Rumination on first order electrical circuit
Contents
1.1 Definition of linear operator
1.2 Relationship between kernel and image
1.3 Structure of the solution set of a linear equation
2 Solution of a first order linear time invariant (LTI) circuit
2.1 Definition of steady state
2.2 Meaning of general solution and particular solution
2.3 Transient process and switching theorem
1 General theory about linear operator
1.1 Definition of linear operator
Let \(L\) be a linear operator from a vector space \(V\) to \(W\), i.e. \(L\in \mathcal {L}(V,W)\). For all \(x,y\) in \(V\) and \(\alpha \in \mathbb {R}\), \begin{equation} \begin {aligned} L(x+y)&=L(x)+L(y) \\ L(\alpha x)&=\alpha L(x) \end {aligned}. \end{equation} Examples of linear operators:
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Linear operator between Cartesian spaces: \(A: \mathbb {R}^n \rightarrow \mathbb {R}^{m}\), hence \(A\) is a matrix in \(\mathbb {R}^{m\times n}\).
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Linear operator in a first order ordinary differential equation (ODE): \(D = a_1\frac {\diff }{\diff t}+a_0\).
1.2 Relationship between kernel and image
Definition 1 (Kernel and image of a linear operator) Let \(L\in \mathcal {L}(V,W)\). Its kernel is the subspace of \(V\) \begin{equation} \kernel (L) = \left \{ v\in V \vert Lv = 0 \right \}. \end{equation} Its image is the subspace of \(W\) \begin{equation} \image (L) = \left \{ w\in W \vert \exists v\in V, \suchthat Lv = w \right \}. \end{equation}
Theorem 1 (Rank plus nullity theorem) Let \(L\in \mathcal {L}(V,W)\). \begin{equation} \mathrm {dim}(\kernel (L)) + \mathrm {dim}(\image (L)) = \mathrm {dim}(V) \end{equation}
Proof The domain of the linear operator \(L\) can be decomposed as a direct sum of the kernel and its orthogonal complement \begin{equation} V = \kernel (L) \oplus \kernel (L)^{\perp }. \end{equation} Therefore \begin{equation} \mathrm {dim}(V) = \mathrm {dim}(\kernel (L)) + \mathrm {dim}(\kernel (L)^{\perp }). \end{equation} If we restrict the domain of \(L\) from \(V\) to \(\kernel (L)^{\perp }\) and let \(B = \kernel (L)^{\perp }\), we get a new operator \(\tilde {L}: B \rightarrow \image (L)\). We will show that this map is bijective.
Because \(\tilde {L}\) is a restriction of \(L\), \(\kernel (\tilde {L})\) is also a restriction of \(\kernel (L)\) to \(B\), i.e. \begin{equation} \kernel (\tilde {L}) = \kernel (L)\cap B = \kernel (L)\cap \kernel (L)^{\perp } = \left \{ 0 \right \}. \end{equation} Therefore, \(\tilde {L}\) is injective.
For any \(w\in \image (L)\), there exists \(v\in V\) such that \(Lv = w\), where \(v\) can be decomposed as \(v = v_0 + v_1\) with \(v_0\in \kernel (L)\) and \(v_1\in \kernel (L)^{\perp }\). Hence \(Lv = L(v_0+v_1) = L(v_0) + L(v_1) = L(v_1) = \tilde {L}(v_1) = w\). Therefore, \(v_1\) is the pre-image of \(w\) under \(\tilde {L}\). \(\tilde {L}\) is surjective. Because it is also injective, \(\tilde {L}\) is bijective and \(\mathrm {dim}(\kernel (L)^{\perp }) = \mathrm {dim}(\image (L))\). Then the theorem is proved.
1.3 Structure of the solution set of a linear equation
The inhomogeneous linear equation is \(Lx = b\), where \(b\in W\) is the given data. To find the set of solutions \(x\in X \subset V\), it is equivalent to find the domain space of the restricted operator \(\tilde {L}: X \rightarrow \left \{ b \right \}\). According to Theorem 1, the dimension of the solution set \(X\) is equal to the dimension of the kernel of \(\tilde {L}\). Because \(\tilde {L}\) represents the same operation as \(L\), it is also the kernel of \(L\). This kernel is just the solution set of the homogeneous equation \(Lx = 0\).
Let \(x_\mathrm {p}\) be a particular solution of \(Lx = b\), i.e. \(Lx_\mathrm {p} = b\). Then the solution space is \begin{equation} X = \kernel (L) + x_\mathrm {p}. \end{equation} For any \(x_\mathrm {h}\in \kernel (L)\), it is called a general solution of the homogeneous equation \(Lx = 0\). Therefore, a specific solution of the inhomogeneous linear equation is the sum of a general solution and a particular solution. Due to the linearity of the operator \(L\), it is obvious that such a decomposed formulation is really the solution of \(Lx=b\): \begin{equation} L(x_{\mathrm {h}} + x_{\mathrm {p}}) = Lx_{\mathrm {h}} + Lx_{\mathrm {p}} = 0 + b = b. \end{equation} Which one of \(x_{\mathrm {h}}\) should be selected from \(\kernel (L)\) depends on additional constraints.
2 Solution of a first order linear time invariant (LTI) circuit
The equation for a first order LTI circuit is just a specific example of the above general linear equation \(Lx=b\), where \(L= D = \frac {\diff }{\diff t} - \lambda \). Here we write the original differential operator \(a_1\frac {\diff }{\diff t} + a_0\) as this equivalent form \(\frac {\diff }{\diff t}-\lambda \) is because we want to cast the homogeneous equation into an eigen equation form.
2.1 Definition of steady state
If a device or a circuit is operating under a steady state, according to the semantics of steady, the state of the device or circuit is either time invariant or periodic. So we have two situations:
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The state of the circuit is a constant function, which can be handled by the switching theorem and three-factor theorem.
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The state of the circuit is a periodic function.
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If the excitation is a special periodic function, i.e. sinusoidal, phasor method will be used.
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If the excitation is an arbitrary periodic function, Fourier series expansion will be applied to the excitation and phasor method will be adopted for each independent frequency. Based on the principle of superposition for LTI circuit, the total response is a linear combination of the responses at all frequencies with the same expansion coefficients in the former Fourier series. Therefore, the result usually takes the form of an infinite sum.
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In the following, we only consider the first case of steady state.
2.2 Meaning of general solution and particular solution
According to the linear operator theory, the solution set \(X\) of a first order LTI circuit equation \(Dx=b\) should have the form \begin{equation} X = \kernel (D) + x_{\mathrm {p}}(t), \end{equation} where the particular solution \(x_{\mathrm {p}}(t)\) represents the steady state of the circuit. In reality, the circuit has fixed parameters and a certain initial state \(x(0)\). It is this initial state \(x(0)\) that the solution \(x(t)\) can be uniquely found in the set \(X\).
The kernel of \(D\) is the just the solution set of the homogeneous equation \begin{equation} \frac {\diff x_{\mathrm {h}}(t)}{\diff t} - \lambda x_{\mathrm {h}}(t) = 0. \end{equation} This is just the eigen equation for the differential operator \(\frac {\diff }{\diff t}\): \begin{equation} \frac {\diff }{\diff t} x_{\mathrm {h}}(t) = \lambda x_h(t). \end{equation} Because the eigen function 1 of \(\frac {\diff }{\diff t}\) is \(\exp (\lambda t)\), the 1-dimensional eigen space 2 associated with the eigen value \(\lambda \) is \begin{equation} C \exp (\lambda t), \end{equation} where \(C\) is an unknown constant. This eigen space is just \(\kernel (D)\).
Therefore, the general form of the solution for the inhomogeneous equation \(Dx = b\) is \begin{equation} x(t) = x_{\mathrm {h}}(t) + x_{\mathrm {p}}(t) = C\exp (\lambda t) + x_{\mathrm {p}}(t). \end{equation} The constant \(C\) can be determined from the initial condition \(x(0)\) of the circuit by solving the equation below \begin{equation} x(0) = C\exp (\lambda 0) + x_{\mathrm {p}}(0) = C + x_{\mathrm {p}}(0). \end{equation} Since we only consider the first case of steady state, \(x_{\mathrm {p}}(t)\) is a constant \(x_{\mathrm {p}}\). Therefore \(C = x(0) - x_{\mathrm {p}}\). In classical electrical circuit book, \(x_{\mathrm {p}}\) is written as \(x_{\infty }\).
2.3 Transient process and switching theorem
We still consider the first case of steady state. Therefore, circuit switching means when the topology of the circuit changes, its state changes from the first steady state \(x_{\mathrm {p}_1}\) to the second steady state \(x_{\mathrm {p}_2}\). The transient process in-between is described by the solution \(x(t)\) under the initial condition \(x(0)\). And the switching theorem is a method to compute this initial condition. The basic idea is energy cannot suddenly change.