Transients in Electric Circuit
Foundations of network analysis with dependent sources and evaluation of switching initial conditions in dynamic R-L-C circuits.
1.1 Characteristics of Various Network Elements
Concept & Classification
Electric circuits are composed of interconnected elements that store, dissipate, or deliver energy. Understanding their mathematical and physical classifications is essential before setting up system equations.
| Classification | Definition & Criterion | Examples |
|---|---|---|
| Linear vs. Non-linear | Satisfies the principles of superposition and homogeneity: \(f(a x_1 + b x_2) = a f(x_1) + b f(x_2)\). V-I curve is a straight line through the origin. | Linear: Pure resistors, ideal inductors, ideal capacitors. Non-linear: Diodes, transistors, saturable magnetic cores. |
| Bilateral vs. Unilateral | V-I characteristics are independent of current direction. Conduction behavior is identical in both directions. | Bilateral: Resistors, inductors, capacitors. Unilateral: Diodes, thyristors, operational amplifiers. |
| Active vs. Passive | An active element can deliver net average energy to the external circuit over an infinite time interval (\(\int_{-\infty}^t p(\tau)d\tau \ge 0\) is violated). | Active: Voltage/current sources, op-amps, BJT/FET. Passive: Resistors (absorb energy), Inductors/Capacitors (store energy). |
| Lumped vs. Distributed | Physical dimensions are much smaller than electrical signal wavelength (\(d \ll \lambda\)). KCL/KVL apply directly. | Lumped: Standard discrete R, L, C components. Distributed: Long transmission lines, waveguides. |
Energy Storage Element Equations
1.2 & 1.3 Nodal and Mesh Analysis with Dependent Sources
Core Principles
Dependent (controlled) sources have their output voltage or current determined by a voltage or current elsewhere in the circuit. There are four types:
1. VCVS: Voltage-Controlled Voltage Source (\(v = \mu v_x\))
2. CCVS: Current-Controlled Voltage Source (\(v = r i_x\))
3. VCCS: Voltage-Controlled Current Source (\(i = g v_x\))
4. CCCS: Current-Controlled Current Source (\(i = \beta i_x\))
When an ideal voltage source (independent or dependent) is connected between two non-reference nodes:
1. Enclose the voltage source and its two bounding nodes into a generalized boundary called a supernode.
2. Apply KCL to the external terminals of this supernode.
3. Supplement with the internal constraint equation: \(v_{\text{pos}} - v_{\text{neg}} = V_s\).
When an ideal current source (independent or dependent) is shared between two adjacent meshes:
1. Temporarily remove the branch containing the current source to form a combined supermesh.
2. Write KVL around the perimeter of the supermesh.
3. Supplement with the internal branch constraint equation: \(i_{\text{mesh, aiding}} - i_{\text{mesh, opposing}} = I_s\).
1.4 Matrix Formulation in Network Analysis
For linear networks, mesh equations are formulated systematically as:
\[\begin{bmatrix} R_{11} & R_{12} & \dots & R_{1n} \\ R_{21} & R_{22} & \dots & R_{2n} \\ \vdots & \vdots & \ddots & \vdots \\ R_{n1} & R_{n2} & \dots & R_{nn} \end{bmatrix} \begin{bmatrix} I_1 \\ I_2 \\ \vdots \\ I_n \end{bmatrix} = \begin{bmatrix} V_1 \\ V_2 \\ \vdots \\ V_n \end{bmatrix}\]
Where:
• \(R_{kk}\) = Self-resistance of mesh \(k\) (sum of all resistances in mesh \(k\)).
• \(R_{jk} = -(\text{Mutual resistance shared between mesh } j \text{ and } k)\).
• \(V_k\) = Algebraic sum of source voltages in mesh \(k\) aiding mesh current direction.
In purely passive networks, \([R]\) is always symmetrical (\(R_{jk} = R_{kj}\)). However, dependent sources introduce non-symmetrical off-diagonal terms once the controlling variable is substituted. Always move dependent terms to the left-hand matrix before solving!
1.5, 1.6 & 1.7 Procedure for Evaluating Initial Conditions & Derivatives
Universal Step-by-Step Procedure at Switching
- Analyze at \(t = 0^-\) (Pre-switching steady state):
Under DC excitation for a long time:
• All inductors behave as short circuits (\(v_L = 0\)). Find initial currents \(i_L(0^-)\).
• All capacitors behave as open circuits (\(i_C = 0\)). Find initial voltages \(v_C(0^-)\). - Apply Continuity Rules at \(t = 0^+\) (Post-switching instant): \[i_L(0^+) = i_L(0^-), \quad v_C(0^+) = v_C(0^-)\]
- Draw the Equivalent Circuit at \(t = 0^+\):
• Replace each inductor with an ideal current source of value \(i_L(0^+)\) (if \(i_L(0^+) = 0\), replace with an open circuit).
• Replace each capacitor with an ideal voltage source of value \(v_C(0^+)\) (if \(v_C(0^+) = 0\), replace with a short circuit).
• Solve purely resistive network for all currents \(i(0^+)\) and voltages \(v(0^+)\). - Evaluate First Derivatives \(\frac{di_L}{dt}(0^+)\) and \(\frac{dv_C}{dt}(0^+)\):
From element v-i relations: \[\frac{di_L}{dt}(0^+) = \frac{v_L(0^+)}{L}, \quad \frac{dv_C}{dt}(0^+) = \frac{i_C(0^+)}{C}\] Write loop KVL or node KCL at \(t = 0^+\) to find \(v_L(0^+)\) and \(i_C(0^+)\). - Evaluate Second Derivatives \(\frac{d^2 i}{dt^2}(0^+)\):
Write KVL/KCL equations as general time-varying differential functions for \(t > 0\). Differentiate once with respect to \(t\), then evaluate at \(t = 0^+\) by substituting previously calculated initial values and first derivatives.
Worked Master Example (IOE Exam Problem)
Common Mistakes to Avoid in Chapter 1
- Assuming inductor current is always zero at \(t=0^+\): If a switch has been closed for a long time before moving at \(t=0\), the inductor will carry a non-zero steady-state current \(i_L(0^-) = I_0\) which persists at \(t=0^+\)!
- Forgetting internal supernode constraint: In nodal analysis, forgetting to write the voltage difference relation (\(v_a - v_b = V_s\)) leaves the system with fewer equations than unknowns.
- Differentiating constants incorrectly: Remember that a fixed DC voltage source has \(\frac{dV}{dt} = 0\).