AExE0101 Basic concept¶
Electrical quantities¶
| Quantity | Meaning | Relation | SI unit |
|---|---|---|---|
| Charge \(q\) | Amount of electric charge | Fundamental quantity | coulomb (C) |
| Current \(i\) | Rate of charge flow | \(i=dq/dt\) | ampere (A) |
| Voltage \(v\) | Energy transferred per unit charge | \(v=dw/dq\) | volt (V) |
| Power \(p\) | Instantaneous rate of energy transfer | \(p=vi\) | watt (W) |
| Energy \(w\) | Accumulated power over time | \(w=\int p\,dt\) | joule (J) |
For constant DC quantities:
\[
P=VI=I^2R=\frac{V^2}{R}, \qquad W=Pt.
\]
Sign recognition: under the passive sign convention, an element absorbs power when current enters its labelled positive-voltage terminal. A negative result means the element is delivering power.
Ohm's law¶
For an ohmic element with constant resistance,
\[
V=IR.
\]
- \(R=V/I\) is measured in ohms (\(\Omega\)).
- Conductance is \(G=1/R\) and is measured in siemens (S).
- Ohm's law is a device relation, not a universal definition of resistance.
- A nonlinear device may have a changing ratio \(V/I\) and does not obey one constant-\(R\) line over its full operating range.
Conductors and insulators¶
| Material class | Charge-carrier behavior | Electrical behavior |
|---|---|---|
| Conductor | Many mobile charge carriers | Low resistivity; current flows readily |
| Insulator | Very few mobile charge carriers | High resistivity; strongly opposes current |
| Semiconductor | Carrier population controllable by doping, field, light, or heat | Conductivity lies between and is controllable |
Resistance also depends on geometry and material:
\[
R=\rho\frac{\ell}{A},
\]
where \(\rho\) is resistivity, \(\ell\) is length, and \(A\) is cross-sectional area.
Series and parallel resistors¶
| Connection | Same quantity | Divided quantity | Equivalent resistance |
|---|---|---|---|
| Series | Current | Voltage | \(R_s=\sum R_i\) |
| Parallel | Voltage | Current | \(1/R_p=\sum 1/R_i\) |
Fast checks:
- A series equivalent is greater than any individual positive resistance.
- A parallel equivalent is less than the smallest positive branch resistance.
- Two parallel resistors: \(R_p=R_1R_2/(R_1+R_2)\).
- Equal resistors: \(n\) in series give \(nR\); \(n\) in parallel give \(R/n\).
Voltage divider, for unloaded series resistors:
\[
V_k=V_s\frac{R_k}{\sum R_i}.
\]
Current divider for two parallel resistors:
\[
I_1=I_T\frac{R_2}{R_1+R_2}, \qquad
I_2=I_T\frac{R_1}{R_1+R_2}.
\]
The branch with the lower resistance carries the larger current.
Kirchhoff's laws¶
| Law | Conservation basis | Equation | Analysis method |
|---|---|---|---|
| KCL | Charge | \(\sum i=0\) at a node | Nodal analysis |
| KVL | Energy | \(\sum v=0\) around a closed loop | Mesh/loop analysis |
- KCL: total current entering a node equals total current leaving it.
- KVL: algebraically add rises and drops using one consistent traversal direction.
- A negative solved current means the real direction is opposite to the assumed arrow.
- Mesh analysis is based primarily on KVL; nodal analysis is based primarily on KCL.
Star-delta conversion¶
Let the delta branches be \(R_{AB}\), \(R_{BC}\), and \(R_{CA}\), and define
\[
S_\Delta=R_{AB}+R_{BC}+R_{CA}.
\]
Delta to star:
\[
R_A=\frac{R_{AB}R_{CA}}{S_\Delta},\quad
R_B=\frac{R_{AB}R_{BC}}{S_\Delta},\quad
R_C=\frac{R_{BC}R_{CA}}{S_\Delta}.
\]
Each star arm is the product of the two adjacent delta arms divided by the sum of all three delta arms.
For star to delta, define
\[
P_Y=R_AR_B+R_BR_C+R_CR_A.
\]
Then
\[
R_{AB}=\frac{P_Y}{R_C},\quad
R_{BC}=\frac{P_Y}{R_A},\quad
R_{CA}=\frac{P_Y}{R_B}.
\]
Each delta arm uses the star arm opposite that delta branch in the denominator.
Balanced shortcut:
\[
R_\Delta=3R_Y, \qquad R_Y=\frac{R_\Delta}{3}.
\]
Circuit classifications¶
These are independent classification axes. Do not treat the words as synonyms.
| Pair | Decisive distinction | Typical examples |
|---|---|---|
| Linear / nonlinear | Linear circuits satisfy additivity and homogeneity; nonlinear circuits do not | Constant \(R,L,C\) / diode |
| Bilateral / unilateral | Terminal reversal leaves / changes the element's V-I behavior | Ideal resistor / diode |
| Active / passive | Can deliver net energy or provide gain / only absorbs or stores energy | Source, biased transistor / \(R,L,C\) |
Important traps:
- A diode is normally classified as passive, nonlinear, and unilateral.
- A resistor is normally passive, linear, and bilateral.
- Linearity, not bilateral behavior, is the controlling requirement for superposition.
- An active circuit does not create energy; an external supply provides the amplified output power.
Basic-concept examples¶
- Two resistors \(6\ \Omega\) and \(3\ \Omega\) in parallel give \(R_p=(6\times3)/(6+3)=2\ \Omega\).
- A balanced \(12\ \Omega\) delta converts to a balanced \(R_Y=12/3=4\ \Omega\) star.
- If \(V=12\ \text{V}\) across \(R=4\ \Omega\), then \(I=3\ \text{A}\) and \(P=36\ \text{W}\).
Basic-concept revision box¶
- Loop -> KVL -> mesh. Node -> KCL -> nodal.
- Series keeps current; parallel keeps voltage.
- Parallel resistance is below the smallest branch.
- Delta to star: adjacent product over delta sum.
- Linear/nonlinear and bilateral/unilateral are different questions.