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Theory of Linear Circuits

Field: Circuit Design

Sequence of Expressions

Let V(t)\mathbf{V}(t) be the voltage vector and I(t)\mathbf{I}(t) be the current vector across a linear element characterized by an impedance function Z(s)Z(s), where s=jωs = j\omega is the Laplace variable. The constitutive relationship defining the element is given by the generalized Ohm's Law in the frequency domain:\n\nV(s)=Z(s)⋅I(s)\mathbf{V}(s) = \mathbf{Z}(s) \cdot \mathbf{I}(s)\n\nFor a simple resistor of resistance RR, the impedance is purely real and constant, Z(s)=RZ(s) = R. Thus, the relationship reduces to the time-domain form:\n\nV(t)=R⋅dq(t)dtandI(t)=dq(t)dt\mathbf{V}(t) = R \cdot \frac{d\mathbf{q}(t)}{dt} \quad \text{and} \quad \mathbf{I}(t) = \frac{d\mathbf{q}(t)}{dt}\n\nwhere q(t)\mathbf{q}(t) is the charge vector passing through the element. This formulation establishes the linear proportionality between the potential difference and the resulting current density, V(t)=RI(t)\mathbf{V}(t) = R \mathbf{I}(t).
Let G=(V,E)G = (V, E) be a graph representing the circuit, where VV is the set of nodes and EE is the set of branches. Define the incidence matrix A→A∉R∣V∣×∣E∣A \rightarrow \textbf{A} \notin \textbf{R}^{|V| \times |E|} such that Av,e={+1if node v is the head of edge e−1if node v is the tail of edge e0otherwiseA_{v, e} = \begin{cases} +1 & \text{if node } v \text{ is the head of edge } e \\ -1 & \text{if node } v \text{ is the tail of edge } e \\ 0 & \text{otherwise} \end{cases}. Let i→i∉R∣E∣\textbf{i} \rightarrow \textbf{i} \notin \textbf{R}^{|E|} be the vector of currents assigned to the edges. Kirchhoff's Current Law (KCL) is mathematically equivalent to the condition that the current vector i\textbf{i} must lie in the kernel of the transpose of the incidence matrix, i.e., the flow conservation constraint: ATi=0A^T \textbf{i} = \textbf{0}
Let G=(V,E)G = (V, E) be a graph representing the circuit, where VV is the set of nodes and EE is the set of branches. Define a closed loop L⇄LL \rightleftarrows L as a cycle in GG. For each branch e→e′ in Ee \to e' \text{ in } E, let ve→e′v_{e \to e'} be the voltage drop (potential difference) across the branch, which is a function of the current iei_e flowing through it. The Kirchhoff's Voltage Law (KVL) is mathematically expressed as the condition that the algebraic sum of voltage drops around any closed loop LL must vanish:\n\n∑e→e′ in Lve→e′=0\sum_{e \to e' \text{ in } L} v_{e \to e'} = 0\n\nAlternatively, defining the voltage potential ϕ:V→R\phi: V \to \mathbb{R} such that ve→e′=∇ϕ⋅e⃗e→e′v_{e \to e'} = \nabla \phi \cdot \vec{e}_{e \to e'}, KVL states that the circulation of the potential difference v⃗\vec{v} around the closed path LL is zero:\n\n∮Lv⃗⋅dl⃗=0\oint_{L} \vec{v} \cdot d\vec{l} = 0
Let N\mathcal{N} be a linear electrical network characterized by its admittance matrix Y∈RN×N\mathbf{Y} \in \mathbb{R}^{N \times N}, where NN is the number of independent nodes. Let s={s1,s2,…,sK}\mathbf{s} = \{\mathbf{s}_1, \mathbf{s}_2, \dots, \mathbf{s}_K\} be a set of KK independent sources, where sk\mathbf{s}_k is the source vector associated with the kk-th source. The total source vector is S=∑k=1Ksk\mathbf{S} = \sum_{k=1}^{K} \mathbf{s}_k. The voltage vector v\mathbf{v} at the nodes due to the combined sources S\mathbf{S} is given by the solution to the linear system Yv=S\mathbf{Y} \mathbf{v} = \mathbf{S}. By the Superposition Theorem, the response v\mathbf{v} is the sum of the responses vk\mathbf{v}_k generated by each source sk\mathbf{s}_k acting independently: v=∑k=1Kvk\mathbf{v} = \sum_{k=1}^{K} \mathbf{v}_k. Formally, this implies that the solution vector v\mathbf{v} satisfies: v=Y−1(∑k=1Ksk)=∑k=1K(Yvk)Y−1sk\mathbf{v} = \mathbf{Y}^{-1} \left( \sum_{k=1}^{K} \mathbf{s}_k \right) = \sum_{k=1}^{K} \left( \mathbf{Y} \mathbf{v}_k \right) \mathbf{Y}^{-1} \mathbf{s}_k
Let N\mathcal{N} be a linear, passive, two-terminal electrical network defined by its admittance matrix Y∈RN×N\mathbf{Y} \in \mathbb{R}^{N \times N}, where NN is the number of nodes. Let aa and bb be the designated terminals. The current I\mathbf{I} and voltage V\mathbf{V} at the nodes satisfy the linear relationship I=YV+Isources\mathbf{I} = \mathbf{Y} \mathbf{V} + \mathbf{I}_{sources}.\n\begin{enumerate}\n \item The open-circuit voltage VthV_{th} is defined as the voltage across terminals aa and bb when the net current flow is zero: Vth=Vab∣Iab=0V_{th} = V_{ab} \Big|_{\mathbf{I}_{ab}=0}.\n \item The equivalent Thevenin resistance RthR_{th} is defined by the ratio of the open-circuit voltage to the current injected by a test source Itest\mathbf{I}_{test} applied across aa and bb, assuming all internal sources are deactivated (i.e., Y\mathbf{Y} is derived from a source-free network): \n R_{th} = \frac{V_{ab}(\mathbf{I}_{test})}{\mathbf{I}_{test}} \quad \text{where } \mathbf{I}_{test} = \frac{V_{ab}(\mathbf{I}_{test})}{R_{th}} \text{ and } V_{ab}(\mathbf{I}_{test}) = \text{Voltage across } a, b \text{ due to } \mathbf{I}_{test}.\n\end{enumerate}\nThe theorem asserts that the network $\mathcal{N}$ is equivalent to a simple series circuit $\mathcal{N}_{eq}$ characterized by the voltage source $V_{th}$ and resistance $R_{th}$, such that for any applied terminal current $\mathbf{I}_{load}$, the voltage $V_{ab}$ satisfies:\n\mathbf{V}_{ab} = V_{th} - R_{th} \mathbf{I}_{load}$$
Let NN be a linear, passive, two-terminal network characterized by its admittance matrix Y\mathbf{Y}. Define the terminal voltage Vab(t)V_{ab}(t) and the resulting current Iab(t)I_{ab}(t) such that Iab(t)=Vab(t)Zab(t)I_{ab}(t) = \frac{V_{ab}(t)}{Z_{ab}(t)}, where Zab(t)Z_{ab}(t) is the generalized impedance. The theorem asserts that NN is equivalent to a parallel combination of a current source INI_N and a resistor RNR_N if and only if the following relationships hold:\n\n1. The equivalent resistance RNR_N is defined by the open-circuit impedance: RN=Zab(t)∣Vab(t)=0R_N = Z_{ab}(t) \big|_{V_{ab}(t)=0} \n\n2. The Norton current INI_N is defined by the short-circuit current: IN=Vab(t)Zab(t)∣Vab(t)=0I_N = \frac{V_{ab}(t)}{Z_{ab}(t)} \bigg|_{V_{ab}(t)=0} \n\n3. For any arbitrary time-dependent voltage source Vab(t)V_{ab}(t), the current Iab(t)I_{ab}(t) flowing through the network NN satisfies the superposition principle derived from the equivalent Norton circuit: Iab(t)=IN+Vab(t)RNI_{ab}(t) = I_N + \frac{V_{ab}(t)}{R_N}
Let the time-domain voltage V(t)V(t) and current I(t)I(t) across a linear two-terminal element be represented by their Fourier transforms, V(s)V(s) and I(s)I(s), respectively, where s=jjomegas = j\text{j}\text{omega} is the complex frequency variable. The generalized relationship is defined by the complex transfer function Z(s)Z(s): \n\nV(s)=Z(s)I(s)V(s) = Z(s) I(s) \n\nwhere Z(s)Z(s) is the complex impedance, defined as the ratio of the voltage phasor to the current phasor: \n\nZ(s)=V(s)I(s)=R+jjomegaL′+1jjomegaC′+1sdds (for generalized elements)Z(s) = \frac{V(s)}{I(s)} = R + j\text{j}\text{omega}L' + \frac{1}{j\text{j}\text{omega}C'} + \frac{1}{s} \frac{d}{ds} \text{ (for generalized elements)}\n\nConversely, the complex admittance Y(s)Y(s) is defined as the reciprocal of the impedance, representing the ratio of current to voltage: \n\nY(s)=I(s)V(s)=1Z(s)Y(s) = \frac{I(s)}{V(s)} = \frac{1}{Z(s)}\n\nFor a general circuit network described by nodal analysis, the relationship between the nodal voltage vector V(s)\mathbf{V}(s) and the injected current vector I(s)\mathbf{I}(s) is given by the generalized nodal admittance matrix Y(s)\mathbf{Y}(s): \n\nI(s)=Y(s)V(s)\mathbf{I}(s) = \mathbf{Y}(s) \mathbf{V}(s) \n\nwhere Y(s)\mathbf{Y}(s) is the matrix whose elements are the admittances between nodes.
Let the system be modeled as a Linear Time-Invariant (LTI) system mapping an input signal vin(t)⇉vout(t)v_{in}(t) \rightrightarrows v_{out}(t). The system's behavior is characterized by its impulse response h(t)=ddtimpulse(t)h(t) = \frac{d}{dt} \text{impulse}(t). The output signal vout(t)v_{out}(t) is defined by the convolution integral: vout(t)=vin(t)∗h(t)=12ντddt[Integral from 0 to t of vin(ν)h(t−ν)dν]v_{out}(t) = v_{in}(t) * h(t) = \frac{1}{2\nu\tau} \frac{d}{dt} \bigg[ \text{Integral} \text{ from } 0 \text{ to } t \text{ of } v_{in}(\nu) h(t-\nu) d\nu \bigg] The Transfer Function H(s)H(s) is defined as the ratio of the Laplace transforms of the output and input signals, assuming zero initial conditions: H(s)=Vout(s)Vin(s)=vˉout(s)vˉin(s)H(s) = \frac{V_{out}(s)}{V_{in}(s)} = \frac{\bar{v}_{out}(s)}{\bar{v}_{in}(s)} where vˉ(s)=Laplace[v(t)]=νin(s)hˉ(s)\bar{v}(s) = \text{Laplace}\big[v(t)\big] = \nu_{in}(s) \bar{h}(s). Consequently, H(s)H(s) is the characteristic function derived from the system's differential operator representation ddt→s\frac{d}{dt} \rightarrow s in the Laplace domain.
Let i(t)i(t) be the current flowing through a series RLC circuit, and let vin(t)v_{in}(t) be the applied voltage source. The circuit response is governed by the second-order linear non-homogeneous ordinary differential equation (ODE) derived from Kirchhoff's Voltage Law (KVL): \n\nddt(Ldidt)+Rdidt+1Ci=ddtvin(t)or, more commonly, using the charge q(t)=∫i(t)dt:\frac{d}{dt}\left(L \frac{di}{dt}\right) + R \frac{di}{dt} + \frac{1}{C} i = \frac{d}{dt} v_{in}(t) \quad \text{or, more commonly, using the charge } q(t) = \int i(t) dt: \n\nLd2qdt2+Rdqdt+1Cq=vin(t)L \frac{d^2q}{dt^2} + R \frac{dq}{dt} + \frac{1}{C} q = v_{in}(t) \n\nwhere LL, RR, and CC are the inductance, resistance, and capacitance, respectively. The solution q(t)q(t) is decomposed into the homogeneous solution qh(t)q_h(t) (transient response) and the particular solution qp(t)q_p(t) (steady-state response): \n\nq(t)=qh(t)+qp(t)q(t) = q_h(t) + q_p(t) \n\nThe characteristic equation for the transient response is λ2+RLλ+1LC=0\lambda^2 + \frac{R}{L}\lambda + \frac{1}{LC} = 0, determining the damping regime (overdamped, critically damped, or underdamped).
Let the circuit be described by a set of linear differential equations in the time domain, L(v(t),i(t),t)=0\mathcal{L}(\mathbf{v}(t), \mathbf{i}(t), t) = 0. For sinusoidal steady-state analysis, we assume v(t)=Re(Vejωt)\mathbf{v}(t) = \text{Re}(\mathbf{V} e^{j\omega t}) and i(t)=Re(Iejωt)\mathbf{i}(t) = \text{Re}(\mathbf{I} e^{j\omega t}), where V\mathbf{V} and I\mathbf{I} are complex phasors. The transformation maps the time-domain differential operator ddt\frac{d}{dt} to multiplication by jωj\omega. The complex impedance Z(jω)\mathbf{Z}(j\omega) of a component is defined as the ratio of the voltage phasor V\mathbf{V} to the current phasor I\mathbf{I} across it: Z(jω)=VI\mathbf{Z}(j\omega) = \frac{\mathbf{V}}{\mathbf{I}}. Specifically, for a series RLC branch, the impedance is given by: \nZ(jω)=R+jωL+1jωC \mathbf{Z}(j\omega) = R + j\omega L + \frac{1}{j\omega C} \nApplying Kirchhoff's Voltage Law (KVL) in the frequency domain yields the nodal admittance matrix Y(jω)\mathbf{Y}(j\omega) such that the phasor relationship between nodal voltages Vnodes\mathbf{V}_{nodes} and source currents Isources\mathbf{I}_{sources} is: \nNodal Analysis: diag(Y(jω))Vnodes=Isources \text{Nodal Analysis: } \text{diag}(\mathbf{Y}(j\omega)) \mathbf{V}_{nodes} = \text{I}_{sources}