On the Calculation of Transient Regimes in a Circuit with Sinusoidal EMFs
Abstract Generated abstract
This paper addresses the calculation of transient currents and voltages in linear electrical circuits driven by sinusoidal electromotive forces. Starting from matrix loop equations with inductive, resistive, and capacitive terms, it separates the operational current into components due to applied sources and to initial stored energy, treating the latter by standard methods. For the source driven component, the authors use complex amplitudes and the displacement theorem to replace the operational image of sinusoidal excitation by a constant complex vector and introduce a shifted operational admittance obtained by substituting p plus j omega for p. The method reduces the transient problem with sinusoidal sources to an analogous operational calculation for constant complex excitations, after which inverse transformation yields the time dependent complex current amplitude and the actual current.
Full Text
On the Calculation of Transient Regimes in a Circuit with Sinusoidal Electromotive Forces
B. A. Borkovsky, G. E. Pukhov
1. Let us consider the possibilities of simplifying the solution of problems of finding currents and voltages in a linear circuit during a transient regime for the case when the electromotive forces of the sources are sinusoidal functions of time.
The loop equations of the circuit have the form:
\[ L \frac{di}{dt} + ri + k \int_{-\infty}^{t} i\,dt = e, \tag{1} \]
where \(L\), \(r\), and \(k\) are matrices of the form:
\[ L = \begin{vmatrix} L_{11} & L_{12} & \cdot & L_{1n} \\ L_{21} & L_{22} & \cdot & L_{2n} \\ \cdot & \cdot & \cdot & \cdot \\ L_{n1} & L_{n2} & \cdot & L_{nn} \end{vmatrix}, \qquad r = \begin{vmatrix} r_{11} & r_{12} & \cdot & r_{1n} \\ r_{21} & r_{22} & \cdot & r_{2n} \\ \cdot & \cdot & \cdot & \cdot \\ r_{n1} & r_{n2} & \cdot & r_{nn} \end{vmatrix}, \]
\[ k = \begin{vmatrix} k_{11} & k_{12} & \cdot & k_{1n} \\ k_{21} & k_{22} & \cdot & k_{2n} \\ \cdot & \cdot & \cdot & \cdot \\ k_{n1} & k_{n2} & \cdot & k_{nn} \end{vmatrix}. \tag{2} \]
(\(n\) is the number of independent loops), while the current \(i\) and the e.m.f. \(e\) are vectors
\[ i= \begin{vmatrix} i_1\\ i_2\\ \cdot\\ i_n \end{vmatrix}, \qquad e= \begin{vmatrix} e_1\\ e_2\\ \cdot\\ e_n \end{vmatrix} = \begin{vmatrix} {\mathcal E}_1\sin(\omega t+\alpha_1)\\ {\mathcal E}_2\sin(\omega t+\alpha_2)\\ \cdot\\ {\mathcal E}_n\sin(\omega t+\alpha_n) \end{vmatrix} \tag{3} \]
The solution of equation (1), written in operational form, will be
\[ I(p)=\left(pL+r+\frac{1}{p}k\right)^{-1} \left[{\mathcal E}(p)+pLi(0)-u(0)\right] \tag{4} \]
Here
\({\mathcal E}(p)\) is the operational image of the vector \(e\),
\(I(p)\) is the same for the current \(i\),
\(i(0)\) is the initial value of the current \(i\),
\(u(0)\) is the initial value of the voltage
\[ u=k\int_{-\infty}^{0} i\,dt. \tag{5} \]
From equation (4) it is seen that the operational current of the circuit consists of three parts, respectively equal to
\[ I_1(p)=Y(p){\mathcal E}(p), \tag{6} \]
\[ I_2(p)=Y(p)pLi(0), \tag{7} \]
\[ I_3(p)=-Y(p)u(0), \tag{8} \]
where
\[ Y(p)=\left(pL+r+\frac{1}{p}k\right) \tag{9} \]
is the operational admittance of the circuit.
The currents \(i_2 \doteq I_2(p)\) and \(i_3 \doteq I_3(p)\), caused by the store of energy in the inductive and capacitive elements of the circuit at the initial instant of time, do not depend on the form of the applied e.m.f.’s and must be determined in the usual way.
As for the current \(i_1 \doteq I_1(p)\), its calculation, as shown below, can be somewhat simplified by replacing the operational function \({\mathcal E}(p)\) by a certain constant complex number.
- The current \(i_1\) and the e.m.f. \(e\) can be represented in the form of half-sums of complex-conjugate vectors
\[ i'=\hat{i}(t)e^{j\omega t}= \begin{vmatrix} \hat{i}_1(t)\\ \hat{i}_2(t)\\ \cdot\\ \hat{i}_n(t) \end{vmatrix} e^{j\omega t}, \qquad i''=\hat{i}(t)e^{-j\omega t}, \tag{10} \]
\[ e'=\dot{\mathcal{E}}e^{j\omega t}= \begin{bmatrix} \dot{\mathcal{E}}_{1}\\ \dot{\mathcal{E}}_{2}\\ \cdot\\ \dot{\mathcal{E}}_{n} \end{bmatrix} e^{j\omega t},\quad e''=\hat{\dot{\mathcal{E}}}e^{-j\omega t}. \tag{11} \]
Indeed,
\[ i_1=\frac{1}{2}(i'+i'') \tag{12} \]
and
\[ e=\frac{1}{2}(e'+e'') \tag{13} \]
The complex amplitude of the current \(\dot{I}(t)\), owing to the nonsinusoidal character of \(i_1\), will be a function of time, while the complex amplitude of the emf will be a constant vector equal to:
\[ \dot{\mathcal{E}}=-j \begin{bmatrix} \mathcal{E}_{1}e^{j\alpha_1}\\ \mathcal{E}_{2}e^{j\alpha_2}\\ \cdot\\ \mathcal{E}_{n}e^{j\alpha_n} \end{bmatrix} = \begin{bmatrix} \dot{\mathcal{E}}_{1}\\ \dot{\mathcal{E}}_{2}\\ \cdot\\ \dot{\mathcal{E}}_{n} \end{bmatrix}. \tag{14} \]
It is known that if any linear operations are performed on a complex number \(\bar z\), and then exactly the same operations are performed on the conjugate number \(\hat z\), the results will be mutually conjugate numbers.
Therefore equation (1) will also be valid for the current \(i'\) when \(e\) is replaced by \(e'\), i.e.,
\[ L\frac{di'}{dt}+ri'+k\int i'\,dt=e'. \tag{15} \]
If from this the current \(i'\) is determined, then the actual current is found from the equation
\[ i_1=\frac{1}{2}(i'+\hat{i'}), \tag{16} \]
since \(i''=\hat{i}'\) (\(i'\) and \(i''\) are complex numbers).
We shall seek the solution of equation (15) in the form
\[ i'=\dot{I}(t)e^{j\omega t}. \tag{17} \]
Substituting into (15), we obtain
\[ L\frac{d}{dt}\left[\dot{I}(t)e^{j\omega t}\right] +r\left[\dot{I}(t)e^{j\omega t}\right] +k\int\left[\dot{I}(t)e^{j\omega t}\right]dt =\dot{\mathcal{E}}e^{j\omega t}. \]
On the basis of the displacement theorem, the operational image of the last equation may be written in the form:
\[ \left(Lp+r+\frac{1}{p}k\right)\dot{I}(p-j\omega)=\dot{\mathcal{E}}. \tag{18} \]
In deriving this expression the initial conditions were taken to be zero, since, in accordance with what was said above, we agreed to determine the circuit currents arising from the initial store of energy in the usual way.
We shall find the complex amplitude of the operational current by replacing \(p\) by \(p+j\omega\) in the equation. We have
\[ \dot I(p)=Y(p+j\omega)\dot{\mathcal E}, \tag{19} \]
where the quantity
\[ Y(p+j\omega)=\left[\,L(p+j\omega)+r+\frac{1}{p+j\omega}\,k\,\right]^{-1} \tag{20} \]
is naturally called the shifted operational admittance, since it is obtained from the ordinary admittance \(Y(p)\) by replacing \(p\) by \(p+j\omega\).
Thus, the problem of calculating transient regimes in a circuit with sinusoidal e.m.f.’s has formally been reduced to the problem of a circuit with constant e.m.f.’s, in the role of which the complex amplitudes of the sinusoids appear. For \(\omega=0\), formula (19) in general becomes identical with the formula for constant e.m.f.’s.
The determination of the current \(i_1\) is practically reduced to carrying out the following operations:
a) the image of the complex amplitude of the current \(\dot I(p)\) is determined; for this purpose the vector of complex amplitudes of the sinusoidal e.m.f.’s \(\dot{\mathcal E}\) is multiplied by the shifted operational admittance \(Y(p+j\omega)\);
b) by means of the expansion theorem or formulas based on the theory of residues, the transition is made from the operational function \(\dot I(p)\) to the original, which will be the complex amplitude of the current \(\dot I(t)\);
c) from the formula
\[ i_1=\frac{1}{2}\left[\dot I(t)e^{j\omega t}+\hat I(t)e^{-j\omega t}\right] =\operatorname{Im}\left[\dot I(t)e^{j\omega t}\right] \]
the current \(i_1\) is found.
After determining the current \(i_1\), the value of the actual current of the circuit at any instant of time is determined as the sum of the currents \(i_1\), \(i_2\), and \(i_3\).
We note that expression (19) is a generalization of the formula obtained earlier for a simple circuit with one sinusoidal e.m.f. [1].
REFERENCES
- Pukhov G. E. On a Certain Generalization of the Symbolic Method. Scientific Notes of the LPI, issue II, 1948.