First-year courses · Engineering

Intro electric circuits: DC analysis and RC, RL basics

Electric circuits flashcards: Ohm's law, Kirchhoff's laws, dividers, nodal and mesh analysis, Thevenin, capacitors, inductors and time constants, with numbers.

42 cards5 topicsAbout 15 minutes a roundTypical first-year engineering electric circuits (DC analysis)

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About this deck

This deck supports a first course in electric circuits for engineering students, usually taken in first or early second year. It covers the vocabulary of charge, current, voltage and power; Ohm's law and the passive sign convention; Kirchhoff's current and voltage laws; series and parallel combinations with voltage and current dividers; systematic nodal and mesh analysis; superposition, source transformation, Thevenin and Norton equivalents and maximum power transfer; and the DC behaviour of capacitors and inductors, including RC and RL time constants.

You should be able to solve two or three simultaneous linear equations, handle exponentials, and work with metric prefixes (milli, micro, kilo, mega) without slipping a factor of 1000. A high school physics unit on electricity is helpful but not required.

Treat each example as a small exercise: redraw the circuit, label node voltages and branch currents with directions, then solve before checking. Once a card is easy, change one value and predict the effect before computing it. A quick check is conservation of power: the power delivered by the sources should equal the power absorbed by everything else.

Common mistakes: adding parallel resistors as if they were in series; mixing up the sign of a voltage drop when walking around a loop in KVL; and forgetting that, at DC steady state, a capacitor acts like an open circuit and an inductor like a short circuit, not the other way round.

Before you start: Grade 12 physics and mathematics, including algebra with exponents and solving systems of linear equations.

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Cover the definitions with a sheet of paper and test yourself, or use the study modes above.

Quantities and laws

Electric current: The rate at which charge flows past a point, i = dq/dt, measured in amperes (1 A = 1 C/s).Example: 3 C passing a point in 2 s is an average of 3/2 = 1.5 A.
Voltage: Energy transferred per unit charge between two points, also called potential difference, measured in volts (1 V = 1 J/C).Example: Moving 2 C through 12 V transfers 24 J.
Electrical power: The rate of energy transfer, p = v i, in watts. Under the passive sign convention a positive value means the element absorbs power.Example: A 12 V lamp drawing 2 A absorbs 12 x 2 = 24 W.
Passive sign convention: Current is drawn entering the + terminal of an element; then p = v i is the power the element absorbs, and a negative result means it supplies power.Example: A source with v = 9 V and current entering its + terminal of -0.5 A absorbs -4.5 W, so it supplies 4.5 W.
Energy in kilowatt-hours: Energy equals power times time; one unit of this kind is 1 kW sustained for 1 h, equal to 3.6 MJ.Example: A 1500 W heater on for 2 h uses 1.5 x 2 = 3 kWh.
Ohm's law: For a linear resistor, voltage is proportional to current: v = i R, with R in ohms.Example: 9 V across 450 ohms drives 9/450 = 0.02 A = 20 mA.
Conductance: The reciprocal of resistance, G = 1/R, measured in siemens; handy in nodal analysis because currents become G times voltage.Example: A 50 ohm resistor has G = 1/50 = 0.02 S = 20 mS.
Resistance of a wire: R = rho L/A, where rho is the material's resistivity, L the length and A the cross-sectional area.Example: Copper (rho = 1.68 x 10^-8 ohm·m), 10 m long, 1 mm^2 area: R = 1.68e-8(10)/1e-6 = 0.168 ohm.
Power absorbed by a resistor: Combining p = v i with Ohm's law gives p = i^2 R = v^2/R; it is always zero or positive.Example: 2 A through 10 ohms: 2^2 x 10 = 40 W.
Open circuit: A break with infinite resistance: no current flows, but a voltage can appear across the gap.Example: The voltage measured across an unconnected battery's terminals is its open-circuit voltage.
Short circuit: A zero-resistance connection: the voltage across it is zero, whatever current flows.Example: A wire across a resistor forces 0 V across it, so the resistor carries no current.
Ideal voltage source: An element that holds a fixed voltage between its terminals regardless of the current drawn from it.Example: An ideal 5 V source keeps 5 V across a 1 ohm or a 1 Mohm load.
Ideal current source: An element that forces a fixed current through its branch regardless of the voltage needed.Example: A 2 mA source into 3 kohms produces 6 V; into 1 kohm it produces 2 V.
Node: A point where two or more circuit elements connect; everything joined by plain wire counts as a single connection point.Example: Three resistors joined at one terminal strip share a single node voltage.
Ground reference: The node chosen as 0 V; all other node voltages are measured relative to it.Example: Picking the node with the most connections as the 0 V node usually shortens nodal analysis.

Kirchhoff and dividers

Kirchhoff's current law (KCL): Charge is conserved at a node: the sum of currents entering equals the sum leaving.Example: 3 A and 2 A enter, 1 A and I leave: 5 = 1 + I, so I = 4 A.
Kirchhoff's voltage law (KVL): Energy is conserved around any closed loop: the algebraic sum of voltage rises and drops is zero.Example: A 12 V source drives drops of 5 V and 4 V, so the third element drops 12 - 5 - 4 = 3 V.
Resistors in series: Elements carrying the same current; their resistances add: Req = R1 + R2 + ...Example: 100 + 220 + 330 = 650 ohms.
Resistors in parallel: Elements sharing the same two nodes and so the same voltage; reciprocals add, and for two, Req = R1 R2/(R1 + R2).Example: 6 ohms and 3 ohms: 18/9 = 2 ohms. Four 100 ohm resistors give 100/4 = 25 ohms.
Voltage divider: Two series resistors split a voltage in proportion to resistance: Vout = Vin R2/(R1 + R2) across R2.Example: 12 V with R1 = 1 kohm and R2 = 2 kohm: Vout = 12(2/3) = 8 V.
Current divider: Two parallel resistors split a current inversely to resistance: I1 = I R2/(R1 + R2).Example: 6 A into R1 = 2 ohms and R2 = 4 ohms: I1 = 6(4/6) = 4 A and I2 = 2 A.
Wheatstone bridge: Four resistors in a diamond with a detector across the middle; it reads zero (balanced) when R1/R2 = R3/Rx, which lets an unknown be found.Example: R1 = 100, R2 = 200, R3 = 150 ohms: balance when Rx = 150(200)/100 = 300 ohms.

Analysis methods

Nodal analysis: Choose a ground, name the other node voltages, write KCL at each using Ohm's law for every branch current, then solve the equations.Example: Node V linked to 12 V by 4 ohms, and to ground by 4 ohms and 2 ohms: (V - 12)/4 + V/4 + V/2 = 0, so V = 3 V.
Mesh analysis: Assign a current to each window of a planar circuit, write KVL around each, and solve; shared resistors carry the difference of mesh currents.Example: Meshes I1 and I2 share 2 ohms: in mesh 1 that resistor's drop is 2(I1 - I2).
Supernode: When a voltage source joins two non-reference nodes, enclose both in one surface, write KCL for the surface, and add the source's voltage constraint.Example: A 5 V source from node a to b gives Va - Vb = 5 plus one combined KCL equation.
Superposition: In a linear circuit, a response equals the sum of responses to each independent source acting alone, with voltage sources shorted and current sources opened.Example: 10 V source alone gives 1 A, current source alone gives 0.5 A: total 1.5 A. Power cannot be added this way.
Thevenin equivalent: Any linear two-terminal network can be replaced by a voltage Vth (the open-circuit voltage) in series with Rth (seen with sources turned off).Example: 12 V behind two 4 ohm resistors forming a divider: Vth = 6 V, Rth = 4 parallel 4 = 2 ohms.
Norton equivalent: A current source In (the short-circuit current) in parallel with Rn = Rth; it is interchangeable with the Thevenin form.Example: Vth = 6 V and Rth = 2 ohms give In = 6/2 = 3 A with 2 ohms in parallel.
Source transformation: A voltage source V in series with R can be swapped for a current source V/R in parallel with the same R, and the reverse.Example: 10 V in series with 5 ohms becomes 2 A in parallel with 5 ohms.
Maximum power transfer: A load receives the most power when its resistance equals the Thevenin resistance; that power is Vth^2/(4 Rth).Example: Vth = 6 V, Rth = 2 ohms: RL = 2 ohms takes 36/8 = 4.5 W.
Dependent source: A source whose value is set by a voltage or current elsewhere in the circuit; it models transistors and amplifiers and stays on during Thevenin analysis.Example: A current source of 3 Ix delivers 6 mA when Ix = 2 mA.

Capacitors and inductors

Capacitor: Two conductors separated by an insulator that store energy in an electric field; i = C dv/dt and stored energy is C v^2/2.Example: 100 uF charged to 12 V stores 0.5(100e-6)(144) = 7.2 mJ.
Capacitors in series and parallel: Parallel capacitances add; series ones combine by reciprocals, the reverse of resistors.Example: Two 10 uF capacitors give 20 uF in parallel and 5 uF in series.
Inductor: A coil that stores energy in a magnetic field; v = L di/dt and stored energy is L i^2/2.Example: 2 mH with current rising at 500 A/s has 0.002 x 500 = 1 V across it.
RC time constant: tau = R C, the time for a charging capacitor to reach about 63.2% of its final voltage (or a discharging one to fall to 36.8%).Example: 10 kohm with 100 uF: tau = 10000 x 0.0001 = 1 s.
Charging curve of an RC circuit: From zero, v(t) = Vs(1 - e^(-t/tau)); after about 5 tau the capacitor is practically fully charged.Example: Vs = 10 V: at t = tau, v = 10(1 - e^-1) = 6.32 V; at 5 tau, 9.93 V.
RL time constant: tau = L/R for a series inductor and resistor; the current approaches its final value along the same exponential shape.Example: 0.5 H with 100 ohms: tau = 0.005 s = 5 ms.
DC steady state: After transients die out with constant sources, capacitor currents and inductor voltages are zero, so a capacitor looks open and an inductor looks shorted.Example: A capacitor across 4 ohms in a 12 V series loop with 2 ohms settles at 12(4/6) = 8 V.

Measurement and AC preview

Ammeter connection: An ammeter goes in series with the branch being measured and ideally has zero resistance.Example: Break the wire to a lamp and insert the meter to read its current.
Voltmeter connection: A voltmeter goes in parallel across the element and ideally has infinite resistance so it draws no current.Example: Touch the probes to both ends of a resistor to read its voltage.
RMS value: The DC value that would deliver the same average power; for a sinusoid it is the peak divided by sqrt(2).Example: A 170 V peak sinusoid has an RMS value of 170/1.414 = 120.2 V.
Frequency and period: Frequency f counts cycles per second in hertz; the period T = 1/f is the time for one cycle.Example: 60 Hz mains has T = 1/60 = 16.7 ms.