ELEC
Formulas
Electrical current - time rate of el. charge flow through a cross section of a
conductor or circuit element - Unit : [A] ampere = [C/s] coulomb/ second
Voltage - energy transferred per unit charge that flows from one point in an
electrical circuit to a second point - Unit : [V] volt = [J/C] joule/coulomb
Power – rate of energy transfer - Unit : [W] watt = [J/s] joules/second
Energy delivered to an element in time - Unit : [J] joule = [Ws] watt*second
Kirchhoff’s Current Law (KCL) - Sum of currents entering a node equals the sum of currents
leaving it. 𝑛𝑜𝑑𝑒 𝑖 = 0
Series circuit – all
the elements have
the same current
Parallel circuit – all the
elements have voltages
with same magnitude
and polarity
Kirchhoff’s Voltage Law (KVL) - Sum of voltages in a closed path/loop in an electrical circuit is 0.
𝑙𝑜𝑜𝑝 𝑣 = 0
Ohm’s Law : The voltage across an ideal resistor is proportional to the current
passing through it.
The constant of proportionality is the resistance R - Unit : [V/A] = [Ω] ohm
Voltage divider
Current divider
𝑖1 =
𝑣
1
=
𝑖
𝑅1 𝑅 1 + 1 𝑡𝑜𝑡𝑎𝑙
1 𝑅
𝑅2
1
Node voltage analysis – write KCL for each node in the circuit 𝑮 𝒗 = [𝑰]
- Shortcut : (only for circuits with resistances and independent current sources!)
• diagonal elements of [G] = sums of 1/R connected to the corresponding nodes
• off-diagonal elements of [G] = negative of 1/R between the corresponding nodes
• [I] = current sources in the corresponding nodes (if current goes out of the node, multiply
with -1)
- In case of voltage sources – write the KCL for the supernode containing it
Mesh current analysis – write KVL for each loop of the circuit
𝑹 𝑰 = [𝑽]
- Shortcut : (only for circuits with resistances and independent voltage sources!)
• diagonal elements of [R] = sums of R contained in the corresponding mesh
• off-diagonal elements of [R] = negative of R shared by the corresponding meshes
• [V] = voltage sources in the corresponding loops (in clockwise passage through the mesh, if
you enter the voltage source from the positive polarity, multiply the voltage by -1!)
- In case of current sources – write the KVL for the supermesh containing it
Thévenin & Norton equivalent circuits
1. Step : Zero the sources to find Thévenin resistance (only if no dependent sources!)
replace voltage sources by a short circuit
replace current sources by an open circuit
2. Step : Find the resistance between the terminals 𝑹𝒕 (Thévenin resistance)
3. Step : Find 𝑖𝑠𝑐 for the circuit  𝑰𝒏 (Norton current)
4. Step : Find 𝑣𝑜𝑐 of the circuit  𝑽𝒕 (Thévenin voltage)
Maximum power transfer
purely resistive (only R) circuit:
reactive (containing also L and/or C) circuit & purely resistive load:
reactive (containing also L and/or C) circuit & reactive load:
Superposition principle - total response is the sum of responses to each of the independent
sources acting individually
Capacitance
Inductance
Frequency [Hz=1/s]
𝑉𝑚 = peak value
𝜔 = angular frequency [rad/s]
𝜃 = phase angle [°]
𝑡
Phase angle 𝜃 = −360° ∗ 𝑚𝑎𝑥
𝑇
Angular frequency
Root-mean-square (rms) or Effective value of the periodic signal
for sinusoidal signals 𝑥 𝑡 = 𝑋𝑚 cos(𝜔𝑡 + 𝜃) rms value is :
Calculating with sinusoids:
1. Step: Write sine signals in cosine form
2. Step: Write the terms as phasors : 𝑥 𝑡 = 𝑋 cos(𝜔𝑡 + 𝜃 )
𝑿𝟏 = 𝑋1 ∠𝜃1
1
1
1
3. Step: Write the phasors as complex numbers in rectangular form and sum them up:
𝑧 = 𝑧 ∠𝜃°
𝑧 = 𝑧 (𝑐𝑜𝑠𝜃 + 𝑗𝑠𝑖𝑛𝜃)
4. Step: Write the solution as a phasor:
𝑧 = 𝑥 + 𝑗𝑦
𝑧=
𝑥 2 + 𝑦 2 ∠ 𝑎𝑟𝑐𝑡𝑎𝑛
5. Step: Write the solution as sinusoidal signal:
𝑥1 𝑡 = 𝑋1 cos(𝜔𝑡 + 𝜃1 )
𝑦
𝑥
𝑿𝟏 = 𝑋1 ∠𝜃1
Resistance:
Sinusoidal steady state
Inductance:
Capacitance:
𝑍𝑅 = 𝑅
Pure R load
𝒁=𝑅
𝜃=0
𝑃>0
𝑄=0
Real (Active, Average) Power
Apparent Power
Pure L load
𝒁 = 𝜔𝐿∠90°
𝜃 = 90°
𝑃=0
𝑄>0
Reactive Power
Complex Power
Pure C load
1
𝒁=
∠ − 90°
𝜔𝐶
𝜃 = −90°
𝑃=0
𝑄<0
Steps for solving any first order circuit (RC, RL):
Solution is a sum of complementary and particular solution
1. KCL and KVL to write the circuit equation
2. If needed, differentiate circuit equation until it is purely differential
3. Find the particular solution – for DC source use the steady state (C open circuited, L
short circuited)
– for AC suppose the solution in the form of forcing
function - for sinusoidal AC source use phasors or suppose Asin(wt)+Bcos(wt); for
exponential suppose K*exp(st)
4. Define complementary solution as 𝑥𝑐 𝑡 = 𝐾𝑒 𝑠𝑡 by solving homogenous equation
(which is obtained by substituting 0 for forcing function)  use it to find s
5. Obtain complete solution by adding the particular solution to the complementary
6. Use initial conditions on the complete solution to find K
7. Write the final solution
Damping Undamped
Second order RLC circuit:
coefficient resonant
The same procedure, except for Step 4, which is as follows: 𝜶
𝑹 frequency
𝜶
=
Complementary solution depends on the damping ratio 𝜻 =
𝟏
𝟐𝑳
𝝎𝟎
𝝎𝟎 =
𝑳𝑪
1. 𝜻>1 Overdamped case
2. 𝜻=1 Critically damped case
3. 𝜻<1 Underdamped case
natural frequency 𝜔𝑛