Calculate switching operations in the time domain

This chapter examines switching processes in electrical circuits in the time domain. A distinction is made between switching processes for direct current and alternating current. The examination covers RC and RL elements, as well as RLC elements.

1 With direct voltage

This chapter calculates and illustrates the charging and discharging processes of capacitors and inductors in series with a resistor at DC voltage.

Switching processes with sinusoidal excitation are covered in Chapter 2.

1.1 In terms of capacity

Example ?? shows the calculation method for determining the voltage curve \(u_C(t)\) of a capacitance \(C\) that is initially discharged when charging with DC voltage for \(t \geq 0\) across a resistor \(R\) and the time curve of the charging current \(i(t)\). The calculation method used corresponds to the procedure for ODE in the time domain according to note ??.

The results show the exponential convergence of \(u_C(t)\) to the excitation \(U_q\): \begin {align} u_C(t) &= U_q \cdot \left ( 1 - \mathrm {e}^{-1/{t}{\tau }} \right ) \end {align}

and the exponential decay of \(i(t)\) and \(u_R(t)\) respectively from their maximum values \(U_q/R\) and \(U_q\): \begin {align} i(t) &= \frac {U_q}{R} \cdot \mathrm {e}^{-1/{t}{\tau }}\\[4pt] u_R(t) &= U_q \cdot \mathrm {e}^{-1/{t}{\tau }} \end {align}

until the capacitance is charged. The time constant \(\tau \) determines the steepness of the exponential curves. For the RC element, \(\tau = RC\). The unit for \(\tau \) is the same as that for time.

As shown in Figure 1, the tangent of \(u_C\) intersects the terminal value \(U_q\) at the point \(u_C(t=0)\) at the time \(t=\tau \). In general, at every point of exponential approximation/decline, the tangent reaches the terminal value after \(\tau \).

Table 1 shows typical values for exponential approximations/declines after multiples of \(\tau \). The values are relative to the maximum value (starting or target limit value). Such a curve is considered to be „stabilised“ after \(5\,\tau \).

Example ?? shows the calculation method for determining the voltage curve \(u_C(t)\) of a capacitance \(C\) when discharging across a resistor \(R\) and the corresponding time curves. Similar to the charging process, this results in exponential curves with the same time constant but with different start and end values.

The calculation method in examples ?? and ?? is essentially identical. The only differences between the charging and discharging processes are the initial condition \(u_C(0)\) (at switching time \(t=0\)) and the boundary condition \(u_{C,\mathrm {e}}\) (for \(t \to \infty \)). This results in different particular solutions and different constants \(K\) in the general solution.

In general terms, the voltage curve \(u_C\) of a capacitance \(C\) during charging/discharging with DC voltage across a resistor \(R\) from a switching point \(t_0\) can be described by: \begin {equation} \label {eq:rc:charge:dc:allg:parthom} u_C(t-t_0) = \underbrace { u_{C,\mathrm {e}} \vphantom {\Big |}}_{u_{C,\mathrm {p}}} + \underbrace { \big (u_C(t_0)-u_{C,\mathrm e}\big ) \cdot \mathrm {e}^{-\frac {t-t_0}{\tau }} \vphantom {\Big |}}_{u_{C,\mathrm {h}}} \end {equation} Rewritten slightly, the voltage curve can also be described quite clearly as follows: \begin {equation} \label {eq:rc:charge:dc:allg:startend} u_C(t-t_0) = \underbrace {u_C(t_0) \vphantom {\Big |}}_{\text {Initial value}} + \big (\underbrace {u_{C,\mathrm e} - u_C(t_0) \vphantom {\Big |}}_{\text {Difference}}\big ) \cdot \bigg ( \underbrace { 1- \mathrm {e}^{-\frac {t-t_0}{\tau }} \vphantom {\Big |}}_{\text {\clap {Exp. approximation}}} \bigg ) \end {equation} The general solution in equation 1 or 2 only applies to excitations of equal magnitude. The solution is typical for lossy, first-order linear systems (with an energy storage device), which is why it also occurs in a similar form in RL elements.

Digression: First-order ODE and time constant \(\tau \)

The exponential form is typical for first-order linear systems with energy storage and results from the solution of the linear, ordinary first-order differential equation with constant coefficients. By transformation (\(a_0=1\)), the time constant \(\tau \) can be read directly from the differential equation (\(a_1=\tau \)).: \begin {equation} \label {eq:dgl:tau} \tau \cdot \mathrm{d}t \,y(t) + y(t) = b(t) \end {equation} This allows the homogeneous solution to be specified directly with: \begin {equation} y_{\mathrm {h}}(t) = K\cdot \mathrm {e}^{-\frac {t}{\tau }} \end {equation} The time constant is a measure of the speed at which a system responds to a change.

RC element DC charging processrc:chargedc Determination of voltage \(u_C\) and charging current \(i\) of capacitance \(C\) when charging via resistor \(R\) with DC voltage \(U_q\) for \(t \geq 0\). The capacitance is discharged at the start.

\begin {equation*} \begin {aligned} \text {given:}&& &i(0),\ U_q,\ R,\ C \\ \text {wanted:}&& &i(t), u_L(t) \end {aligned} \end {equation*}

1.) Set up ODE: von \(u_C(t)\) for \(t>0\)

Based on the mesh equation, \(u_R\) is expressed as a function of \(u_C\):\begin {align*} &\text {Mesh}:&&& u_R + u_C &= U_q&&& &\text {with} u_R = R \cdot i \\ &&&& R \cdot {\color {red} i} + u_C &= U_q&&& &\text {with} i = C \cdot \mathrm{d}t u_C \\ &\text {ODE}:&&& R \cdot C \cdot \mathrm{d}t u_C + u_C &= U_q \\[-15pt]&\hphantom {R1ooooooo}&&&\hphantom {L3oooooooooooooooooooo}&\hphantom {R3ooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

2.) Homogeneous solution, gen.: (fleeting) (without interference)\begin {align*} &\text {ODE}_{h}:&&& RC \cdot \mathrm{d}t u_{C,\mathrm {h}} + u_{C,\mathrm {h}} &= 0 &&& &\text {Approach:} u_{C,\mathrm {h}} = K \cdot \mathrm {e}^{\lambda t}\\ &&&& K \cdot (RC \cdot \lambda \mathrm {e}^{\lambda t}\ +\ \mathrm {e}^{\lambda t}) &= 0 \\ &\text {char. Pol.:}&&& RC \cdot \lambda + 1 &= 0 &&& &\Rightarrow \lambda = -\frac {1}{\tau } = -\frac {1}{RC}\\ &\text {hom. Solu.}:&&& u_{C,\mathrm {h}} = K \cdot \mathrm {e}^{-1/{t}{\tau }}&&&& &\text {with} \tau = RC \\[-15pt]&\hphantom {R1ooooooo}&&&\hphantom {L3oooooooooooooooooooo}&\hphantom {R3ooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

3.) Particular solution: (settled in) (for \(t \to \infty \))\begin {align*} &\text {ODE}_{p}:&&& \cancel {RC \cdot \mathrm{d}t u_{C,p}} + u_{C,p} &= U_q &&& &\text {with} u_{C,p} = const. \\ &\text {part. Lsg.}:&&& u_{C,p} &= U_q \\[-15pt]&\hphantom {R1ooooooo}&&&\hphantom {L3oooooooooooooooooooo}&\hphantom {R3ooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

4.) Inhomogeneous solution, gen.: (Superposition)\begin {align*} &&&& u_C(t) &= u_{C,\mathrm {h}}(t) + u_{C,p}(t) &&& &u_C(t)= u_{C,f}(t) + u_{C,e}(t) \\ &\text {general sol.:}&&& u_C(t) &= K \cdot \mathrm {e}^{-1/{t}{\tau }} + U_q \\[-15pt]&\hphantom {R1oooooooooooo}&&&\hphantom {L3ooo}&\hphantom {R3oooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

5.) Determine constants: (Set initial conditions)\begin {align*} &\text {initial condition}&&& u_C(0) &= K + U_q \overset {!}{=} \frac {0}{1} &&& &\Rightarrow K = -U_q \\ &&&& u_C(t) &= -U_q \cdot \mathrm {e}^{-1/{t}{\tau }} + U_q \\ &\text {special general solu.:}&&& u_C(t) &= U_q \left ( 1 - \mathrm {e}^{-1/{t}{\tau }} \right ) \\[-15pt]&\hphantom {R1oooooooooooo}&&&\hphantom {L3ooo}&\hphantom {R3oooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

Determine current \(i\) from voltage \(u_C\):\begin {align*} &\text {Current:}&&& i &= \frac {U_q}{R} \cdot \mathrm {e}^{-1/{t}{\tau }} &&& &\text {with} i = C \cdot \mathrm{d}t u_C \\[-15pt]&\hphantom {R1oooooooooooo}&&&\hphantom {L3ooo}&\hphantom {R3oooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

RC element discharge processrc:discharge Determination of voltage \(u_C\) and current \(i_C\) of a capacitance \(C\) when discharging through a resistor \(R\) for \(t \geq 0\). The capacitance is initially charged to \(U_q\).

\begin {equation*} \begin {aligned} &\text {given:}& &u_C(0),\ U_q,\ R,\ C \\ &\text {wanted:}& &u_C(t) \end {aligned} \end {equation*}

1.) Set up ODE: from \(u_C(t)\) for \(t>0\), analogous to example ?? \begin {align*} &\text {Mesh}:&&& u_R + u_C &= 0&&& &\text {with} u_R = R \cdot i \\ &\text {ODE}:&&& RC \cdot \mathrm{d}t u_C + u_C &= 0 &&& &\text {with} i = C \cdot \mathrm{d}t u_C \\[-15pt]&\hphantom {R1ooooooo}&&&\hphantom {L3oooooooooooooooooooo}&\hphantom {R3ooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

2.) Homogeneous solution, gen.: (fleeting) Identical to example ??\begin {align*} &\text {hom. Lsg.}:&&& u_{C,\mathrm {h}} = K \cdot \mathrm {e}^{-1/{t}{\tau }}&&&& &\text {with} \tau = RC \\[-15pt]&\hphantom {R1ooooooo}&&&\hphantom {L3oooooooooooooooooooo}&\hphantom {R3ooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

The particular solution \(u_{C,p}\) is omitted, since \(u_C\) becomes zero when \(t \to \infty \).
3. Part. Lsg.: (settled in) and 4. Superposition:\begin {align*} &\text {general sol.:}&&& u_C &= u_{C,\mathrm {h}} + \cancel {u_{C,p}}&&& &\text {with} u_{C,p}=0 \\[-15pt]&\hphantom {R1oooooooooooo}&&&\hphantom {L3ooo}&\hphantom {R3oooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

5.) Determine constants: (Set initial conditions)\begin {align*} &\text {Initial condition.}&&& u_C(0) &= K \cdot \mathrm {e}^{0} \overset {!}{=} U_q &&&& \Rightarrow K = U_q\\ &\text {special general sol.:}&&& u_C(t) &= U_q \cdot \mathrm {e}^{-1/{t}{\tau }} \\[-15pt]&\hphantom {R1oooooooooooo}&&&\hphantom {L3ooo}&\hphantom {R3oooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

Determine current \(i\) from voltage \(u_C\):\begin {align*} &\text {Current:}&&& i &= - \frac {U_q}{R} \cdot \mathrm {e}^{-1/{t}{\tau }} &&& &\text {with} i = C \cdot \mathrm{d}t u_C \\[-15pt]&\hphantom {R1oooooooooooo}&&&\hphantom {L3ooo}&\hphantom {R3oooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

1.2 For Inductances

The calculation method for determining the current or voltage curve for inductors when excited with DC voltage is identical to the calculation for capacitors in Chapter 1.1. The resulting time behaviour with exponential curves and time constants is analogous to that described there.

Examples are given in Example ?? and ?? the charging current \(i(t)\) and the inductance voltage \(u_L(t)\) of an inductance \(L\) during charging via a resistor \(R\) with DC voltage \(U_q\) are calculated and displayed for \(t \geq 0\).

The examples differ only in the calculation method. In example ??, the differential equation for \(i(t)\) is set up and solved, and in example ??, the differential equation for \(u_L(t)\) is set up and solved.

RL element DC charging process - Via ODE from \(i\)rl:chargedc:dgli Determination of charging current \(i\) and voltage \(u_L\) of inductance \(L\) when charging via resistor \(R\) with DC voltage \(U_q\) from switch-on time \(t=0\). No current flows before this point.[-6pt]

\begin {equation*} \begin {aligned} \text {given:}&& &i(0),\ U_q,\ R,\ L \\ \text {wanted:}&& &u_L(t),\ i(t) \end {aligned} \end {equation*}

1.) Set up ODE: von \(i(t)\) for \(t \geq 0\)

Based on the mesh equation, \(u_R\) and \(u_L\) are rewritten according to \(i\):\begin {align*} &\text {Mesh}:&&& u_R + u_L &= U_q&&& &\text {with} u_R = R \cdot i \\ &&&& R \cdot i + L \cdot \mathrm{d}t i &= U_q &&& &\text {with} u_L = L \cdot \mathrm{d}t i \\ &\text {ODE}:&&& \frac {L}{R} \cdot \mathrm{d}t i + i &= \frac {U_q}{R} \\[-15pt]&\hphantom {R1oooooooooo}&&&\hphantom {L3ooooooooooo}&\hphantom {R3ooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooo}\nonumber \end {align*}

2.) Homogeneous solution, gen.: (fleeting) (without interference)\begin {align*} &\text {ODE}_{h}:&&& \frac {L}{R} \cdot \mathrm{d}t i_{\mathrm h} + i_{\mathrm h} &= 0 &&& &\text {Approach:} i_{\mathrm h} = K \cdot \mathrm {e}^{\lambda t}\\ &\text {char. Pol.:}&&& \frac {L}{R} \cdot \lambda + 1 &= 0 &&& &\Rightarrow \lambda = -\frac {1}{\tau } = -\frac {R}{L}\\ &\text {hom. sol.}:&&& i_{\mathrm h} &= K \cdot \mathrm {e}^{-1/{t}{\tau }}&&& &\text {with} \tau = \frac {L}{R} \\[-15pt]&\hphantom {R1oooooooooo}&&&\hphantom {L3ooooooooooo}&\hphantom {R3ooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooo}\nonumber \end {align*}

3.) Particular solution: (settled in) with \(t \to \infty \)\begin {align*} &\text {ODE}_{p}:&&& \cancel {\frac {L}{R} \cdot \mathrm{d}t i_{\mathrm p}} + i_{\mathrm p} &= \frac {U_q}{R} &&& &\text {with} i_{\mathrm p} = const. \\ &\text {part. sol.}:&&& i_{\mathrm p} &= \frac {U_q}{R} \\[-15pt]&\hphantom {R1oooooooooo}&&&\hphantom {L3ooooooooooo}&\hphantom {R3ooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooo}\nonumber \end {align*}

4.) Non-homogeneous solution, gen.: (superposition)\begin {align*} &\text {general sol.:}&&& i(t) &= \underbrace {K \cdot \mathrm {e}^{-1/{t}{\tau }} \vphantom {\frac {U_q}{R}}}_{i_{\mathrm h}} + \underbrace { \frac {U_q}{R} }_{i_{\mathrm p}} &&& &\text {with} i_{\mathrm h}=i_f, i_{\mathrm p}=i_e \\[-15pt]&\hphantom {R1oooooooooo}&&&\hphantom {L3ooooooooooo}&\hphantom {R3ooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooo}\nonumber \end {align*}

5.) Determine constants: (Apply initial conditions)\begin {align*} &\text {Initial condition}&&& i(0) &= K \cdot \cancel {\mathrm {e}^0} + \frac {U_q}{R} \overset {!}{=} 0 &&& &\Rightarrow K = -\frac {U_q}{R}\\ &\text {clear sol.:}&&& i(t) &= \frac {U_q}{R} \cdot \left ( 1 - \mathrm {e}^{-1/{t}{\tau }} \right ) \\[-15pt]&\hphantom {R1oooooooooo}&&&\hphantom {L3ooooooooooo}&\hphantom {R3ooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooo}\nonumber \end {align*}

Determine voltage \(u_L\) from current \(i\):\begin {align*} &\text {Voltage:}&&& u_L &= U_q \cdot \mathrm {e}^{-1/{t}{\tau }} &&& &\text {with} u_L = L \cdot \mathrm{d}t i \\[-15pt]&\hphantom {R1oooooooooo}&&&\hphantom {L3ooooooooooo}&\hphantom {R3ooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooo}\nonumber \end {align*}

RL element DC charging process - Via ODE of \(u_L\)rl:chargedc:dglu Determination of voltage \(u_L\) and charging current \(i\) of inductance \(L\) when charging via resistor \(R\) with DC voltage \(U_q\) for \(t \geq 0\). The circuit is interrupted before \(t=0\).

\begin {equation*} \begin {aligned} \text {given:}&& &i(0),\ U_q,\ R,\ L \\ \text {wanted:}&& &u_L(t),\ i(t) \end {aligned} \end {equation*}

1.) Set up ODE: from \(u_L(t)\) for \(t \geq 0\)

Based on the mesh equation, \(u_R\) is expressed as a function of \(u_L\):\begin {align*} &\text {Mesh}:&&& u_R + u_L &= U_q&&& &\text {with} u_R = R \cdot i \\ &&&& R \cdot i + u_L &= U_q &&& &\left | \cdot \mathrm{d}t \right . \\ &&&& R \cdot \frac {u_L}{L} + \mathrm{d}t u_L &= \mathrm{d}t U_q &&& &\text {with} u_L = L \cdot \mathrm{d}t i \\ &\text {ODE}:&&& \frac {L}{R} \cdot \mathrm{d}t u_L + u_L &= 0 \\[-15pt]&\hphantom {R1ooooooo}&&&\hphantom {L3oooooooooooooooooooo}&\hphantom {R3ooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

2.) Homogeneous solution, gen.: (fleeting) (without interference)\begin {align*} &\text {ODE}_{h}:&&& \frac {L}{R} \cdot \mathrm{d}t u_{L,h} + u_{L,h} &= 0 &&& &\text {Approach:} u_{L,h} = K \cdot \mathrm {e}^{\lambda t}\\ &\text {char. Pol.:}&&& \frac {L}{R} \cdot \lambda + 1 &= 0 &&& &\Rightarrow \lambda = -\frac {1}{\tau } = -\frac {R}{L}\\ &\text {hom. sol.}:&&& u_{L,h} = K \cdot \mathrm {e}^{-1/{t}{\tau }}&&&& &\text {with} \tau = \frac {L}{R} \\[-15pt]&\hphantom {R1ooooooo}&&&\hphantom {L3oooooooooooooooooooo}&\hphantom {R3ooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

3.) Particular sol.: (settled in) with \(t \to \infty \)\begin {align*} &\text {ODE}_{p}:&&& \cancel {\frac {L}{R} \cdot \mathrm{d}t u_{L,p}} + u_{L,p} &= 0 &&& &\text {with} u_{L,p} = const. \\ &\text {part. sol.}:&&& u_{L,p} &= 0 \\[-15pt]&\hphantom {R1ooooooo}&&&\hphantom {L3oooooooooooooooooooo}&\hphantom {R3ooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

4.) Inhomogeneous solution, gen.: (superposition)\begin {align*} &&&& u_L(t) &= u_{L,h}(t) + \cancel {u_{L,p}(t)} &&& &u_L(t)= u_{L,f}(t) + \cancel {u_{L,e}(t)} \\ &\text {clear sol.:}&&& u_L(t) &= K \cdot \mathrm {e}^{-1/{t}{\tau }} \\[-15pt]&\hphantom {R1oooooooooooo}&&&\hphantom {L3ooo}&\hphantom {R3oooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

5.) Determine const.: (Set initial conditions)\begin {align*} &\text {Initial condition.}&&& i(0) &= \frac {u_R(0)}{R} \overset {!}{=} 0 &&& &\Rightarrow u_L(0) = U_q - \cancel {u_R(0)}\\ &&&& u_L(0) &= K \cdot \mathrm {e}^{-0} \overset {!}{=} U_q &&& &\Rightarrow K = U_q\\ &\text {clear sol.:}&&& u_L(t) &= U_q \cdot \mathrm {e}^{-1/{t}{\tau }} \\[-15pt]&\hphantom {R1oooooooooooo}&&&\hphantom {L3ooo}&\hphantom {R3oooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

Determine current \(i\) from voltage \(u_L\):\begin {align*} &\text {Current:}&&& i &= \frac {U_q}{R} \cdot \left ( 1 - \mathrm {e}^{-1/{t}{\tau }} \right ) &&& &\text {with} i = \frac {u_R}{R} = \frac {U_q - u_L}{R} \\[-15pt]&\hphantom {R1oooooooooooo}&&&\hphantom {L3ooo}&\hphantom {R3oooooooooooooooo}&&&&\hphantom {R5oooooooooooooooooooooo}\nonumber \end {align*}

The discharge process of an inductance \(L\) via a resistor \(R\) is similar to the charging process via a resistor \(R\) with a DC voltage \(U_q\). The short circuit across \(R\) corresponds to a voltage source with \(U_q=0\ \mathrm {V}\). The discharge process and the charging process differ only in the initial conditions and the disturbance term of the respective differential equation. The time constant \(\tau \) is identical for the charging and discharging processes. A detailed calculation is therefore not provided here.

Figure 2 shows the circuit diagram (a) and the voltage and current curve (b) of the inductance \(L\) when discharging via a resistor \(R\) from switching time \(t=0\). Before switching, the inductance is connected in series with the resistor \(R\) to a DC voltage source \(U_q\). At the switching point, the inductance is fully charged (steady state) and conducts the constant current \(U_q/R\) before switching.

Real inductances can be modelled by a series connection of an ideal inductance \(L\) and an ohmic resistor \(R_{\text {Cu}}\), which corresponds to the copper losses. connecting an ohmic resistor \(R_{\text {Fe}}\) in parallel directly to the inductance \(L\), iron losses can also be modelled.

Figure 3 shows the circuit diagram (a) of a real coil with inductance \(L\) in series with a resistor \(R_{\text {Cu}}\) (here \(R_1\)). Immediately parallel to the inductance is the ohmic resistor \(R_{\text {Fe}}\) (corresponding to \(R_2\)). The inductance is charged from \(t=0\) via an ideal constant voltage source \(U_q\) and is completely discharged beforehand. The corresponding voltage and current curves (b) are shown alongside.

Compared to the charging process of a pure RL series connection in Example ??, the voltage across the inductance does not jump to the full source voltage but to a lower value limited by the voltage divider \(\frac {R_2}{R_1+R_2}\). Due to iron losses, the input current jumps at the switching point and then exponentially approaches the current \(\frac {U_q}{R_1}\). The current through \(R_2\) jumps to the value \(\frac {U_q}{R_1+R_2}\) when switching and then drops exponentially. The current through the inductance approaches the current \(\frac {U_q}{R_1}\) exponentially. The time constant is \(\tau =\frac {L}{R_1||R_2}\).

2 With sinusoidal excitation

The procedure for calculating switching operations with sinusoidal excitation is essentially the same as the procedure for uniform excitation. Therefore, this chapter only deals with the special features of sinusoidal excitation and explains the calculation using an example.

The differential equations for circuits with alternating voltage or current sources differ from those for direct voltage or current sources only in terms of the right-hand side of the differential equation, the disturbance term. The disturbance term results from the excitation and is also sinusoidal in the case of sinusoidal excitation of linear, time-invariant systems.

Due to the time dependency of sinusoidal excitation, the exact switching point is decisive for the behaviour of the system. More precisely, the phase position of the excitation at the switching point is decisive.

Using the example of an RC element, Chapter 2.1 shows how to calculate the switching process for sinusoidal excitation. It explains in detail how the switching process depends on the switching time and the possibility of transient peak currents and voltages.

Example ?? summarises the calculation of a switching operation for an RL element when excited with alternating voltage.

2.1 In terms of capacity

To explain the special features of switching processes with sinusoidal excitation, we consider the example of an RC circuit. The calculation is performed as in Example ?? according to Rule ??.

A capacitance \(C\) is connected via a resistor \(R\) from the switching time \(t=t_0\) to an ideal AC voltage source with voltage \(u_q(t) = \hat {U}_q \cdot \cos (\omega t + \varphi _q)\). The capacitance \(C\) is completely discharged at the start. Figure 4 shows the circuit diagram for this arrangement.

1.) Set up a differential equation:

The inhomogeneous ODE of \(u_C\) for \(t \geq t_0\) is obtained analogously to Example ?? as: \begin {align} RC \cdot \mathrm{d}t u_C(t) + u_C(t) ={}& u_q(t) \\ \text {mit} &u_q(t) = \hat {U_q} \cdot \cos (\omega t + \varphi _q) \end {align}

Compared to example ??, only the disturbance term has been changed. The DC voltage term \(U_q\) has been replaced here by the AC voltage term \(u_q\).

2.) Homogeneous solution, general (fleeting state):

The homogeneous solution \(u_{C,h}\) is independent of the disturbance term and thus of the excitation. The solution is taken from Example ??, taking into account the variable switching time \(t_0\): \begin {align} u_{C,\mathrm {h}} &= K \cdot \mathrm {e}^{-\frac {t'}{\tau }} \qquad \text {with} t' = t-t_0 \nonumber \\ &= K \cdot \mathrm {e}^{-\frac {t-t_0}{\tau }} \qquad \text {with} \tau = RC \label {eq:rc:charge:ac:homogeneloesung} \end {align}

Since the exponential approach only applies from switching time \(t_0\), this is subtracted from time \(t\).

3.) Particular solution (steady state):

The particular solution \(u_{C,p}\) also corresponds to a sinusoidal quantity with the same frequency as the excitation in the case of sinusoidal excitation, as described in Chapter ??. The steady state can be described using the complex amplitude indicator \(\hat {\underline {U}}_{C,p}\) as follows: \begin {align} u_{C,p} &= \hat {U_{C,p}} \cdot \cos (\omega t + \varphi _{C,\mathrm {p}}) \label {eq:rc:charge:ac:partikulaereloesung}\\ &= \Re {\left \{ \hat {\underline {U}_{C,p}} \cdot \mathrm {e}^{\mathrm {j} \omega t} \right \}} \qquad \text {with} \hat {\underline {U}_{C,p}} = \hat {U_{C,p}} \cdot \mathrm {e}^{\mathrm {j} \varphi _{C,\mathrm {p}}} \end {align}

The amplitude \(\hat {U}_{C,p}\) and the (load) phase angle \(\varphi _{C,\mathrm {p}}\) can be determined according to complex voltage dividers. \begin {align} \hat {\underline {U}_{C,p}} &= \hat {\underline {U}_q} \cdot \frac {\underline {Z}_C}{R + \underline {Z}_C} = \hat {\underline {U}_q} \cdot \frac {\frac {1}{\mathrm {j}\omega C}}{R + \frac {1}{\mathrm {j} \omega C}} \nonumber \\ &= \frac {\hat {\underline {U}_q}}{1 + \mathrm {j} \omega R C} \qquad \text {with} \hat {\underline {U}_q} = \hat {U_q} \cdot \mathrm {e}^{\mathrm {j} \varphi _q} \label {eq:rc:charge:ac:komplex} \end {align}

For the amplitude \(\hat {U}_{C,p}\) and the phase \(\varphi _{C,\mathrm {p}}\), the following follows from Eq. ??: \begin {align} \hat {U_{C,p}} = \frac {\hat {U_q}}{\sqrt {1 + \left (\omega RC\right )^2}} \qquad \text {und}\qquad \varphi _{C,\mathrm {p}} = \varphi _q - \arctan \left (\omega RC\right ) \label {rc:charge:ac:amplitudephase} \end {align}

4.) Inhomogeneous solution, generally (superposition):

The superposition of Eq. ?? and ?? yields the general solution of the inhomogeneous differential equation: \begin {align} u_C &= u_{C,\mathrm {h}} + u_{C,p} \nonumber \\ &= K \cdot \mathrm {e}^{-\frac {t-t_0}{\tau }} + \hat {U_{C,p}} \cdot \cos (\omega t + \varphi _{C,\mathrm {p}}) \label {eq:rc:charge:ac:allg} \end {align}

5.) Determine constants (Initial condition):

From the initial condition that the capacitance is discharged at the start, the following applies to the constant \(K\): \begin {align} u_C(t_0) &\overset {!}{=} 0 \nonumber \\ K \cdot \cancel {\mathrm {e}^{0}} + \hat {U_{C,p}} \cdot \cos (\omega t_0 + \varphi _{C,\mathrm {p}}) &= 0 \nonumber \\ K &= -\hat {U_{C,p}} \cdot \cos (\omega t_0 + \varphi _{C,\mathrm {p}}) \label {eq:rc:charge:ac:k} \end {align}

Using Eq. ?? inserted into ??, we obtain the unique solution for \(u_C(t)\): \begin {align} u_C(t) &= \underbrace {-\hat {U_{C,p}} \cdot \cos (\omega t_0 + \varphi _{C,\mathrm {p}}) \cdot \mathrm {e}^{-\frac {t-t_0}{\tau }}}_{u_{C,\mathrm {h}}} + \underbrace {\hat {U_{C,p}} \cdot \cos \left (\omega t + \varphi _{C,\mathrm {p}}\right )}_{u_{C,p}} \label {eq:rc:charge:ac:uc} \end {align}

Equation ?? shows that the transient state \(u_{C,f}=u_{C,h}\) tends exponentially towards zero from \(t_0\). In contrast to the DC case (example ??), the starting value of the exponential curve \(K\) from equation ?? depends on \(t_0\), since the steady state \(u_{C,e}=u_{C,p}\) is time-dependent.

Behaviour when changing the switching point/time constant:

Figure 4a shows the voltage curve of the excitation \(u_q\) at a zero phase angle \(\varphi _q\) of zero and the voltage curve of the capacitance \(u_C\). For comparison, \(u_C\) is shown for two different switching times \(t_0\). One is at the negative peak value and the other is at the zero crossing of \(u_{C,p}\). \(u_C\) is shown as a superposition of the transient state \(u_{C,h}\) (black, dashed) and the steady state \(u_{C,p}\) (black).

Was can be seen, the compensation process is maximal (\(|K|=max\)) when the difference \(|u_{C,p}(t_0)- u_C(t_0)|\) is maximal. If \(u_{C,p}\) already satisfies the initial condition with \(u_{C,p}(t_0)\overset {!}{=}u_C(t_0)\), the transient state is completely eliminated \(K=0 \Rightarrow u_{C,h}=0\) and no compensation process occurs.

Figure 4b shows the voltage curve \(u_C\) for different time constants \(\tau \). In one case, \(\tau =\frac {2\pi }{\omega }\), and in the other, \(\tau =\frac {20\pi }{\omega }\). In both cases, \(t_0\) is chosen so that the equalisation process is maximised.

With large time constants, the transient state decays more slowly, which can lead to overvoltages/overcurrents at unfavourable switching times. If \(u_C\) is discharged at the start according to the initial condition in the example (\(u_C(t_0)=0\)), \(u_C\) can rise to twice the peak value of \(u_{C,p}\). If, on the other hand, \(u_C\) is already charged to the negative/positive peak value of \(u_{C,p}\) at the start, \(u_C\) can even rise to three times the peak value of \(u_{C,p}\).

2.2 For Inductors

Example ?? shows an example of the calculation of a switching operation for an RL element when excited with alternating voltage according to Merksatz ??.

The calculation of switching processes for inductors with sinusoidal excitation is analogous to the calculation for capacitors in Chapter 2.1. The same special features apply with regard to switching times and possible transient peak currents and voltages as for capacitors.

In the example, the DGL of the particular solution is transformed according to equation ??: \begin {align*} &\text {ODE$_{\mathrm p}$:}& \frac {L}{R} \cdot \mathrm{d}t i_{\mathrm p} + i_{\mathrm p} &= \frac {u_q}{R} & &\text {with} u_q = \Re \left \{\underline {\hat {U}_q}\cdot \mathrm {e}^{\mathrm {j}\omega t}\right \} \vphantom {\Bigg |}\\ &\text {transformed:}& \Leftrightarrow \frac {L}{R} \cdot \mathrm {j}\omega \, \underline {\hat {I}_{\mathrm p}} + \underline {\hat {I}_{\mathrm p}} &= \frac {\underline {\hat {U}_q}}{R} & &\text {with} \underline {\hat {U}_q} = \hat {U_q}\cdot \mathrm {e}^{\mathrm {j}\varphi _q} \vphantom {\Bigg |} \end {align*}

The approach offers a simple conversion from the complex amplitude pointer of the particular solution and is in principle equivalent to the derivation using complex voltage dividers as shown in equation ?? using the example of the RC element.

RL element, switching operation with AC voltage sourcerl:chargeac Determination of the current \(i\) through the inductance \(L\) in series with the ohmic resistor \(R\) when the ideal AC voltage source \(u_q = \hat {U_q} \cdot \cos (\omega t + \varphi _q)\) is switched on at the switching time \(t = t_0\). No current flows prior to this time.

\begin {equation*} \begin {aligned} \text {given:}&& &i(t_0),\ u_q,\ R,\ L \\ \text {wanted:}&& &i(t),\ u_L(t) \end {aligned} \end {equation*}

1.) Set up ODE: from \(i\) for \(t \geq t_0\), Solution from example. ?? \begin {align*} &\text {ODE}:&&& \frac {L}{R} \cdot \mathrm{d}t i + i &= \frac {u_q}{R}&&& &\text {with} u_q = \hat {U_q} \cdot \cos (\omega t + \varphi _q) \\[-15pt]&\hphantom {R1oooooooooo}&&&\hphantom {L3ooooooooooo}&\hphantom {R3ooooooooooooooo}&&&&\hphantom {R5ooooooooooooooooooooooo}\nonumber \end {align*}

2.) Homogeneous solution, gen.: (fleeting), Solution from example ?? for switching time \(t_0\) \begin {align*} &\text {hom. sol.:} &&& i_{\mathrm {h}}&= K \cdot \mathrm {e}^{-\frac {t-t_0}{\tau }} &&& &\text {with} \tau = \frac {L}{R} \\[-15pt]&\hphantom {R1oooooooooo}&&&\hphantom {L3ooooooooooo}&\hphantom {R3ooooooooooooooo}&&&&\hphantom {R5ooooooooooooooooooooooo}\nonumber \end {align*}

3.) Particular solution: (settled in) with \(t \to \infty \), complete AC calculation\begin {align*} &\text {transf.}:&&& \frac {L}{R} \cdot \mathrm {j}\omega \, \underline {\hat {I}_{\mathrm p}} + \underline {\hat {I}_{\mathrm p}} &= \frac {\underline {\hat {U}_q}}{R} &&& &\text {with} \underline {\hat {U}_q}=\hat {U_q} \cdot \mathrm {e}^{\mathrm {j}\varphi _q}\\ &\text {Pointer:}&&& \underline {\hat {I}_{\mathrm p}} &= \frac {\underline {\hat {U}_q}}{R} \cdot \frac {1}{1 + \mathrm {j} \omega \frac {L}{R}} &&& &\text {with} \underline {\hat {I}_{\mathrm p}} = \hat {I_{\mathrm p}}\cdot \mathrm {e}^{\mathrm {j}\varphi _I}\\ &&&& \hat {I_{\mathrm p}} &= \frac {\hat {U_q}/R}{\sqrt {1 + \left (\omega \frac {L}{R}\right )^2}} &&& & \varphi _I = \varphi _q - \arctan \left (\omega \frac {L}{R}\right ) \\ &\text {part. sol.}:&&& i_{\mathrm p} &= \hat {I_{\mathrm p}} \cdot \cos (\omega t + \varphi _I) &&& &\text {with} i_{\mathrm p} = \Re \left \{\underline {\hat {I}_{\mathrm p}} \cdot \mathrm {e}^{\mathrm {j}\omega t}\right \} \vphantom {\Bigg |} \\[-15pt]&\hphantom {R1oooooooooo}&&&\hphantom {L3ooooooooooo}&\hphantom {R3ooooooooooooooo}&&&&\hphantom {R5ooooooooooooooooooooooo}\nonumber \end {align*}

4.) Inhomogeneous solution, gen.: (Superposition)\begin {align*} &\text {gen. sol.:}&&& i &= \underbrace { \vphantom {\big |} K \cdot \mathrm {e}^{-\frac {t-t_0}{\tau }} }_{i_{\mathrm h}} + \underbrace { \vphantom {\big |} \hat {I_{\mathrm p}} \cdot \cos (\omega t + \varphi _I) }_{i_{\mathrm p}}&&& \\[-15pt]&\hphantom {R1oooooooooo}&&&\hphantom {L3oooooo}&\hphantom {R3ooooooooooooooooooooooooooooooooooooooooooo}&&&&\hphantom {R5}\nonumber \end {align*}

5.) Determine constants: (Insert AB)\begin {align*} &\text {Set initial cond.:}&&& i(t_0) &= K \cdot \cancel {\mathrm {e}^0} + \hat {I_{\mathrm p}} \cdot \cos (\omega t_0 + \varphi _I) \overset {!}{=} 0 \\ &&&& \Rightarrow K &= -\hat {I_{\mathrm p}} \cdot \cos (\omega t_0 + \varphi _I) \\ &\text {clear sol.:}&&& i(t) &= -\hat {I_{\mathrm p}} \cdot \cos (\omega t_0 + \varphi _I) \cdot \mathrm {e}^{-\frac {t-t_0}{\tau }} + \hat {I_{\mathrm p}} \cdot \cos (\omega t + \varphi _I) \\[-15pt]&\hphantom {R1oooooooooo}&&&\hphantom {L3oooooo}&\hphantom {R3ooooooooooooooooooooooooooooooooooooooooooo}&&&&\hphantom {R5}\nonumber \end {align*}

Calculate voltage \(u_L\) across inductance \(L\) using \(u_L = L \cdot \mathrm{d}t i\). \begin {align*} &\text {Voltage:}&&& u_L(t) &= R \cdot \hat {I_{\mathrm p}} \cdot \cos (\omega t_0 + \varphi _I) \cdot \mathrm {e}^{-\frac {t-t_0}{\tau }} - \omega L \cdot \hat {I_{\mathrm p}} \cdot \sin (\omega t + \varphi _I) \end {align*}

3 RLC series resonant circuits

Systems with two different energy storage devices (inductance \(L\) and capacitance \(C\)) are fundamentally capable of oscillation. For this reason, RLC circuits are also referred to as oscillating circuits or resonance circuits. The oscillatory capability can be determined and described using the eigenvalues. Another approach is to consider the (oscillating circuit) quality factor \(Q\). [See Module 8]

Figure 6 shows the circuit diagram of an RLC element charging with a constant DC voltage \(U_q\). The time curves of the capacitor voltage for the three possible cases are also shown. These are the aperiodic case, the aperiodic limit case and the periodic case which are described in more detail in Chapter 3.1.

The calculation of switching operations for RLC elements is analogous to the examples for RC and RL elements from the previous chapters.

1.) Set up ODE: for \(u_C\) for \(t \geq 0\) \begin {align} u_L + u_R +u_C &= u \nonumber \\ L \cdot \mathrm{d}t i + R \cdot i + u_C&= U_q \nonumber \\ LC \cdot \mathrm{d}t [2] u_C + RC \cdot \mathrm{d}t u_C + u_C &= U_q \label {eq:dgl:rlc:dc} \end {align}

Due to the two different energy storage devices, a second-order ODE results.

2.) Homogeneous solution, gen.: (fleeting) (without interference)

The form of the homogeneous solution depends on the type of eigenvalues. \begin {align} &\text {hom. ODE:}& LC \cdot \mathrm{d}t [2] u_{C,\mathrm {h}} + RC \cdot \mathrm{d}t u_{C,\mathrm {h}} + u_{C,\mathrm {h}} &= 0 &&\text {with} u_{C,\mathrm {h}} = K \cdot \mathrm {e}^{\lambda t} \label {eq:dgl:rlc:homo}\\[6pt] &\text {char. Polynom:}& \lambda ^2 + \frac {R}{L} \cdot \lambda + \frac {1}{LC} &= 0 \nonumber \\[6pt] &\text {Eigenvalues:}& \lambda _{1/2} = -\frac {R}{2L} \pm \sqrt {\left (\frac {R}{2L}\right )^2 - \frac {1}{LC}}& &&\text {with} D = \left (\frac {R}{2L}\right )^2 - \frac {1}{LC} \label {eq:dgl:rlc:lambda} \end {align}

Applying the \(p\)-\(q\) formula (midnight formula) yields either two different or two identical values for the eigenvalues (EW) \(\lambda _{1/2}\), depending on the root term. The homogeneous solution therefore takes one of the following two forms, as described in Chapter ??: \begin {equation} \text {hom. sol.:}\qquad \qquad \qquad u_{C,\mathrm {h}} = \begin {cases} K_1 \cdot \mathrm {e}^{\lambda _1 t} + K_2 \cdot \mathrm {e}^{\lambda _2 t} & \text {for} \lambda _1 \neq \lambda _2 \\ (K_1 + K_2 \cdot t) \cdot \mathrm {e}^{\lambda t} & \text {for} \lambda _1 = \lambda _2 = \lambda \end {cases} \qquad \qquad \qquad \label {eq:dgl:rlc:homolsg:cases} \end {equation} The cases can be distinguished using the so-called discriminant \(D\) .

3.1 Case differentiation for RLC series resonant circuits

The discriminant \(D\) is the term under the root in equation ?? for determining \(\lambda _{1/2}\). \begin {align} &&\lambda _{1/2} &= -\frac {R}{2L} \pm \sqrt {\smash {\underbrace {\left (\frac {R}{2L}\right )^2 - \frac {1}{LC}}_{{\text {Discriminant }D}}}\vphantom {\left (\frac {R}{L}\right )^2} }\label {eq:dgl:rlc:lambda:allg}\\[-10pt]\nonumber \\[-15pt] &\hphantom {1rrrrrrrrrrrrrrrr} & \hphantom {2llllll}&\hphantom {\mathrel {\phantom {=}}2rrrrrrrrrrrrrrrrrrrrrrrrrrr} &&\hphantom {3rrrrrrrrrrrrrrrrrrrrrrrr} \nonumber \end {align}

Table 2 lists all three cases with the type of eigenvalues and relative statements regarding the damping of the oscillating circuit. The value ranges of the discriminant \(D\) and the oscillating circuit quality factor \(Q_S\) (see Module 8) are also given as distinguishing features.

A locus diagram of the eigenvalues is shown in the optional excursus in Chapter 4.2. The relationship between \(\lambda \) and \(Q_S\) is explained in more detail in the optional excursus in Chapter 4.3.

3.2 Aperiodic case in the RLC series resonant circuit

If the discriminant \(D\) is positive, the root term is real and always smaller than the absolute value of the term \(\frac {R}{2L}\) before the root. As a result, both eigenvalues are different, real and negative.\begin {align*} &\text {with} D > 0 \text {applies}& &\phantom {\mathrel {\phantom {=}}-}\frac {R}{2L} \overset {!}{>} \sqrt {\left (\frac {R}{2L}\right )^2 - \frac {1}{LC}} &&\Rightarrow \text {2 real eigenvalues} \\[-10pt] &\hphantom {1rrrrrrrrrrrrrrrr} & \hphantom {2llllll}&\hphantom {\mathrel {\phantom {=}}2rrrrrrrrrrrrrrrrrrrrrrrrrrr} &&\hphantom {3rrrrrrrrrrrrrrrrrrrrrrrr} \nonumber \end {align*}

2.) Homogeneous solution: (fleeting) \begin {align} && u_{C,\mathrm {h}} &= K_1 \cdot \mathrm {e}^{\lambda _1 t} + K_2 \cdot \mathrm {e}^{\lambda _2 t} &&\text {with} \lambda _1 \neq \lambda _2,\ \lambda < 0 \label {eq:dgl:rlc:homolsg:aperiodisch} \\[-15pt] &\hphantom {1rrrrrrrrrrrrrrrr} & \hphantom {2llllll}&\hphantom {\mathrel {\phantom {=}}2rrrrrrrrrrrrrrrrrrrrrrrrrrr} &&\hphantom {3rrrrrrrrrrrrrrrrrrrrrrrr} \nonumber \end {align}

The constants \(K_1\) and \(K_2\) must both be real so that the solution remains real-valued. The transient state thus corresponds to the addition of two exponential functions. The slower exponential function dominates, i.e. the one with the larger time constant (see excursus in chapter 4.2).

3.) Particular solution: (settled, identical for all three cases)\begin {align*} && u_{C,\mathrm {p}} &= U_q \\[-15pt] &\hphantom {1rrrrrrrrrrrrrrrr} & \hphantom {2llllll}&\hphantom {\mathrel {\phantom {=}}2rrrrrrrrrrrrrrrrrrrrrrrrrrr} &&\hphantom {3rrrrrrrrrrrrrrrrrrrrrrrr} \nonumber \end {align*}

4.) General solution: (Superposition)\begin {align} && u_C(t) &= K_1 \cdot \mathrm {e}^{\lambda _1 t} + K_2 \cdot \mathrm {e}^{\lambda _2 t} + U_q \nonumber \\[-14pt] &\hphantom {1Rooooooo}&\hphantom {2Loooooo}&\hphantom {2Roooooooooooooooooooooooooooooooooo}&\hphantom {3Loooooo}&\hphantom {3Roooooooooo} \nonumber \end {align}

5.) Determine constants: (as is known)

Be \(u_C(0)=0\) and \(i(0)=0 \Rightarrow \mathrm{d}t \, u_C(0) = 0 \) as initial conditions (IC) given, then it follows that

\begin {align*} &\hphantom {1Rooooooo}&\hphantom {2Loooooo}&\hphantom {2Roooooooooooooooooooooooooooooooooo}&\hphantom {3Loooooo}&\hphantom {3Roooooooooo}\\[-15pt] &\text {1st IC:}& u_C(0) &= K_1 \cdot \cancel {\mathrm {e}^{\lambda _1 0}} + K_2 \cdot \cancel {\mathrm {e}^{\lambda _2 0}} + U_q \overset {!}{=} 0 & \Rightarrow K_1 &= -K_2 -U_q\\ &\text {2nd IC:}& \mathrm{d}t u_C(0) &= K_1 \cdot \lambda _1 \cdot \cancel {\mathrm {e}^{\lambda _1 0}} + K_2 \cdot \lambda _2 \cdot \cancel {\mathrm {e}^{\lambda _2 0}} + \cancel {0} \overset {!}{=} 0 & \Rightarrow K_1 &= - K_2 \frac {\lambda _2}{\lambda _1} \nonumber \\ &\Longrightarrow & K_1 &= \frac {U_q}{\frac {\lambda _1}{\lambda _2}-1} \text {and} K_2 = \frac {U_q}{\frac {\lambda _2}{\lambda _1}-1} \end {align*}

3.3 Aperiodic boundary case in the RLC series resonant circuit

If the discriminant \(D\) is exactly zero, both eigenvalues are identical, real and negative. The double zero point results in a special form of the homogeneous solution by superimposing the two approaches (\(K_1 \cdot \mathrm {e}^{\lambda _1 t}\) and \(K_2 \cdot t \cdot \mathrm {e}^{\lambda _2 t}\)).\begin {align*} \vphantom {\sqrt {\left (\frac {R}{2L}\right )^2}} &\text {for} D = 0 \text {applies}& \lambda _{1/2} &= -\frac {R}{2L} \cancel { {} \pm {} \sqrt {0}} = \lambda &&\Rightarrow \text {1 real Eigenvalue} \\[-15pt] &\hphantom {1rrrrrrrrrrrrrrrr} & \hphantom {2llllll}&\hphantom {\mathrel {\phantom {=}}2rrrrrrrrrrrrrrrrrrrrrrrrrrr} &&\hphantom {3rrrrrrrrrrrrrrrrrrrrrrrr} \nonumber \end {align*}

2.) Homogeneous solution: (fleeting) \begin {align} && u_{C,\mathrm {h}} &= \left ( K_1 + K_2 \cdot t \right ) \cdot \mathrm {e}^{\lambda t} &&\text {with} \lambda _1 = \lambda _2 < 0 \label {eq:dgl:rlc:homolsg:grenzfall} \\[-15pt] &\hphantom {1rrrrrrrrrrrrrrrr} & \hphantom {2llllll}&\hphantom {\mathrel {\phantom {=}}2rrrrrrrrrrrrrrrrrrrrrrrrrrr} &&\hphantom {3rrrrrrrrrrrrrrrrrrrrrrrr} \nonumber \end {align}

3.) Particular solution: (settled in)\begin {align*} && u_{C,\mathrm {p}} &= U_q \\[-15pt] &\hphantom {1rrrrrrrrrrrrrrrr} & \hphantom {2llllll}&\hphantom {\mathrel {\phantom {=}}2rrrrrrrrrrrrrrrrrrrrrrrrrrr} &&\hphantom {3rrrrrrrrrrrrrrrrrrrrrrrr} \nonumber \end {align*}

4.) General solution: (settled in)\begin {align} && u_C(t) &= \left ( K_1 + K_2 \cdot t \right ) \cdot \mathrm {e}^{\lambda t} + U_q \nonumber \\[-14pt] &\hphantom {1Rooooooo}&\hphantom {2Loooooo}&\hphantom {2Roooooooooooooooooooooooooooooooooo}&\hphantom {3Loooooo}&\hphantom {3Roooooooooo} \nonumber \end {align}

5.) Determine constants: (as is known)

Let \(u_C(0)=0\) and \(i(0)=0 \Rightarrow \mathrm{d}t \, u_C(0) = 0 \) be given as initial conditions (IC), then the following applies:

\begin {align*} &\hphantom {1Rooooooo}&\hphantom {2Loooooo}&\hphantom {2Roooooooooooooooooooooooooooooooooo}&\hphantom {3Loooooo}&\hphantom {3Roooooooooo}\\[-15pt] &\text {1st IC:}& u_C(0) &= \left ( K_1 + \cancel {K_2 \cdot 0 }\right ) \cancel {\cdot \mathrm {e}^{\lambda 0}} + U_q \overset {!}{=} 0 & \Rightarrow K_1 &= -U_q\\ &\text {2nd IC:}& \mathrm{d}t u_C(0) &= \left ( K_1 + \cancel {K_2 \cdot 0} \right ) \cdot \lambda \cdot \cancel {\mathrm {e}^{\lambda 0}} + K_2 \cdot \cancel {\mathrm {e}^{\lambda 0}} \overset {!}{=} 0 & \Rightarrow K_2 &= -K_1 \cdot \lambda \nonumber \\ &\Longrightarrow & K_1 &= -U_q \text {and} K_2 = +U_q \cdot \lambda \end {align*}

3.4 Periodic case for the RLC series resonant circuit

If the discriminant is negative, there are two complex conjugate eigenvalues with negative real parts. Transformation with \(\sqrt {-1}=\mathrm {j}\cdot \sqrt {1}\) and derivation for \(u_{C,h}\) in chapter 4.1:\begin {align} &\text {for} D < 0 \text {applies}& \lambda _{1/2} &= -\underbrace { \vphantom {\sqrt {\left (\frac {R}{2L}\right )^2}} \frac {R}{2L} }_{\mathrm{d}elta } \pm \, \mathrm {j} \cdot \underbrace { \sqrt {\frac {1}{LC} - \left (\frac {R}{2L}\right )^2} }_{\omega _d} &&\Rightarrow \text {2 compl. EW} \label {eq:dgl:rlc:lambda:periodisch} \\[-15pt] &\hphantom {1rrrrrrrrrrrrrrrr} & \hphantom {2llllll}&\hphantom {\mathrel {\phantom {=}}2rrrrrrrrrrrrrrrrrrrrrrrrrrr} &&\hphantom {3rrrrrrrrrrrrrrrrrrrrrrrr} \nonumber \end {align}

2.) Homogeneous solution: (fleeting) \begin {align} && u_{C,\mathrm {h}} &= \mathrm {e}^{-\mathrm{d}elta t} \cdot C_0 \cdot \sin (\omega _d t + \varphi _0) &&\text {with} \lambda _{1/2} = -\mathrm{d}elta \pm \mathrm {j}\,\omega _d \label {eq:dgl:rlc:homolsg:periodisch} \\[-15pt] &\hphantom {1rrrrrrrrrrrrrrrr} & \hphantom {2llllll}&\hphantom {\mathrel {\phantom {=}}2rrrrrrrrrrrrrrrrrrrrrrrrrrr} &&\hphantom {3rrrrrrrrrrrrrrrrrrrrrrrr} \nonumber \end {align}

The homogeneous solution corresponds to a decaying oscillation with the decay constant \(\mathrm{d}elta \) (negative real part of the EW) and the damped natural frequency \(\omega _d\) (positive imaginary part of the EW). A derivation for Eq. ?? from Eq. ?? is given in Chapter 4.1.

3.) Particular solution: (settled in)\begin {align*} && u_{C,\mathrm {p}} &= U_q \\[-15pt] &\hphantom {1rrrrrrrrrrrrrrrr} & \hphantom {2llllll}&\hphantom {\mathrel {\phantom {=}}2rrrrrrrrrrrrrrrrrrrrrrrrrrr} &&\hphantom {3rrrrrrrrrrrrrrrrrrrrrrrr} \nonumber \end {align*}

4.) General solution: (Superposition)\begin {align} && u_C(t) &= \mathrm {e}^{-\mathrm{d}elta t} \cdot C_0 \cdot \sin (\omega _d t + \varphi _0) + U_q \nonumber \\[-14pt] &\hphantom {1Rooooooo}&\hphantom {2Loooooo}&\hphantom {2Roooooooooooooooooooooooooooooooooo}&\hphantom {3Loooooo}&\hphantom {3Roooooooooo} \nonumber \end {align}

5.) Determine constants: (as is known) \begin {align*} &\hphantom {1Rooooooo}&\hphantom {2Loooooo}&\hphantom {2Roooooooooooooooooooooooooooooooooo}&\hphantom {3Loooooo}&\hphantom {3Roooooooooo}\\[-15pt] &\text {1st IC:}& u_C(0) &= \cancel {\mathrm {e}^{-\mathrm{d}elta 0}} \cdot C_0 \cdot \sin (\cancel {\omega _d t } + \varphi _0) + U_q \overset {!}{=} 0\\ &\text {2nd IC:}& \mathrm{d}t u_C(0) &= -\mathrm{d}elta \cdot \cancel {\mathrm {e}^{-\mathrm{d}elta 0}} \cdot C_0 \cdot \sin (\varphi _0) + {\cancel {\mathrm {e}^{-\mathrm{d}elta 0}} \cdot C_0 \cdot \omega _d \cdot \cos (\varphi _0) \overset {!}{=} 0 }\nonumber \\ &\Longrightarrow & C_0 &= -\frac {U_q}{\sin (\varphi _0)} \text {and} \varphi _0 = \arctan \left (\frac {\omega _d}{\mathrm{d}elta }\right ) \end {align*}

Figure 7 shows the time curve of the voltage \(u_C(t)\) for \(t=0\) to \(t=5\cdot \frac {1}{\mathrm{d}elta }\). In the selected scaling, \(\omega _0\) is four times the decay constant \(\mathrm{d}elta \) (quality factor \(Q_S = 2\)).

4 Digressions on series resonant circuits

This chapter contains supplementary digressions on the behaviour of oscillating circuits, which serve to deepen understanding of the topic and build bridges to other subject areas. The content of the digressions is not essential for understanding the rest of the module and can be skipped if necessary.

4.1 Derivation of the homogeneous solution in the periodic case

From the general homogeneous solution in the periodic case (Eq. 5), it follows with \(\lambda _{1/2}\) (Eq. ??): \begin {align} && u_{C,\mathrm {h}} &= \mathrm {e}^{-\mathrm{d}elta t} \cdot \left ( K_1 \cdot \mathrm {e}^{+\mathrm {j}\omega _d t} + K_2 \cdot \mathrm {e}^{-\mathrm {j}\omega _d t} \right ) \nonumber \\ \end {align}

\(K_1\) and \(K_2\) must be complex conjugates so that \(u_{C,\mathrm {h}}\) is real! [?, cf. p. 379] \begin {align} &\text {with}& K_1 &= a + \mathrm {j} b { \text {and} K_2 = a - \mathrm {j} b \text {with} a,b \in \mathbb {R}} \nonumber \\[6pt] && u_{C,\mathrm {h}} &= \mathrm {e}^{-\mathrm{d}elta t} \cdot \left [ (a + \mathrm {j} b) \cdot \mathrm {e}^{+\mathrm {j}\omega _d t} + (a - \mathrm {j} b) \cdot \mathrm {e}^{-\mathrm {j}\omega _d t} \right ] \nonumber \\[2pt] && u_{C,\mathrm {h}} &= \mathrm {e}^{-\mathrm{d}elta t} \cdot \left [\vphantom {\Big |} \smash { a \cdot \underbrace {\left ( \mathrm {e}^{+\mathrm {j}\omega _d t} + \mathrm {e}^{-\mathrm {j}\omega _d t} \right )}_{2\cos (\omega _d t)} + \mathrm {j} b \cdot \underbrace {\left ( \mathrm {e}^{+\mathrm {j}\omega _d t} - \mathrm {e}^{-\mathrm {j}\omega _d t} \right )}_{2\mathrm {j}\sin (\omega _d t)} }\right ] \nonumber \\[20pt] &\text {with}& \mathrm {e}^{+\mathrm {j}x} + \mathrm {e}^{-\mathrm {j}x} &= \cos (x) + \cancel {\mathrm {j}\sin (x)} + \cos (x) - \cancel {\mathrm {j}\sin (x)} = 2\cos (x) \nonumber \\ && \mathrm {e}^{+\mathrm {j}x} - \mathrm {e}^{-\mathrm {j}x} &= \cancel {\cos (x)} + \mathrm {j}\sin (x) - \cancel {\cos (x)} + \mathrm {j}\sin (x) = 2\mathrm {j}\sin (x) \nonumber \\[6pt] && u_{C,\mathrm {h}} &= \mathrm {e}^{-\mathrm{d}elta t} \cdot \left [\vphantom {\Big |}\smash { \underbrace {2a}_{C_1} \cdot \cos (\omega _d t) - \underbrace {2b}_{C_2} \cdot \sin (\omega _d t) }\right ] \label {eq:dgl:rlc:homolsg:periodisch:herleitung} \end {align}

In periodic cases, the homogeneous solution corresponds to a decaying (\(\mathrm {e}^{-\mathrm{d}elta t}\)) superposition of the oscillations \(z_1\) and \(-z_2\) with the same angular frequency \(\omega _d\) and a phase shift of \(1/{\pi }{2}\) between them.

Figure 8 shows the superposition in the form of added time courses (a) and in the form of added complex amplitude pointers in the pointer diagram (b). Mathematically expressed:\begin {align} &&&{\textit {Time domain}}&&&&{\textit {Polarform}}&&{\textit {Cartesian}} &&\nonumber \\ && z_1(t) &= +C_1 \cdot \cos (\omega _d t)& &\Leftrightarrow & \underline {z}_1 &= C_1 \cdot \mathrm {e}^{\mathrm {j}0} & \underline {z}_1 &= C_1 \nonumber \\ && z_2(t) &= -C_2 \cdot \sin (\omega _d t)& &\Leftrightarrow & \underline {z}_2 &= C_2 \cdot \mathrm {e}^{-\mathrm {j}\frac {\pi }{2}} & \underline {z}_2 &= \hphantom {C_1} - \mathrm {j} \cdot C_2 \nonumber \\ && z_0(t) &= z_1(t) + z_2(t)& &\Leftrightarrow & \underline {z}_0 &= \underline {z}_1 + \underline {z}_2 & \underline {z}_0 &= C_1 - \mathrm {j} \cdot C_2 \nonumber \\ && &= C_0 \cdot \cos (\omega _d t + \psi )& &\Leftrightarrow & \hphantom {z_0} &= C_0 \cdot \mathrm {e}^{\mathrm {j}\psi } \nonumber \\[4pt] &\text {with}& C_0 &= \sqrt {C_1^2 + C_2^2}& &\text {and}& \psi &= \arctan \left (-\frac {C_2}{C_1}\right ), &\varphi _0 &= \psi + \frac {\pi }{2} \nonumber \\[4pt] && z_0(t) &= C_0 \cdot \sin (\omega _d t + \varphi _0) \label {eq:dgl:rlc:homolsg:periodisch:zeigeraddition} \end {align}

The homogeneous solution in the periodic case from Eq. ?? can be described according to Eq. ?? as a decaying, phase-shifted sine curve, as shown below (cf. Eq. ??): \begin {align*} u_{C,\mathrm {h}} &= \mathrm {e}^{-\mathrm{d}elta t} \cdot C_0 \cdot \sin (\omega _d t + \varphi _0) &&\text {with} \lambda _{1/2} = -\mathrm{d}elta \pm \mathrm {j} \cdot \omega _d \end {align*}

4.2 Digression: Case differentiation for RLC series resonant circuits, zero points

The zero points of the characteristic polynomial, i.e. the eigenvalues \(\lambda _{1/2}\) of the RLC series oscillating circuit, depend on the discriminant \(D\).

Figure 8a shows the location of the zero points in the complex number space as a locus curve as a function of \(D\) for \(\mathrm{d}elta = \frac {R}{2L} = const.\). The course of the NST corresponds to an increase in the undamped natural frequency \(\omega _0 = \frac {1}{\sqrt {LC}}\). [See Eq. ??]

In the aperiodic case (\(D>0\)), the two negative real zeros for \(\mathrm{d}elta =const.\) and increasing \(\omega _0\) move towards each other. The zero points closer to the imaginary axis dominates the transient behaviour of the system (slower decay process). In the aperiodic limit case (\(D=0\)), the zeros merge into a double zero at \(\lambda _{1/2}=-\mathrm{d}elta \). In the periodic limit case (\(D<0\)), the conjugate complex zeros for \(\mathrm{d}elta =const.\) and increasing \(\omega _0\) move parallel to the imaginary axis and away from the real axis.

Comparison: Pole locations of the transfer function (system theory)

The position of the eigenvalues in the locus diagram allows conclusions to be drawn about the transient behaviour of the RLC series resonant circuit. This procedure is common in the stability analysis of systems in control engineering and originates from system theory.

For vanishing initial conditions (\(\mathrm{d}t [i]\,u_C(0)=0\ \forall i\)), the differential equation from Eq. ?? can be described directly in the image domain using the Laplace transform. By rearranging the transformed output variable \(U_C(s)\) according to the input variable \(U(s)\), the transfer function \(G(s)\) of the RLC series resonant circuit is obtained:

\begin {align} u(t) &= LC \cdot \mathrm{d}t [2]\,u_{C}(t) + RC \cdot \mathrm{d}t \,u_{C}(t) + u_{C}(t)& &\bigg |\,\mathscr {L}\\ U(s) &= LC \cdot s^2 \cdot U_C(s) + RC \cdot s \cdot U_C(s) + U_C(s) \vphantom {\bigg |}\\ G(s) &= \frac {U_C(s)}{U(s)} = \frac {\text {Output variable}(s)}{\text {Input variable}(s)} \vphantom {\bigg |}\\ &= \frac {1}{LC \cdot s^2 + RC \cdot s + 1} \end {align}

The poles of the transfer function \(G(s)\) correspond to the eigenvalues \(\lambda _{1/2}\).

4.3 Decay constant, natural and resonance circuit frequency, quality factor

In the periodic case, the (damped) natural frequency \(\omega _d\) and the decay constant \(\mathrm{d}elta \) are derived from the eigenvalues \(\lambda _{1/2}\). These quantities are closely related to the quality factor \(Q_{\mathrm {S}}\) and the resonant frequency \(\omega _0\) of an oscillating circuit (resonant circuit). Both quantities are used as parameters for oscillating circuits and are discussed in more detail in Module 8 (Frequency-variable circuits).

The relationship between \(\omega _d\), \(\mathrm{d}elta \), \(\omega _0\) and \(Q_{\mathrm {S}}\) will be explained using the RLC series oscillating circuit from Figure 5a. According to Equation ??, the eigenvalues \(\lambda _{1/2}\) are given by: \begin {align} \lambda _{1/2} &= -\underbrace { \vphantom {\sqrt {\left (\frac {R}{2L}\right )^2}} \frac {R}{2L} }_{\mathrm{d}elta } \pm \, \mathrm {j} \cdot \underbrace { \sqrt {\frac {1}{LC} - \left (\frac {R}{2L}\right )^2} }_{\omega _d} \label {eq:dgl:rlc:lambda:periodisch:exkurs} \end {align}

The negative real part \(-\mathrm{d}elta \) determines the decay rate and the imaginary part \(\omega _d\) determines the angular frequency of the linearly damped harmonic oscillation. The natural angular frequency \(\omega _d\) can also be expressed as a function of \(\omega _0\) and \(\mathrm{d}elta \) using Eq. ??.\begin {align} \omega _d &= \sqrt {\frac {1}{LC} - \left (\frac {R}{2L}\right )^2} = \sqrt {\omega _0^2 - \mathrm{d}elta ^2} \label {eq:dgl:rlc:eigenkreisfrequenz} \end {align}

This has the advantage that the undamped natural frequency \(\omega _0\) and the decay constant \(\mathrm{d}elta \) can be read directly from the coefficients of the differential equation. The following form is suitable for this purpose: \begin {equation} \mathrm{d}dot y + 2\mathrm{d}elta \cdot \mathrm{d}ot y + \omega _0^2 \cdot y = b \end {equation}

for ordinary, homogeneous, linear second-order differential equations. The differential equation describes the behaviour of linear damped second-order systems (with two energy stores). The undamped natural frequency corresponds to the resonance frequency \(\omega _0\).

Resonance occurs when an oscillating circuit is excited at its resonance frequency \(f_0\). In the case of resonance, the reactive component of the total impedance \(\underline {Z}(\omega )\) disappears (resonance condition). [See Module 8]

With the total impedance: \begin {align} \underline {Z}(\omega ) &= R + \mathrm {j} \left ( \omega L - \frac {1}{\omega C} \right ) \nonumber \\ \end {align}

and the resonance condition: \begin {align} \Im \{\underline {Z}(\omega _0)\} &= \omega _0 L - \frac {1}{\omega _0 C} \overset {!}{=} 0 \nonumber \\ \end {align}

follows for the resonance circuit frequency: \begin {align} \omega _0 &= \sqrt {\frac {1}{LC}} \label {eq:dgl:rlc:resonanzkreisfrequenz} \end {align}

The quality factor \(Q_{\mathrm {S}}\), which measures the oscillation capability of an oscillating circuit, can also be derived from \(\omega _0\) and \(\mathrm{d}elta \). It is defined as the ratio of the reactive power oscillating within the circuit to the power loss within the circuit in the case of resonance. In the case of an RLC series oscillating circuit, this corresponds to the ratio of the reactive resistance of the inductance \(X_{L,0}\) at the resonance frequency to the series resistance \(R\) (without derivation). This results in the following relationship:

\begin {align} Q_{\mathrm {S}} &= \frac {X_{L,0}}{R} = \frac {\omega _0 L}{R} = \frac {\omega _0}{2\mathrm{d}elta } \label {eq:dgl:rlc:guete} \end {align}

The quality is proportional to the ratio of the undamped natural frequency to the damping constant.