Electric Circuits

Network theorems and equivalent circuits

Thevenin's theorem states that a linear two-terminal network can be replaced by an ideal voltage source V_{th} in series with an impedance Z_{th}. The Thevenin voltage is the open-circuit voltage between the two terminals, and the Thevenin impedance is the impedance seen looking into those terminals with all independent sources deactivated, that is, voltage sources replaced by short circuits and current sources replaced by open circuits. When the network contains dependent sources, they must stay active while the sources are deactivated, so Z_{th} is found by injecting a test source at the terminals and taking the ratio of the test voltage to the test current, or by computing Z_{th}=V_{oc}/I_{sc}. Norton's theorem is the dual form, with a current source I_N in parallel with R_N, where R_N=R_{th} and I_N=V_{th}/R_{th}. A load connected to a linear source attains maximum power when Z_L=Z_{th}^{*}; if the load is purely resistive, the condition becomes R_L=|Z_{th}|. The same rule applies to a resistive two-port, where R_L equals the output resistance found after deactivating the sources. Ideal voltage sources that are deactivated become shorts, so if they tie two nodes together, any resistor joining those nodes is shorted and the Thevenin resistance between them is zero. The Thevenin method also applies in the phasor domain, where the open-circuit phasor is converted back to an instantaneous waveform. A branch containing a current source in parallel with a resistor has a straight-line i-v characteristic, with intercept equal to the source current and slope equal to the conductance of the resistor. Finally, a branch carrying a current source does not change the node voltage fixed by a voltage source elsewhere, so such branches can be ignored when finding that node voltage.

Sinusoidal steady-state AC analysis

In sinusoidal steady state, every voltage and current is represented by a phasor, and each element is described by its impedance Z_R=R, Z_L=j\omega L, and Z_C=1/(j\omega C). A series or parallel combination of elements is analysed with the same rules as resistive circuits, using complex arithmetic. A capacitive divider formed by a source resistance and a capacitor across the output shows that the output amplitude ratio is X_C/\sqrt{R_s^2+X_C^2}, so loading the source with a capacitor reduces the terminal voltage. For a balanced three-phase system with a star or delta load, the line-to-line voltage is \sqrt{3} times the phase voltage, and phase sequence determines the angular position of each phasor. When the neutral points of source and load are not connected, the displacement voltage between them is found from a weighted sum of phasor voltages divided by the sum of admittances. An unbalanced load connected in star with a grounded or isolated neutral causes a shift of the neutral point, and the displacement voltage can be made zero by choosing a suitable impedance in one phase. A branch whose neutral is connected directly to one line of the supply can be treated by applying Kirchhoff's current law at the neutral node, which fixes the phase currents. Converting a delta load to an equivalent star divides each impedance by three, which is useful when combining balanced branches in parallel. The average power of a single-phase sinusoidal voltage and current with phase difference \theta is V_{rms}I_{rms}\cos\theta, and over a half cycle the average of the double-frequency term is zero.

Switched RL/RC circuit transient

The inductor current and the capacitor voltage cannot change instantaneously at a switching instant, so i_L(0^+)=i_L(0^-) and v_C(0^+)=v_C(0^-). These values are found from the steady state that exists before the switch operates. In steady state with a DC source, an inductor behaves as a short circuit and a capacitor behaves as an open circuit, which gives the initial conditions for the transient analysis. After the switch changes position, the derivative of the other variable at t=0^+ is found from Kirchhoff's voltage or current law, so the inductor voltage at 0^+ follows from the continuity of current. For a first-order circuit the solution has the form x(t)=x_\infty+[x(0^+)-x_\infty]e^{-t/\tau}, where \tau=RC for an RC circuit and \tau=L/R for an RL circuit, with R the Thevenin resistance seen by the energy storage element. The capacitor voltage at t=0^+ may differ from the source voltage because a resistor in the path drops a voltage before the capacitor, and the final value is the voltage obtained by treating the capacitor as an open circuit. When a sinusoidal source is switched in, a transient-free switching instant requires the capacitor voltage at the switching instant to equal the forced response value, so the forced response and the switching time must be matched. A circuit with an independent DC source that is switched in while an element is charged from a different source has an initial condition determined by the state before switching and an exponential approach to the new final value. Solving for the time at which a charging capacitor reaches a given voltage uses the logarithm of the ratio of the remaining difference to the initial difference. When two inductive branches are connected in parallel to a common source and the switch changes the topology, the transient is governed by the characteristic equation of the combined network, which gives one effective time constant.

Mutual inductance and coupled coils

When two coils are magnetically coupled, the voltage across one coil contains a term proportional to the rate of change of the current in the other coil, with proportionality constant M. For an open-circuited secondary, the induced voltage is |V_2|=\omega M|I_1|, which gives a direct way to measure M from the primary current and secondary voltage. The coupling coefficient satisfies k=M/\sqrt{L_1L_2} and cannot exceed unity, and unity coupling means M^2=L_1L_2. When both currents enter the dotted terminals, the mutual flux aids the self flux and the mutual term is added, but if one current leaves a dotted terminal the sign of the mutual term reverses. The input impedance seen at the primary terminals of a coupled circuit is found by reflecting the secondary impedance, giving Z_{in}=Z_{11}-(\omega M)^2/Z_{22} for the case where both currents enter the dots. Reflected impedance is a useful way to remove the coupled coil from the analysis, and the resulting expression is used to find the Thevenin impedance between two terminals. A coupled-coil network with unity coupling has stored energy written as a quadratic form in the two currents, and the energy is a perfect square when the currents are in the ratio fixed by the self inductances and mutual inductance.

Laplace-domain circuit analysis

The Laplace transform converts a linear time-invariant circuit into an algebraic circuit in which an inductor of inductance L becomes an impedance sL, a capacitor of capacitance C becomes 1/(sC), and initial conditions appear as independent sources. The transformed circuit is solved with the same network theorems used for phasor analysis. For coupled coils, the transformed impedance of the primary is sL_1 and the mutual term becomes sM, so the input impedance of a coupled network is a rational function of s obtained after eliminating the secondary current. The reflected impedance term in the Laplace domain is (sM)^2/(sL_2+Z_L) and is subtracted from the primary impedance when both currents enter the dotted terminals. An initial inductor current is represented as a source Li(0^-) in series with sL, and a switched source is represented by its Laplace transform, so the response of a switched circuit follows directly from partial fractions of the transformed current. The inverse transform of a sum of exponentially decaying terms gives the time-domain response, where each pole determines a time constant of the circuit.

Resonant circuit Q-factor

The quality factor of a resonant circuit is defined as the ratio of reactive power to real power, and for a series RLC circuit at resonance Q=\omega_0L/R=1/(\omega_0CR). The quality factor of a component is Q=X/R for its reactance X and series loss resistance R. When the inductor and capacitor are both lossy, their quality factors do not simply add. The total series loss resistance is the sum of the individual resistances, and since the reactance at resonance is the same for both elements, the reciprocals of the quality factors add: 1/Q=1/Q_L+1/Q_C. For a resonant circuit, the bandwidth is related to the resonant frequency by Q=f_0/\text{BW}, where BW is the half-power bandwidth, and this relation holds independently of how the losses are distributed. A high Q means a narrow bandwidth and a sharp resonance, so the selectivity of a tuned circuit is determined directly by its quality factor.

Nodal analysis of resistive circuits

Nodal analysis writes Kirchhoff's current law at each non-reference node, expressing each branch current as the voltage difference across the branch divided by its resistance. A node connected to an ideal voltage source has its voltage fixed directly, so it is not an unknown in the equation set. When a dependent source appears, its controlling variable is expressed in terms of the node voltages, and the constraint is added to the node equations before solving. A branch containing an ideal current source contributes its source current to the KCL equation of the node it enters, and the source does not contribute a conductance. If a node is connected to a current source with the rest of the branch in series with an ideal voltage source, the node voltage is determined by KCL alone, and any branch carrying no current produces zero power. The power delivered by a source is the product of its voltage and the current leaving its positive terminal, so a source whose current is zero delivers zero power regardless of its voltage.