⚡Electrical Circuits and Systems I Unit 7 – First–Order Circuits

Key Concepts and Definitions

• First-order circuits contain one energy storage element (capacitor or inductor) and exhibit exponential responses to input changes
• Time constant $\tau$ characterizes the response speed of a first-order circuit and is defined as $\tau = RC$ for capacitive circuits and $\tau = L/R$ for inductive circuits
• Transient response refers to the circuit's behavior during the transition from one steady state to another, while steady-state response describes the circuit's behavior after the transient has settled
• Kirchhoff's Current Law (KCL) states that the sum of currents entering a node equals the sum of currents leaving the node
• Kirchhoff's Voltage Law (KVL) states that the sum of voltages around a closed loop in a circuit is zero
• Thevenin's theorem allows simplification of a linear circuit by replacing it with an equivalent voltage source and series resistance
• Norton's theorem is similar to Thevenin's theorem but uses an equivalent current source and parallel resistance

Circuit Elements and Components

• Resistors oppose the flow of electric current and have a linear relationship between voltage and current described by Ohm's Law: $V = IR$
• Capacitors store energy in an electric field and have a voltage-current relationship given by $I = C \frac{dV}{dt}$, where $C$ is the capacitance measured in farads (F)
  • Capacitors act as open circuits in steady state and short circuits during transients
• Inductors store energy in a magnetic field and have a voltage-current relationship given by $V = L \frac{dI}{dt}$, where $L$ is the inductance measured in henrys (H)
  • Inductors act as short circuits in steady state and open circuits during transients
• Voltage sources provide a constant voltage across their terminals, while current sources provide a constant current through a branch
• Switches control the flow of current in a circuit and can be used to create different circuit configurations (series, parallel)

Kirchhoff's Laws and Circuit Analysis

• Apply KCL at each node by equating the sum of currents entering and leaving the node to zero
• Apply KVL around each closed loop by summing the voltages and setting the result to zero, considering the polarity of voltage drops and rises
• Use Ohm's Law to relate voltages and currents across resistors
• Combine resistors in series by summing their resistances: $R_{eq} = R_1 + R_2 + ... + R_n$
• Combine resistors in parallel using the reciprocal formula: $\frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + ... + \frac{1}{R_n}$
• Apply Thevenin's or Norton's theorem to simplify complex circuits and focus on the portion of interest
• Use mesh analysis or nodal analysis to systematically solve for unknown currents and voltages in a circuit

Time-Domain Analysis of First-Order Circuits

• Write the differential equation describing the circuit's behavior using KVL or KCL and the voltage-current relationships of capacitors and inductors
• Solve the differential equation to obtain the time-domain expression for voltage or current
  • For a first-order RC circuit, the voltage across the capacitor is given by $v_C(t) = V_f + (V_0 - V_f)e^{-t/\tau}$, where $V_0$ is the initial voltage and $V_f$ is the final (steady-state) voltage
  • For a first-order RL circuit, the current through the inductor is given by $i_L(t) = I_f + (I_0 - I_f)e^{-t/\tau}$, where $I_0$ is the initial current and $I_f$ is the final (steady-state) current
• Determine the initial and final conditions based on the circuit configuration and input
• Analyze the transient and steady-state behavior of the circuit using the time-domain expression

Step Response and Time Constants

• The step response of a first-order circuit describes its behavior when subjected to a sudden change in input (e.g., a voltage or current step)
• The time constant $\tau$ determines the speed of the circuit's response to a step input
  • For an RC circuit, $\tau = RC$, where $R$ is the resistance and $C$ is the capacitance
  • For an RL circuit, $\tau = L/R$, where $L$ is the inductance and $R$ is the resistance
• The time constant represents the time required for the voltage or current to reach 63.2% of its final value during a charging or discharging process
• After one time constant, the capacitor voltage or inductor current has changed by 63.2% of the total change
• After five time constants, the transient response is considered settled, and the circuit has reached its steady-state value (within 1% of the final value)

Energy Storage in Capacitors and Inductors

• Capacitors store energy in an electric field between their plates, with the energy given by $E_C = \frac{1}{2}CV^2$, where $C$ is the capacitance and $V$ is the voltage across the capacitor
• Inductors store energy in a magnetic field generated by the current flowing through them, with the energy given by $E_L = \frac{1}{2}LI^2$, where $L$ is the inductance and $I$ is the current through the inductor
• The power delivered to or absorbed by a capacitor is $p_C(t) = Cv(t)\frac{dv(t)}{dt}$, while for an inductor, it is $p_L(t) = Li(t)\frac{di(t)}{dt}$
• During the charging process, energy is stored in the capacitor or inductor, while during the discharging process, energy is released from the component
• The total energy stored in a first-order circuit is the sum of the energies stored in the capacitor and inductor

Applications and Real-World Examples

• RC circuits are used in timing circuits, signal filtering, and power supply decoupling (bypass capacitors)
• RL circuits are found in power electronics, motor control, and electromagnetic relay systems
• First-order circuits are used to model the charging and discharging of batteries, as well as the response of sensors and actuators
• RC and RL circuits are used in audio and video systems for equalizing and shaping frequency responses (tone controls, crossover networks)
• First-order circuits are employed in control systems for smoothing and filtering signals, as well as for implementing lead and lag compensators

Problem-Solving Techniques

• Identify the type of first-order circuit (RC or RL) and the input waveform (step, ramp, sinusoidal)
• Determine the initial and final conditions based on the circuit configuration and input
• Write the differential equation describing the circuit's behavior using KVL or KCL and the voltage-current relationships of capacitors and inductors
• Solve the differential equation to obtain the time-domain expression for voltage or current
• Analyze the transient and steady-state behavior of the circuit using the time-domain expression
• Calculate the time constant and use it to determine the speed of the circuit's response and the time required to reach steady state
• Apply Thevenin's or Norton's theorem to simplify complex circuits and focus on the portion of interest
• Use simulation tools (SPICE, MATLAB) to verify hand calculations and explore the circuit's behavior under different conditions