \cdot t} \cdot \cos (b \cdot t) + K_2 \cdot \mathrm {e}^{a \cdot t} \cdot \sin (b \cdot t) \\ &= C_0 \cdot \mathrm {e}^{a \cdot t} \cdot \cos (b \cdot t + \varphi _0) \end {aligned} \end {equation} If all eigenvalues \(\lambda _i\) are equal, the homogeneous solution is: \begin {equation} \label {eq:grundlagen:homogeneloesung:lambdagleich} y_{\mathrm {h}}(t) = \sum _{i=1}^{n} K_i \cdot t^{i-1} \cdot \mathrm {e}^{\lambda t} \qquad \text {für} \lambda = \lambda _i \ \forall \ i \end {equation} For mixed zeroes, the solution is obtained from a combination of the above solutions, as shown in Example ??.

LLinear independent solutions of a differential equationgrundlagen:linearunabhaengige_loesungen_homo_dgl This example is taken almost unchanged from [?, p. 241]. We are looking for the general solution to the following 5th order homogeneous differential equation: \begin {equation*} \mathrm{d}t [5] y(t) + 7 \mathrm{d}t [4] y(t) + 26 \mathrm{d}t [3] y(t) + 62 \mathrm{d}t [2] y(t) + 8 \mathrm{d}t y(t) + 75 y(t) = 0 \end {equation*} The characteristic polynomial: \begin {equation*} \lambda ^{5} + 7 \cdot \lambda ^{4} + 26 \cdot \lambda ^{3} + 62 \cdot \lambda ^{2} + 85 \cdot \lambda + 75 = 0 \end {equation*} has one simple and two double, conjugate, complex zeros: \begin {equation*} \lambda _1 = -3, \lambda _2 = \lambda _3 = -1+\mathrm {2j}, \lambda _4 = \lambda _5 = -1-\mathrm {2j} \end {equation*} Using the approaches from Eq. 4, 5 and 6, the general solution is: \begin {equation*} y_{\mathrm {h}}(t) = K_1 \cdot \mathrm {e}^{-3 t} + (K_2 + K_3 \cdot t) \cdot \mathrm {e}^{-t} \sin (2t) + (K_4 + K_5 \cdot t) \cdot \mathrm {e}^{-t} \cos (2t) \end {equation*}

4.3 Calculation of the particular solution

Various methods can be chosen to calculate a particular solution \(y_{\mathrm {p}}\) of an inhomogeneous ODE. One option is to choose an approach similar to the suggestion.

Specifically for switching operations, \(y_{\mathrm {p}}\) corresponds to the steady state as derived in section 4.1. For linear, time-invariant switching elements, this implies:

• With DC excitation (constant quantities), the particular solution is also a constant quantity.
• With AC excitation (alternating quantities), the particular solution is also an alternating quantity.

The determination can be carried out using static network calculation methods. For this purpose, Kirchhoff’s rules, the equations for the components \(R\), \(L\) and \(C\) and the resulting current and voltage divider rules are sufficient. For alternating quantities, the complex alternating current equivalents of the methods listed are used.

Transformation of DGLs of sinusoidal quantities into the complex number space:

Since, in the case of AC excitation, the ODE ?? applies to the steady state for alternating quantities, the ODE can be transformed into the complex number space. The procedure corresponds to the transformation of the differential current and voltage relationships for \(R\), \(L\) and \(C\) to their complex current and voltage relationships using their respective impedance \(\underline {Z}\) and admittance \(\underline {Y}\).

A sinusoidal quantity \(y(t)\) with amplitude \(\hat {Y}\) and a zero phase angle \(\varphi \) can be described with a complex amplitude pointer (phasor) \(\underline {\hat {Y}}\) as follows [see Module 7]:

\begin {align*} y(t) &= \hat {Y} \cdot \cos (\omega t + \varphi ) = \Re \left \{ \underline {\hat {Y}} \cdot \mathrm {e}^{\mathrm {j} \omega t} \right \} & &\Longleftrightarrow & \underline {\hat {Y}} &= \hat {Y} \cdot \mathrm {e}^{\mathrm {j} \varphi } \vphantom {\bigg |}\\ \end {align*}

Derivatives with respect to time \(\mathrm{d}t \) are replaced by \(\mathrm {j} \omega \) in the complex number space: \begin {align*} \mathrm{d}t y(t) &= \hat {Y} \cdot \omega \cdot \cos (\omega t + \varphi + \frac {\pi }{2}) & &\Longleftrightarrow & \mathrm {j} \omega \cdot \underline {\hat {Y}} &= \hat {Y} \cdot \omega \cdot \mathrm {e}^{\mathrm {j} \left (\varphi + \frac {\pi }{2}\right )} \end {align*}

The factor \(\mathrm {j}\omega \) corresponds to the prefactor \(\omega \) and a phase shift of \(1/{\pi }{2}\) when transformed back. This means that, in general, for differential equations of sinusoidal quantities (index \(ac\)) of the form in equation ??: \begin {equation} \sum _{i=0}^n a_i \cdot \left (\mathrm{d}t \right )^i y_{ac} = b_{ac} \qquad \Longleftrightarrow \qquad \sum _{i=0}^n a_i \cdot \left (\mathrm {j}\omega \right )^i \cdot \underline {\hat {Y}} = \underline {\hat {B}} \label {eq:grundlagen:dgl:komplex} \end {equation} Alternatively, the imaginary part of complex amplitude vectors can also be used to represent sinusoidal alternating quantities. [Compare Module 7 – Periodic Quantities]

4.4 Summary

Calculating compensation processes in the time domain consists, on the one hand, of setting up a suitable differential equation and, on the other hand, of solving the differential equation that has been set up. Note box 16 summarises the calculation of compensation processes in the time domain in five steps. The steps are closely based on the procedure described by Weißgerber [?, p. 6].

The order of steps 2 and 3 is, in principle, arbitrary. In mathematics, the order shown is more common with the determination of the general homogeneous solution before the solution of the particular solution(s). From an electrical engineering point of view, it may be advisable to first determine the steady state (particular solution).