Circuit analysis connects voltage, current, resistance, capacitance, and inductance so students can predict how electrical networks behave. This cheat sheet covers both steady DC circuits and sinusoidal AC circuits, which are essential in college physics and engineering. It helps students organize sign conventions, circuit laws, equivalent components, and frequency-dependent behavior in one quick reference.
The core tools are Ohm's law, Kirchhoff's junction and loop rules, equivalent resistance and capacitance, and time constants for transient circuits. For AC circuits, phasors and impedance turn sinusoidal circuit problems into algebra with complex quantities. Power calculations depend on whether the circuit is DC, purely resistive AC, or has phase differences between voltage and current.
Key Facts
- Ohm's law relates voltage, current, and resistance by for an ideal resistor.
- Kirchhoff's current law says the algebraic sum of currents at a junction is zero, so .
- Kirchhoff's voltage law says the algebraic sum of potential differences around a closed loop is zero, so .
- Series resistors add directly as , while parallel resistors satisfy .
- Capacitors combine oppositely to resistors: series capacitors satisfy , while parallel capacitors add as .
- The time constant for an RC circuit is , and the time constant for an RL circuit is .
- AC impedance is , , and , where .
- Average AC power is , where is the phase angle between voltage and current.
Vocabulary
- Node
- A node is a connection point in a circuit where two or more elements meet and share the same electric potential.
- Loop
- A loop is any closed path through a circuit that returns to its starting point.
- Impedance
- Impedance is the complex opposition to AC current, written as , and measured in ohms.
- Phasor
- A phasor is a rotating complex representation of a sinusoidal voltage or current with a fixed amplitude and phase.
- Reactance
- Reactance is the frequency-dependent part of impedance caused by inductors or capacitors.
- Resonance
- Resonance occurs in an RLC circuit when inductive and capacitive reactances cancel, so .
Common Mistakes to Avoid
- Adding parallel resistors as is wrong because parallel branches provide multiple current paths, so use .
- Using peak voltage instead of RMS voltage in AC power is wrong unless the formula is written for peak values. For standard power formulas, use and .
- Ignoring phase angle in AC power is wrong for circuits with capacitors or inductors. The correct average power is , not just .
- Treating capacitors and inductors like resistors in AC is wrong because their opposition depends on frequency. Use and .
- Choosing inconsistent loop signs is wrong because it can reverse voltage rises and drops within Kirchhoff's voltage law. Pick a loop direction and apply consistently.
Practice Questions
- 1 A battery is connected to resistors and in series. Find and the current .
- 2 Two resistors, and , are connected in parallel across a source. Find and the total current.
- 3 An AC source has and is connected to a capacitor . Calculate the capacitive reactance .
- 4 In a series RLC circuit, explain why the current is largest at resonance even though the circuit contains both an inductor and a capacitor.
Understanding Circuit Analysis DC and AC
A reliable circuit solution starts with a diagram that is labeled clearly. Mark a reference direction for every current before doing any calculations. Mark voltage polarities across components using the passive sign convention.
In that convention, current enters the terminal marked positive. A negative answer does not mean the work failed. It means the real current or voltage points opposite to the direction first chosen.
This simple habit prevents many sign errors when using loop equations. For larger networks, find unknown node voltages first.
Choose one node as zero volts, then write current expressions for each branch connected to the remaining nodes. This method is often faster than trying to guess every loop current.
Real components do more than obey a single ideal rule. A resistor changes electrical energy into thermal energy, which is why resistors can become hot. A capacitor stores energy in an electric field.
Its voltage cannot change instantly unless an infinite current is available. An inductor stores energy in a magnetic field. Its current cannot change instantly unless an infinite voltage is available.
These limits explain the curved charging and discharging graphs in switching circuits. After one time constant, the change is about sixty three percent complete. After about five time constants, the circuit is usually close enough to its final steady state for most practical work.
Students should always inspect the circuit just before a switch changes position and just after it changes position. Capacitor voltage and inductor current carry information across that instant.
In alternating current circuits, frequency controls how strongly capacitors and inductors oppose current. At low frequency, a capacitor has plenty of time to charge and can block much of the current. At high frequency, it charges and discharges more easily.
An inductor behaves in the opposite way because rapid current changes create a stronger opposing induced voltage. Phasor diagrams show these timing shifts as angles. In a resistor, voltage and current rise and fall together.
In a capacitor, current leads voltage. In an inductor, current lags voltage.
Keeping track of lead and lag matters more than memorizing a complex expression. It tells you whether a circuit is mainly capacitive or mainly inductive.
Resonance occurs in a circuit containing both an inductor and a capacitor when their frequency effects cancel. In a series arrangement, the remaining opposition can become small, so the current can become large. In a parallel arrangement, the supply current can become small while large currents circulate between components.
Radios use this frequency selection to pick one signal from many. Power systems use capacitors to reduce unwanted phase difference caused by inductive motors. When calculating AC power, distinguish between energy actually converted to heat, light, or motion and energy that moves back and forth between the source and reactive components.
Use root mean square values for household voltage ratings and power calculations. Finally, check units, limiting cases, and energy behavior.
A capacitor should act nearly open after a long time in a DC circuit. An inductor should act nearly like a wire after a long time, if its resistance is ignored.