Transients in direct current (DC) circuits are fundamental to understanding how biomedical devices respond when switched on or off. In biomedical engineering, precise control and prediction of transient behaviour are critical for the safe operation of equipment such as electrocardiographs, defibrillators, and medical imaging systems. This chapter explores the mathematical derivation of growth and decay equations for transient currents and voltages in resistor-inductor (R-L) and resistor-capacitor (R-C) circuits, enabling biomedical engineers to model and design circuits with predictable temporal responses.

4.1 Derivation of Growth and Decay Equations in R-L and R-C Circuits

The transient response of R-L and R-C circuits describes how current and voltage evolve over time when a DC supply is suddenly connected or disconnected. In biomedical instrumentation, these transient phenomena affect signal integrity and safety, for example, when powering sensors or controlling pulse generation. Understanding the mathematical foundations of transient growth and decay allows engineers to predict circuit behaviour and avoid harmful surges or delays in device operation.

4.1.1 Growth of Current in an R-L Circuit

The growth of current in an R-L circuit occurs when a DC voltage source is suddenly applied to a series combination of a resistor (R) and an inductor (L). The inductor initially opposes changes in current due to its property of self-inductance, causing the current to increase gradually rather than instantaneously. This transient current growth is crucial in biomedical devices such as implantable pulse generators, where controlled current ramp-up prevents tissue damage.

When the switch closes at time t=0, the governing differential equation from Kirchhoff’s voltage law is:

V = L (di/dt) + R i

Where V is the applied voltage, i is the current at time t, L is inductance, and R is resistance.

Solving this first-order linear differential equation yields the growth equation:

i(t) = (V / R) * (1 - e^(-Rt/L))

This equation shows that current asymptotically approaches the steady-state value V/R as time increases.

Characteristics of Current Growth in R-L Circuits

• Initial Current Zero: At t=0, the current is zero because the inductor resists sudden changes.
• Exponential Increase: The current rises exponentially with a time constant τ = L/R, which determines the speed of growth.
• Steady-State Current: As t → ∞, current stabilizes at V/R, the value determined solely by resistance.
• Impact of Inductance: Larger inductance increases τ, slowing current growth and protecting sensitive biomedical components.
• Energy Stored in Magnetic Field: During growth, energy accumulates in the inductor’s magnetic field, which can affect downstream circuits when switched off.

Example: In a hospital’s patient monitoring system, an R-L circuit may be used to filter noise on sensor leads. The controlled current growth ensures that the sensor circuits stabilize without sudden surges that could distort readings.

4.1.2 Decay of Current in an R-L Circuit

Current decay in an R-L circuit happens when the power supply is disconnected, and the inductor discharges its stored energy through the resistor. This decay phase is important in biomedical contexts such as defibrillator circuits, where controlled energy dissipation prevents harmful voltage spikes.

The governing equation when the source is removed at t=0 is:

0 = L (di/dt) + R i

Rearranged and solved, this yields the decay equation:

i(t) = I₀ * e^(-Rt/L)

Where I₀ is the initial current at t=0.

Features of Current Decay in R-L Circuits

• Exponential Decrease: Current decreases exponentially from I₀ toward zero.
• Time Constant τ: The rate of decay is determined by τ = L/R, similar to growth.
• Energy Dissipation: The inductor’s stored magnetic energy converts to heat in the resistor.
• Voltage Polarity Reversal: Inductor voltage reverses polarity during decay, which can induce voltage spikes.
• Protection Considerations: Suppression circuits or snubbers are often needed in biomedical devices to protect components from voltage surges during decay.

Example: At a university biomedical engineering lab, an R-L decay circuit simulates the discharge behavior in a pulse generator, providing insights into timing and safety margins for implantable devices.

4.1.3 Growth of Voltage in an R-C Circuit

In an R-C circuit, when a DC voltage source is applied, the capacitor initially behaves like a short circuit, and voltage across it grows as it charges through the resistor. The transient voltage growth across the capacitor is critical in devices like electrocardiographs, where capacitive sensors must charge predictably to ensure accurate signal acquisition.

The differential equation governing this process is:

V = R (dq/dt) + q/C

Where q is the charge on the capacitor, C is capacitance, and dq/dt is the current.

Solving leads to the voltage growth equation across the capacitor:

v_c(t) = V * (1 - e^(-t/RC))

The voltage exponentially approaches the source voltage V with a time constant τ = RC.

Aspects of Voltage Growth in R-C Circuits

• Initial Voltage Zero: At t=0, the capacitor voltage is zero due to no initial charge.
• Exponential Rise: Voltage rises gradually with time constant τ = RC.
• Steady-State Voltage: After a long time, voltage equals the supply voltage V.
• Current Behavior: Charging current decreases exponentially as the capacitor approaches full charge.
• Relevance to Signal Conditioning: Proper timing of voltage growth ensures stable baseline voltage in biomedical sensors.

Example: In a county hospital’s EEG equipment, R-C circuits regulate sensor input voltages, allowing smooth voltage transitions that reduce signal artifacts.

4.1.4 Decay of Voltage in an R-C Circuit

Voltage decay in an R-C circuit occurs when the supply is disconnected and the capacitor discharges through the resistor. This decay is essential in timing circuits within biomedical instrumentation, such as in pacemakers, where voltage decay controls pulse duration.

The governing equation when discharging starts at t=0 is:

0 = R (dq/dt) + q/C

Solving yields:

v_c(t) = V₀ * e^(-t/RC)

Where V₀ is the initial capacitor voltage.

Characteristics of Voltage Decay in R-C Circuits

• Exponential Decay: Voltage decreases exponentially from V₀ toward zero.
• Time Constant τ: The speed of decay depends on τ = RC.
• Energy Release: Stored electric energy in the capacitor dissipates as heat in the resistor.
• Influence on Timing: Decay time controls timing intervals in pulse circuits.
• Safety Implications: Residual voltage during decay must be managed to avoid shocks in biomedical devices.

Example: At a retail business specializing in medical equipment, R-C decay circuits are used in battery charging modules to control discharge rates, ensuring device longevity and patient safety.

Chapter Summary

This chapter explored the derivation of growth and decay equations in R-L and R-C circuits, providing the mathematical foundation for understanding transient responses in these circuits. It then examined the sketching of growth and decay curves, emphasizing how these graphical representations illustrate the behavior of voltage and current over time. Calculations involving growth and decay based on time constants were discussed in detail, enabling precise determination of circuit response during transient states. The chapter further applied the concept of time constants to the switching of inductive and capacitive loads, highlighting practical considerations in managing these transient effects in electrical systems. Lastly, it analyzed both passive and active filters, explaining their roles in shaping signal frequencies and improving circuit performance. Throughout the chapter, the interaction between theoretical derivations and practical applications was emphasized to deepen understanding of transient phenomena in DC circuits.