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Current I (ampere) is charge flow; voltage V (volt) is energy per unit charge between two points; power P = V I (watt), positive when a device absorbs energy (passive sign convention: current enters the positive terminal). Elements - Resistor: V = R I (Ohm's law), dissipates power R I^2 as heat; conductance G = 1/R. - Capacitor: I = C dV/dt; stores energy C V^2/2; blocks direct current. - Inductor: V = L dI/dt; stores energy L I^2/2; behaves like a short circuit for direct current. - Independent voltage and current sources, and dependent (controlled) sources used to model transistors. Series connections share the current (resistances add); parallel connections share the voltage (conductances add). Voltage divider: V_out = V_in R2/(R1 + R2); current divider: I1 = I R2/(R1 + R2).

Kirchhoff's current law (KCL): the algebraic sum of currents at a node is zero (charge conservation). Kirchhoff's voltage law (KVL): the algebraic sum of voltages around a closed loop is zero (energy conservation). Nodal analysis: choose a reference node, write KCL at each other node in terms of node voltages, and solve. Mesh analysis: write KVL around each independent loop in terms of mesh currents. Choose nodal analysis when there are few nodes or many current sources, and mesh analysis for the opposite.

1. State KCL and KVL. 2. What is the voltage divider formula? 3. How does an inductor behave for direct current? 4. Two 100 ohm resistors in parallel: what is the equivalent resistance?

1. KCL: the sum of currents into a node is zero. KVL: the sum of voltages around a loop is zero. 2. V_out = V_in R2/(R1 + R2). 3. As a short circuit (zero voltage at steady state). 4. 50 ohm.

Thevenin equivalent: a voltage source V_th in series with a resistance R_th. V_th is the open-circuit voltage; R_th is the resistance seen from the terminals with independent sources turned off (voltage sources replaced by shorts, current sources by opens). Norton equivalent: a current source I_N = V_th/R_th in parallel with R_th. Maximum power transfer to a load occurs when R_load = R_th. Superposition (linear circuits): the response to several independent sources is the sum of the responses to each source acting alone with the others turned off.

First-order circuits (one storage element): an RC circuit charging through R has v(t) = V_final + (V_initial - V_final) exp(-t/tau) with time constant tau = R C; an RL circuit has tau = L/R. After 5 tau the transient is over (99 percent). Second-order (RLC) circuits: response depends on the damping: overdamped (two real exponentials), critically damped, or underdamped (damped oscillation) with natural frequency omega0 = 1/sqrt(LC). Applications: defibrillator pulses, filters and stimulator waveforms.

1. How is R_th determined? 2. State the maximum power transfer condition. 3. What is the time constant of an RC circuit? 4. When is an RLC circuit underdamped?

1. Turn off independent sources and compute the resistance seen from the terminals. 2. The load resistance equals the Thevenin resistance. 3. tau = R C. 4. When the resistance is low compared with 2 sqrt(L/C) for a series RLC, so that the response oscillates.

A sinusoid v(t) = Vm cos(omega t + phi) is represented by the phasor V = Vm exp(j phi) (RMS values are used for power: V_rms = Vm/sqrt(2); mains 230 V is RMS). Impedance Z = V/I: resistor Z = R; inductor Z = j omega L; capacitor Z = 1/(j omega C). All the DC methods (KCL, KVL, Thevenin, dividers) work with impedances. Transfer function H(j omega) = output/input as a function of frequency; its magnitude and phase describe filters. An RC low-pass filter has cut-off frequency f_c = 1/(2 pi R C) (gain down by 3 dB).

Active power P = V_rms I_rms cos(phi) (watt), the average power converted to useful work or heat. Reactive power Q = V_rms I_rms sin(phi) (var), exchanged with inductors and capacitors and not consumed. Apparent power S = V_rms I_rms (volt-ampere), with S squared = P squared + Q squared. Power factor cos(phi) = P/S; a low power factor increases line currents and losses, and is corrected with capacitors. Three-phase systems: three voltages of equal magnitude displaced by 120 degrees, delivering constant power, used for distribution and large equipment (star or delta connection; line voltage is sqrt(3) times the phase voltage in a star connection).

1. Write the impedance of an inductor and a capacitor. 2. What is the cut-off frequency of an RC low-pass filter? 3. Define power factor. 4. What is the relation between line and phase voltage in a star connection?

1. Inductor: j omega L; capacitor: 1/(j omega C). 2. f_c = 1/(2 pi R C). 3. The ratio of active to apparent power, cos(phi). 4. Line voltage equals sqrt(3) times the phase voltage.

Two coils on a common magnetic core: the secondary voltage is V2/V1 = N2/N1 and the currents are in the inverse ratio I2/I1 = N1/N2 (ideal transformer, power conserved). A load impedance seen from the primary is multiplied by (N1/N2) squared. Real transformers have winding resistance, leakage inductance and core losses (eddy current and hysteresis). Isolation transformers galvanically separate the patient environment from the mains, which is a safety feature.

Effects at 50 Hz on the skin surface: perception at about 1 mA, let-go limit about 10 mA (muscles cannot release), ventricular fibrillation from about 50-100 mA (macroshock). Microshock: currents of only about 10 microamperes applied directly to the heart (catheters, electrodes) can cause fibrillation, so much lower leakage limits apply to cardiac-connected devices. Protective measures: protective earth conductor, residual-current (differential) circuit breakers, double insulation, isolation transformers, equipotential bonding in medical rooms and limitation of leakage currents. Standards: the CEI (Italian electrotechnical committee) and IEC 60601 for medical electrical equipment; device applied parts are classified B, BF and CF (CF for direct cardiac contact).

1. How do the voltage and current change through an ideal transformer? 2. What current at 50 Hz is dangerous for the let-go limit? 3. Why are limits lower for microshock? 4. Which classification applies to devices with direct cardiac contact?

1. V2/V1 = N2/N1 and I2/I1 = N1/N2. 2. About 10 mA. 3. Because currents applied directly to the heart can cause fibrillation at only about 10 microamperes. 4. Type CF applied parts.