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Fundamental Concepts & Circuit Elements

Units, circuit elements, laws, theorems, and network analysis techniques.

a) Basic Concepts & Units

SI units, dimensions, electrical quantities, power & energy fundamentals.

1. Units & Dimensions Reference Table

Hover over any row to glow. Click on Quantity for related formulas. Dimensional base: M (Mass), L (Length), T (Time), A (Current)

Quantity Unit Dimension
Resistance Ohm (Ω) [ML²T⁻²A⁻²]
Resistivity Ohm·meter (Ω·m) [ML²T⁻²A⁻²]
Conductivity mho/m or Siemens/m [M⁻¹L⁻²T²A²]
Voltage Volt (V) [ML²T⁻²A⁻²]
Current Ampere (A) [A]
Electric Power Watt (W) [ML²T⁻²]
Electric Energy kWh (or Joule) [ML²T⁻²]
Permittivity Farad/meter (F/m) [M⁻¹L⁻²T⁴A²]
Electric field int. V/m or N/C [MLT⁻²A⁻²]
Electric flux density C/m² [L⁻²TA]
Capacitance Farad (F) [M⁻¹L⁻²T⁴A²]
Inductance Henry (H) [ML²T⁻²A⁻²]
Permeability H/m [MLT⁻²A⁻²]
Magnetic flux density Tesla (Wb/m²) [MT⁻²A⁻¹]
Magnetic field intensity A/m [ML⁻¹T⁻²A⁻²]
mmf (Magneto-Motive Force) AT or Gilbert [A]
Reluctance AT/Wb or H⁻¹ [M⁻¹L⁻²T²A²]
Permeance Wb/AT (H) [ML²T⁻²A⁻²]

2. Definitions & Classifications

Network vs. Circuit

Hover to compare

Network: Any interconnection of electrical elements (may or may not be closed).

Circuit: A network containing at least one closed path for current to flow.

Active vs. Passive

⚡ vs ▱

Hover to reveal

Active: Can deliver power indefinitely (Battery, Generator, Op-Amp).

Passive: Only consumes or stores energy (R, L, C).

Linear vs. Non-Linear

Hover to reveal

Linear: Follows Ohm's Law strictly; V-I characteristic is a straight line through origin (Resistor).

Non-Linear: V-I relationship changes (Diode, Transistor).

3. Fundamental Parameters

Resistance (R)

Ω (Ohms)
$$ R = \rho \frac{L}{A} $$

Inductance (L)

H (Henrys)
$$ V = L \frac{di}{dt} $$
$$ W = \frac{1}{2}LI^2 $$

Capacitance (C)

F (Farads)
$$ I = C \frac{dv}{dt} $$
$$ W = \frac{1}{2}CV^2 $$
+++
---

b) Resistors & Resistance

Ohm's law, resistor networks, temperature effects, colour coding & material properties.

4. Resistor Fundamental Formulas

Resistance from Material Properties

$$ R = \frac{\rho \ell}{A} $$
  • $\rho$ = Resistivity of material (Ω·m)
  • $\ell$ = Length of conductor (m)
  • $A$ = Cross-sectional area (m²)
Stretching & Compressing

When wire is STRETCHED to $n$ times its length:

$$ R' = n^2 R $$

When wire is COMPRESSED to $1/n$ of its length:

$$ R' = \frac{R}{n^2} $$
R (ℓ, A)

Original Resistance: R

Series & Parallel Combinations

Series Combination
$$ R_{eq} = R_1 + R_2 + \cdots + R_n = \sum_{i=1}^{n} R_i $$
Parallel Combination
$$ \frac{1}{R_{eq}} = \frac{1}{R_1} + \frac{1}{R_2} + \cdots + \frac{1}{R_n} $$

For two resistors:

$$ R_{eq} = \frac{R_1 \times R_2}{R_1 + R_2} \quad \text{(product over sum)} $$

For $n$ equal resistors:

$$ R_{eq} = \frac{R}{n} $$
Live Combiner
Ω Ω
$R_{series} = 20.00$ Ω
$R_{parallel} = 5.00$ Ω

5. Delta ↔ Star (Y) Conversion

A B C $R_{AB}$ $R_{BC}$ $R_{CA}$ Neutral $R_A$ $R_B$ $R_C$
Delta to Star Conversion formulas:
$$ R_A = \frac{R_{AB} \times R_{CA}}{R_{AB} + R_{BC} + R_{CA}} $$
$$ R_B = \frac{R_{AB} \times R_{BC}}{R_{AB} + R_{BC} + R_{CA}} $$
$$ R_C = \frac{R_{BC} \times R_{CA}}{R_{AB} + R_{BC} + R_{CA}} $$

Memory rule: $R_{star} = \frac{\text{Product of adjacent } \Delta}{\text{Sum of all } \Delta}$

Live Calculator: Input Δ
RA = 10.00 Ω, RB = 10.00 Ω, RC = 10.00 Ω

6. Resistance vs Temperature

Resistance changes with temperature according to the Temperature Coefficient of Resistance ($\alpha$).

$$ R_t = R_0[1 + \alpha_0(T - T_0)] $$
  • $R_t$ = Resistance at temp T, $R_0$ = Rest. at ref. temp T₀
  • $\alpha$ = Temp. coefficient (per °C)
$$ \alpha_T = \frac{\alpha_0}{1 + \alpha_0 T} $$

Temp. coefficient at any other temperature.

Nature of Coefficients (PTC/NTC)
  • α > 0 (PTC): Metals. Resistance increases with T. (e.g., Cu, Al).
  • α < 0 (NTC): Semiconductors/Insulators. Resistance decreases with T. (e.g., Si, Ge).
  • α ≈ 0: Alloys. Near-zero change. (e.g., Manganin, Eureka).
Resistance vs Temperature Graph
■ Copper (+0.00393) ■ Semiconductor (Exp. Fall) ■ Manganin (~0.00001)

7. Resistivity & Cable Insulation Resistance

Specific Resistance (Resistivity $\rho$)

Resistance of a unit length and unit cross-section of a material.

$$ R = \rho \frac{\ell}{A} $$
  • Depends on the nature of the material and temperature.
  • Does not depend on length or area.
  • Unit: $\Omega \cdot m$
Insulation Resistance of a Cable
r₁ r₂
$$ R_{ins} = \frac{\rho}{2\pi \ell} \ln\left(\frac{r_2}{r_1}\right) $$

Notice: $R_{ins} \propto \frac{1}{\ell}$

8. Colour Coding of Resistors

BB ROY of Great Britain had a Very Good Wife

1.0 kΩ ± 5%

9. Metal Properties & Melting Points

Metal Melting (°C) Metal Melting (°C)
Copper 1084 Chromium 1850
Magnesium 650 Molybdenum 2622
Zinc 419.5 Tungsten 3390
Aluminium 658.6 Iron 1538
Tin 231.8 Cobalt 1490
Lead 327.4 Nickel 1445
Silver 961 Carbon 3550
Melting Point Comparison
Behaviour of Resistor
$$ \text{Resistor is a:} $$
LINEAR
BILATERAL
PASSIVE
$$ \text{Linear: } V \propto I \text{ (Ohm's law)} $$
$$ \text{Bilateral: current flows both ways} $$
$$ \text{Passive: only absorbs energy} $$
Metal
Use description

10. Material & Dielectric Constants

The dielectric constant ($\varepsilon_r$, relative permittivity) defines how well a material can store electrical energy in an electric field.

$$ \varepsilon_0 = 8.854 \times 10^{-12} \text{ F/m} $$

(permittivity of free space)

$$ \varepsilon = \varepsilon_0 \varepsilon_r $$
Material $\varepsilon_r$ Material $\varepsilon_r$
Vacuum 1.0 Polystyrene 2.6
Glass 5-12 Air (100°C) 1.0548
Mica 4-8 Rubber 3.6
Germanium 16 Porcelain 5.5
Water 80.6 Bakelite 2.5
Air (1 atm) 1.00059
Hover over a material to see its application...

c) Capacitors & Capacitance

Capacitor formulas, series-parallel combinations, delta-star transforms & transient response.

11. Capacitor Formulas

(a) Basic Parallel Plate Capacitance

$$ C = \frac{\varepsilon_0 \varepsilon_r A}{d} \quad \text{Farad} $$
  • $\varepsilon_0 = 8.854 \times 10^{-12}$ F/m
  • $\varepsilon_r$ = relative permittivity
  • $A$ = plate area (m²)
  • $d$ = distance between plates (m)
$$ C \propto A, \quad C \propto \frac{1}{d}, \quad C \propto \varepsilon_r $$
d
C = 100 pF

(b) Layered Dielectrics (Series)

Different dielectrics stacked horizontally between plates.

$$ C = \frac{\varepsilon_0 A}{\dfrac{t_1}{\varepsilon_{r1}} + \dfrac{t_2}{\varepsilon_{r2}} + \dfrac{t_3}{\varepsilon_{r3}}} $$

Acts as capacitors in SERIES:

$$ \frac{1}{C} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} $$
εr1, t1 εr2, t2 εr3, t3 C1 C2 C3
(c) Vertical Division (Parallel)
$$ C = \frac{\varepsilon_0 A_1}{d} + \frac{\varepsilon_0 \varepsilon_r A_2}{d} $$
Air εr

$C = C_{air} + C_{dielectric}$

(d) Horizontal Division (Series)
$$ C = \frac{\varepsilon_0 A}{d - t + t/\varepsilon_r} $$
Air εr (t)

$\frac{1}{C} = \frac{1}{C_{air}} + \frac{1}{C_{dielectric}}$

(e) Cylindrical Capacitor

$$ C = \frac{2\pi\varepsilon_0\varepsilon_r \ell}{\ln(b/a)} $$
a b
  • $a$ = inner radius
  • $b$ = outer radius
  • $\ell$ = length
  • $E_{max} = \frac{V}{a\ln(b/a)}$

(f) Multi-plate Variable

$$ C = \frac{(n-1)\varepsilon_0\varepsilon_r A}{d} $$

Where $n$ = total number of plates

12. Series & Parallel Capacitor Combinations

Series Combination

$$ \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + \dots + \frac{1}{C_n} $$

For two: $C_{eq} = \frac{C_1 C_2}{C_1 + C_2}$

  • Charge (Q) is SAME on each capacitor
  • $V_{total} = V_1 + V_2 + V_3$
  • Voltage divider: $V_1 = V \frac{C_2}{C_1+C_2}$
C1 C2 C3 +Q -Q +Q -Q +Q -Q
Ceq = 3.33 µF

Parallel Combination

$$ C_{eq} = C_1 + C_2 + \dots + C_n = \sum C_i $$

  • Voltage (V) is SAME on each capacitor
  • $Q_{total} = Q_1 + Q_2 + Q_3$
  • Charge divider: $Q_1 = Q_{total} \frac{C_1}{C_{eq}}$
C1 C2 C3 q1 q2 q3
Ceq = 30.00 µF

Note: Capacitor combinations are OPPOSITE to resistor rules!

$$ \text{Series Caps: } C_{eq} < C_{min} \quad | \quad \text{Parallel Caps: } C_{eq}> C_{max} $$

13. Delta ↔ Star (Y) Conversion for Capacitors

Delta to Star Star to Delta

Star to Delta ($\Delta$)

Opposite of resistors! Multiply pairwise, divide by opposite.

$$ C_A = \frac{C_1 C_2 + C_2 C_3 + C_3 C_1}{C_1} $$
$$ C_B = \frac{C_1 C_2 + C_2 C_3 + C_3 C_1}{C_2} $$
$$ C_C = \frac{C_1 C_2 + C_2 C_3 + C_3 C_1}{C_3} $$
C1 C2 C3 A B C Ca Cc Cb

14. Capacitor Responses & Transients

Key Behaviors
  • Initial state ($t=0$): Uncharged capacitor acts as a SHORT CIRCUIT.
  • Steady state ($t=\infty$): Fully charged capacitor acts as an OPEN CIRCUIT (blocks DC).
  • Instantaneous Change: Voltage across capacitor cannot change instantaneously. $V_C(0^-) = V_C(0^+)$.
$$ i_c(t) = C \frac{dv_c(t)}{dt} $$
$$ v_c(t) = \frac{1}{C} \int i_c(t) dt $$
Energy Stored
$$ E = \frac{1}{2} C V^2 = \frac{1}{2} Q V = \frac{Q^2}{2C} $$
E

Charging & Discharging (RC Circuit)

Vs t=0 R C
Current $\tau$: 3.0 s
Charging: $V_C = V_S(1-e^{-t/\tau})$

15. Important Points: Capacitor Behaviour

(a) Voltage Opposition
Capacitor OPPOSES change of voltage
$$ V_C(0^+) = V_C(0^-) $$ (voltage cannot jump instantaneously)
Step Input V_c(t)
(b) Energy Storage
Stores energy in ELECTRIC FIELD
$$ E = \frac{1}{2}CV^2 = \frac{Q^2}{2C} = \frac{1}{2}QV \text{ J} $$
Energy
(c) Charging
$I_C(t)$ must be POSITIVE
$$ I_C(t) = C\frac{dV_C}{dt} \Rightarrow \frac{dV_C}{dt} > 0 \Rightarrow I_C > 0 $$
V_c \uparrow +I_c
(d) Discharging
$I_C(t)$ must be NEGATIVE
$$ I_C(t) = C\frac{dV_C}{dt} \Rightarrow \frac{dV_C}{dt} < 0 \Rightarrow I_C < 0 $$
V_c \downarrow -I_c
(e) Fixed Polarity
DC voltages NEVER change polarity
While capacitor charges/discharges
C + - Charge Discharge
(f) Live Energy
$E = \frac{1}{2}CV^2$
Assumes $C = 1\text{ F}$
Energy
50 J

d) Inductors & Inductance

Inductor fundamentals, energy storage, magnetic coupling & RL transient analysis.

16. Inductor Fundamentals

Faraday's Law for Inductor

$$ V_L = L\frac{di}{dt} \quad \text{and} \quad i(t) = \frac{1}{L}\int i\,dt + i(0^-) $$
Where $V_L$ = Voltage (V), $L$ = Inductance (Henry), $\frac{di}{dt}$ = Rate of current change (A/s)

(a) Coefficient of Self Inductance ($L$)

$$ L = \frac{N\phi}{i} = \frac{V_L}{di/dt} = \frac{N^2}{\mathcal{S}} = \frac{\mu_0\mu_r N^2 A}{\ell} \text{ Henry} $$
$N$ = Number of turns
$\phi$ = Magnetic flux (Wb)
$\mathcal{S}$ = Reluctance
$A$ = Core area (m²)
$\ell$ = Coil length (m)
$\mu_0 = 4\pi\times10^{-7}$ H/m
Magnetic Flux $\phi$ i V_L
Induced $V_L$: 50 V

(b) Coefficient of Mutual Inductance ($M$)

$$ M = \frac{N_2\phi_1}{i_1} = \frac{V_{L2}}{di_1/dt} = \frac{\mu_0\mu_r N_1 N_2 A}{\ell} \text{ Henry} $$
$$ \text{Coupling Factor, } K = \frac{M}{\sqrt{L_1 L_2}} \quad (0 \leq K \leq 1) $$ $$ M_{max} = \sqrt{L_1 L_2} \quad \text{(when } K=1 \text{)} $$
Coil 1 ($L_1$) Coil 2 ($L_2$) Mutual Flux $\phi_m$ Coupling rings
Mutual Inductance $M = K\sqrt{L_1L_2}$

17. Inductor with Initial Condition

(a) WITH initial condition — $I_L(0^-) = I_0$
$$ \text{At } t = 0^+: $$ $$ I_L(0^+) = I_L(0^-) = I_0 $$
Acts as CURRENT SOURCE of value $I_0$


$$ \text{At } t \rightarrow \infty \rightarrow \text{acts as SHORT CIRCUIT} $$
(b) WITHOUT initial condition — $I_L(0^-) = 0$
$$ \text{At } t = 0^+: I_L(0^+) = 0 $$
Acts as OPEN CIRCUIT at $t=0^+$


$$ \text{At } t \rightarrow \infty \rightarrow \text{acts as SHORT CIRCUIT} $$
Transient Behaviour Summary
Inductor at $t=0^+$ ($I_0 \neq 0$)

Current Source
(value = $I_0$)

Inductor at $t=0^+$ ($I_0 = 0$)

Open Circuit

Inductor at $t\rightarrow\infty$ (DC)

Short Circuit
(wire)

Capacitor at $t=0^+$ ($V_0 \neq 0$)

Voltage Source
(value = $V_0$)

Capacitor at $t=0^+$ ($V_0 = 0$)

Short Circuit

Capacitor at $t\rightarrow\infty$ (DC)

Open Circuit

$$ \boxed{I_L \text{ cannot change instantaneously}} $$
$$ \boxed{V_C \text{ cannot change instantaneously}} $$

18. Energy Stored in Inductor

$$ E = \frac{1}{2}LI^2 \quad \text{Joules} $$
Where $L$ = Inductance (Henry), $I$ = Current (A)
Inductor stores energy in MAGNETIC FIELD
(Compare: Capacitor stores in ELECTRIC FIELD)

19. Important Points: Inductor Behaviour

(a) Linear & Passive
Inductor is LINEAR, BILATERAL and PASSIVE
(b) Current Opposition
OPPOSES sudden change of current
$$ I_L(0^+) = I_L(0^-) $$ (current cannot jump instantaneously)
Step Input I_L(t)
(c) Energy Storage
Inductor is an ENERGY STORAGE element
$$ E = \frac{1}{2}LI^2 $$
(d) Zero Power
Does NOT dissipate any power
$$ P_{avg} = 0 \quad \text{(ideal inductor)} $$
P(t)
(e) Memory
Initial condition: $I_L(0^-) = I_L(0^+)$
(inductor remembers the current flowing through it)

20. Magnetic Coupling & Equivalent Inductance

Series Aiding vs Opposing
L1 L2 Mutual M
Opposing Aiding
$$ L_{eq} = L_1 + L_2 \pm 2M $$
$$ L_{eq} = L_1 + L_2 + 2M $$
Parallel Aiding vs Opposing
L1 L2 Mutual M
Opposing Aiding
$$ L_{eq} = \frac{L_1 L_2 - M^2}{L_1 + L_2 \mp 2M} $$
$$ L_{eq} = \frac{L_1 L_2 - M^2}{L_1 + L_2 - 2M} $$

21. RL Circuit Transients

DC Source with R & L
+ - Vs Switch R L I_L
Current $\tau$: 1.5 s
Current Rise: $I_L = I_{max}(1-e^{-t/\tau})$
$$ V_L(t) = L\dfrac{di}{dt} = V_S e^{-t/\tau} \quad \text{(exponential decay even during current rise)} $$

e) RLC Circuits & Damping

Series RLC behaviour, natural & forced response, damping types & step response.

22. Series RLC Circuit & Damping

R L C
$\alpha = \frac{R}{2L}$ (Neper frequency)
$\omega_0 = \frac{1}{\sqrt{LC}}$ (Resonant freq)
$\zeta = \frac{\alpha}{\omega_0}$ (Damping ratio)
Response Types based on Damping ($\zeta$)

23. RLC Complete Response Summary

Source-Free Natural Response — All Three Cases
■ Overdamped ($\zeta > 1$) ■ Critically Damped ($\zeta = 1$) ■ Underdamped ($\zeta < 1$)
Series RLC (Source-Free)
$$ \alpha = \frac{R}{2L}, \quad \omega_0 = \frac{1}{\sqrt{LC}} $$
Overdamped ($\alpha > \omega_0$, $C > \frac{4L}{R^2}$):
$i(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}$
Critical ($\alpha = \omega_0$, $C = \frac{4L}{R^2}$):
$i(t) = (A_1 + A_2 t)e^{-\alpha t}$
Underdamped ($\alpha < \omega_0$, $C < \frac{4L}{R^2}$):
$i(t) = e^{-\alpha t}(A_1\cos\omega_d t + A_2\sin\omega_d t)$
Parallel RLC (Source-Free)
$$ \alpha = \frac{1}{2RC}, \quad \omega_0 = \frac{1}{\sqrt{LC}} $$
Overdamped ($\alpha > \omega_0$, $L > 4R^2C$):
$v(t) = A_1 e^{s_1 t} + A_2 e^{s_2 t}$
Critical ($\alpha = \omega_0$, $L = 4R^2C$):
$v(t) = (A_1 + A_2 t)e^{-\alpha t}$
Underdamped ($\alpha < \omega_0$, $L < 4R^2C$):
$v(t) = e^{-\alpha t}(A_1\cos\omega_d t + A_2\sin\omega_d t)$
🔍 RLC Case Classifier

Enter R, L, C and click Classify

ζ<1 Underdamped ζ=1 ζ>1 Overdamped

24. Step Response of RLC Circuits

Series RLC Step Response
Vs t=0 R L C + Vc −
OD: $V_c = V_s + A_1 e^{s_1 t} + A_2 e^{s_2 t}$
CD: $V_c = V_s + (A_1 + A_2 t)e^{-\alpha t}$
UD: $V_c = V_s + e^{-\alpha t}(A_1\cos\omega_d t + A_2\sin\omega_d t)$
Parallel RLC Step Response
Is R L C
OD: $i = I_s + A_1 e^{s_1 t} + A_2 e^{s_2 t}$
CD: $i = I_s + (A_1 + A_2 t)e^{-\alpha t}$
UD: $i = I_s + e^{-\alpha t}(A_1\cos\omega_d t + A_2\sin\omega_d t)$
Complete Solution Decomposition
$$ \text{Complete} = \underbrace{V_s \text{ or } I_s}_{\text{Forced (DC steady-state)}} + \underbrace{\text{transient terms}}_{\text{Natural (decays to 0)}} $$

Initial conditions for $A_1, A_2$:

$$ \frac{dV_C}{dt}\bigg|_{0^+} = \frac{I_C(0^+)}{C} \qquad \frac{dI_L}{dt}\bigg|_{0^+} = \frac{V_L(0^+)}{L} $$

f) Circuit Laws & Classification

Circuit types, Kirchhoff's current & voltage laws (KCL / KVL).

25. Electric Circuit Classification

Active vs Passive Elements
Active Passive
Delivers energy for infinite time Cannot deliver energy infinitely
$V/I$ ratio negative on V-I curve $V/I$ ratio positive
$P = VI < 0$ (delivers power) $P = VI > 0$ (absorbs power)
Sources, Op-amp, BJT, FET R, L, C, Bulb, Transformer
I V Passive Active
Bidirectional vs Unidirectional
Bidirectional Unidirectional
Property independent of current direction Property dependent on current direction
V-I curve symmetric in opposite quadrants V-I curve asymmetric
R, L, C Diode, BJT, Op-amp
I V R (Bidirectional) Diode

26. Kirchhoff's Current Law (KCL)

N I₁ I₂ I₃ I₄ I₁ = I₂ + I₃ + I₄
$$ \sum I_{in} = \sum I_{out} \quad \Rightarrow \quad \sum_{k=1}^{n} I_k = 0 $$

Law of Conservation of Charge

⚡ Interactive KCL Node Solver
6.0 A
2.0 A
1.5 A
I₄ = 2.5 A
✓ BALANCED — ΣI = 0

27. Kirchhoff's Voltage Law (KVL)

CW + − V₁ R₁ +V₂− R₂ +V₃− R₃ +V₄− −V₁ + V₂ + V₃ + V₄ = 0
$$ \sum V = 0 \quad \text{(around any closed loop)} $$

Law of Conservation of Energy

⚡ Interactive KVL Loop Solver
12.0 V
4 Ω
6 Ω
2 Ω
$I = V_s / (R_1 + R_2 + R_3)$ = 1.00 A
$V_{R_1}$ = 4.0 V, $V_{R_2}$ = 6.0 V, $V_{R_3}$ = 2.0 V
✓ −V₁ + VR₁ + VR₂ + VR₃ = 0.0 V

g) Sources & Network Topology

Voltage/current sources, energy conversion, network topology & graph theory.

28. Voltage & Current Sources

Ideal Voltage Source
+ − Vs r = 0
$$ V_{terminal} = V_s \quad \forall \; I $$ $$ I_{sc} \to \infty $$
Practical Voltage Source
+ − Vs r V_term
$$ V_{terminal} = V_s - Ir $$ $$ I_{sc} = \frac{V_s}{r} $$
V-I Characteristic & Max Power Transfer
2.0 Ω
$V_s = 12\text{ V}$
$V_{OC} = V_s = $ 12.0 V
$I_{SC} = V_s/r = $ 6.0 A
$R_L = r \Rightarrow P_{max} = $ 18.0 W
$\eta_{MPT} = 50\%$
Source Transformation
+ − Vs r

Voltage Source + Series r

⇌
Is r

Current Source ‖ r

$$ V_s = I_s \times r \qquad I_s = \frac{V_s}{r} $$

29. Voltage Source Connections

Series Connection

Adding (aiding — same polarity):

$$ V_1 \text{ series } V_2 \rightarrow V_{eq} = V_1 + V_2 $$

Subtracting (opposing — opposite polarity):

$$ V_1 \text{ series } (-V_2) \rightarrow V_{eq} = V_1 - V_2, \quad V_1 > V_2 $$

Series with Resistance:

$$ I = \frac{V_{eq}}{R} = \frac{V_1 + V_2}{R} $$
$V_{eq} = \sum_{k=1}^{n} V_k$ (Algebraic sum)
+ - V₁ (12V) + - V₂ (6V) Eq: 18V
Parallel Connection
In parallel, VOLTAGE SOURCE is dominant

Two equal sources: $V_1 \parallel V_2 = V$

Parallel with resistance: $V \parallel R \rightarrow V_{eq} = V$

Unequal parallel: Violates KVL! (Not allowed)

Short circuit: $V \parallel 0 \approx 0 \text{ V}$ (Source shorted)

+ - + - 12V

30. Current Source (Ideal & Practical)

Ideal Current Source
$$ I = I_s \quad (R_{int} = \infty) $$

V-I Characteristic: Vertical straight line at $I = I_s$.

Open Circuit: $V_{OC} \rightarrow \infty$ (Ideal limit)

Short Circuit: $I_{SC} = I_s$ (Safe)

Is V I I = Is
Voltage $V = I_s R_L$: 50V
Practical Current Source
$$ I_{load} = I_s - \frac{V_{term}}{R_{int}} $$

V-I Characteristic: Downward sloping line.

Note: Internal resistance $R_{int}$ is shunt (parallel).

Rint -1/Rint
As Rint increases, src becomes better.
Current Source Connections
Parallel: $I_{eq} = \sum I_k$

Adding ($Same$) or Subtracting ($Opposite$). Resistance in parallel has no effect on ideal current.

Must be equal or KCL violated! Series R has no effect on current.

31. Dependent (Controlled) Sources

Type Equation Description
VCVS $V = k \cdot v$ Voltage Controlled Voltage Source (Gain $k$: V/V)
VCCS $I = g_m \cdot v$ Voltage Controlled Current Source (Transconductance $g_m$: A/V)
CCVS $V = r_m \cdot i$ Current Controlled Voltage Source (Transresistance $r_m$: V/A)
CCCS $I = \beta \cdot i$ Current Controlled Current Source (Current Gain $\beta$: A/A)
Interactive VCCS (MOSFET Model)
v I = gm · v
Generated Current: 4 mA
Key Rules
  • ◆ Value depends on another circuit variable elsewhere.
  • ◆ Symbol: Diamond shape.
  • ⚠ NEVER zero out dependent sources in Superposition.
  • 📱 Models: MOSFET (VCCS), BJT (CCCS), Op-amp (VCVS).

32. Power Absorbed & Delivered

Sign Convention

1. Absorbing Power (+): Current ENTERING (+) terminal.

+ - I P = +VI

2. Delivering Power (-): Current LEAVING (+) terminal.

+ - I P = -VI
Conservation of Power
$$\sum P_{all} = 0 \quad \Rightarrow \quad P_{del} = P_{abs}$$

In any isolated circuit, total power generated must equal total power absorbed.

Interactive Balance Check

Delivered: 120W
=
Absorbed: 120W
✓ Balanced

33. Source Transformation

MORPHING TRANSFORMATION
A B + - Vs = 12V Rs = 2Ω
Transformation Rules
$$ V_s \text{ (series } R_s) \longleftrightarrow I_s \text{ (parallel } R_s) $$ $$ I_s = \frac{V_s}{R_s} \qquad V_s = I_s \times R_s $$
Direction: Arrow points toward the positive (+) terminal.

Valid only for external circuit variables. Internal power calculations will differ!

34. Voltage Division Rule (VDR)

Direct Proportion (Series)
+ - R₁: 10Ω R₂: 20Ω
V₁: 4V V₂: 8V
The Formula
$$ V_x = \frac{R_x}{R_{total}} \times V_{source} $$

Key Insight: Voltage divides in direct proportion to resistance. Larger R consumes more voltage.

$$ \frac{V_1}{V_2} = \frac{R_1}{R_2} $$

35. Current Division Rule (CDR)

Inverse Proportion (Parallel)
R₁ R₂
I₁: 7A I₂: 3A
The Formula
$$ I_1 = \frac{R_2}{R_1 + R_2} \times I_{total} $$
Note: Opposite resistor in numerator.

Using conductances ($G = 1/R$):

$$ I_k = \frac{G_k}{G_{total}} \times I_{total} $$

36. Nodal Voltage Analysis

Circuit Graph (KCL Solver)
V₁ V₂ V₃ ⏚

Click a node to build KCL equation

Select a node above...
Nodal Equations

Application of KCL at each non-reference node.

$$ \text{Num Equations} = N - 1 $$

Matrix Form: $[G][V] = [I]$

G₁₁ -G₁₂ -G₁₃
-G₂₁ G₂₂ -G₂₃
-G₃₁ -G₃₂ G₃₃

Supernode: Used when a voltage source exists between two nodes. (Constraint: $V_a - V_b = V_s$)

37. Mesh Current Analysis

Loop Graph (KVL Solver)
I₁ I₂

Click a loop to write KVL equation

Select a mesh above...
Mesh Equations

Application of KVL around each independent loop (mesh).

$$ \text{Num Equations} = B - (N-1) $$

Matrix Form: $[R][I] = [V]$

R₁₁ -R₁₂
-R₂₁ R₂₂

Supermesh: Used when a current source exists between two meshes.

Limit: Only for Planar networks.

38. Energy Sources

Ideal Sources

+ - Independent Voltage
V = constant
Independent Current
I = constant
kV Dependent (VCVS, etc.)
V/I depends on network

Source Transformation

Any practical voltage source in series with a resistor can be transformed into a current source in parallel with that same resistor, and vice versa.

$$ V_s = I_s \times R \quad \leftrightarrow \quad I_s = \frac{V_s}{R} $$
+- Series R
⇄
Parallel R

39. Network Topology & Graph Theory

  • Nodes (n): Meeting point of 2 or more elements.
  • Branches (b): Single element representing a path between two nodes.
  • Loops (l): Any closed path in a circuit.
  • Meshes: A loop that contains no other loops inside it.

Fundamental Theorem of Network Topology:

$$ b = l + n - 1 $$
$$ \text{No. of KVL mesh equations} = b - n + 1 $$

Hover over terms to highlight diagram

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