Electrical Circuits and Fields
DC
Current:
The rate of flow of electrical charge.
1 A = 1 C 1 s 1 A = \dfrac{1 C}{1 s} 1 A = 1 s 1 C 1 C = 6.24 ⋅ 10 18 1 C = 6.24 \cdot\ 10^{18} 1 C = 6.24 ⋅ 1 0 18 I ( t ) = d Q ( t ) d t [ A ] I(t) = \dfrac{dQ(t)}{dt}\ [A] I ( t ) = d t d Q ( t ) [ A ] Q ( t ) = ∫ t 0 t i ( t ) d t + q ( t 0 ) Q(t) = \int_{t_0}^{t} i(t) dt + q(t_0) Q ( t ) = ∫ t 0 t i ( t ) d t + q ( t 0 ) Voltage:
The difference in potential energy between two points, for one Coulomb of charge.
V = Δ E p q = W q [ V ] V = \dfrac{\Delta E_p}{q} = \dfrac{W}{q}\ [V] V = q Δ E p = q W [ V ] Resistance:
Is the opposition to the flow of current.
R = ρ L A [ Ω ] R = \frac{\rho L}{A}\ [\Omega] R = A ρ L [ Ω ] ρ = resistivity of the material \rho = \text{resistivity of the material} ρ = resistivity of the material Ohm’s Law:
V = R I V = RI V = R I Direction:
From + to -. V a b V_{ab} V ab a a a b b b I a b I_{ab} I ab
KCL:
The sum of current entering the node is equivalent to the sum of current leaving the node.
∑ k = 1 n I e n t e r i n g = ∑ k = 1 n I l e a v i n g \sum_{k = 1}^{n}\ I_{entering} = \sum_{k = 1}^{n}\ I_{leaving} k = 1 ∑ n I e n t er in g = k = 1 ∑ n I l e a v in g KVL:
The sum of voltages equals zero, for any closed loop.
∑ k = 1 n V k = 0 \sum_{k = 1}^{n}\ V_k = 0 k = 1 ∑ n V k = 0 Power:
P = V I [ W ] P = VI\ [W] P = V I [ W ] Energy:
W = ∫ t 1 t 2 P ( t ) d t W = \int_{t_1}^{t_2} P(t) dt W = ∫ t 1 t 2 P ( t ) d t Equivalent resistance in series:
R e q = R 1 + R 2 + … R N R_{eq} = R_1 + R_2 + \ldots\ R_N R e q = R 1 + R 2 + … R N Equivalent resistance in parallel:
R e q = 1 1 R 1 + 1 R 2 + … + 1 R N R_{eq} = \dfrac{1}{\dfrac{1}{R_1} + \dfrac{1}{R_2} + \ldots\ + \dfrac{1}{R_N}} R e q = R 1 1 + R 2 1 + … + R N 1 1 R e q = R 1 R 2 R 1 + R 2 R_{eq} = \dfrac{R_1 R_2}{R_1 + R_2} R e q = R 1 + R 2 R 1 R 2 R e q = R 1 R 2 R 3 R 1 R 2 + R 2 R 3 + R 1 R 3 R_{eq} = \dfrac{R_1 R_2 R_3}{R_1 R_2 + R_2 R_3 + R_1 R_3} R e q = R 1 R 2 + R 2 R 3 + R 1 R 3 R 1 R 2 R 3 Voltage Divider (Series):
V k = V t o t a l ⋅ R k R 1 + R 2 + … R N V_{k} = V_{total} \cdot\ \dfrac{R_k}{R_1 + R_2 + \ldots\ R_N} V k = V t o t a l ⋅ R 1 + R 2 + … R N R k Current Divider (Parallel):
I k = I t o t a l ⋅ R o t h e r R 1 + R 2 + … R N I_{k} = I_{total} \cdot\ \dfrac{R_{other}}{R_1 + R_2 + \ldots\ R_N} I k = I t o t a l ⋅ R 1 + R 2 + … R N R o t h er For example, for two parallel:
I 1 = I t o t a l ⋅ R 2 R 1 + R 2 I_{1} = I_{total} \cdot\ \dfrac{R_2}{R_1 + R_2} I 1 = I t o t a l ⋅ R 1 + R 2 R 2 I 2 = I t o t a l ⋅ R 1 R 1 + R 2 I_{2} = I_{total} \cdot\ \dfrac{R_1}{R_1 + R_2} I 2 = I t o t a l ⋅ R 1 + R 2 R 1 Node-Voltage analysis
Idea:
Find the nodes
Assign a reference node (usually, we pick the node with most connections)
Assign node voltages (Note, in a circuit with, N N N N − 1 N - 1 N − 1
Then we solve these using KCL on each node (∑ I o u t = ∑ I i n \sum\ I_{out} = \sum\ I_{in} ∑ I o u t = ∑ I in
The convention is also the following:
Consider i o u t i_{out} i o u t
Consider i o u t i_{out} i o u t
V c u r r e n t − V a d j a c e n t V_{current} - V_{adjacent} V c u r r e n t − V a d j a ce n t
Mesh-Current analysis:
Is the opposite of Node-voltage analysis. Therefore, we just apply KVL instead of KCL. Note that this only forks for planar circuits .
Planar circuit: It is possible to draw it in a plane without crossing wires.
Superposition:
As the name suggest, it’s the principle that, given a linear system, the net response caused by two or more stimuli is the sum of these respones.
In our case, the stimuli are voltage/current sources.
So our method is:
Equivalent circuits:
Replace the load, R L R_L R L
Find the short/open circuit current/voltage, V o c / I s c V_{oc} / I_{sc} V oc / I sc
Find the equivalent resistance, R e q R_{eq} R e q
Then:
V T H = V o c V_{TH} = V_{oc} V T H = V oc
I N = I s c I_{N} = I_{sc} I N = I sc
R T H = R N = R e q R_{TH} = R_{N} = R_{eq} R T H = R N = R e q
V T H = I N ⋅ R e q V_{TH} = I_N \cdot\ R_{eq} V T H = I N ⋅ R e q Capacitors:
A capacitor is a device that stores electric charge by creating an electric field between two conductive plates separated by an insulating material.
C = q V C = \dfrac{q}{V} C = V q I = C d V d t I = C \dfrac{dV}{dt} I = C d t d V P = I V = C V d V d t P = IV = CV \dfrac{dV}{dt} P = I V = C V d t d V W ( t ) = ∫ t 0 t P ( t ) d t W ( t ) = ∫ t 0 t C V d V d t d t W ( t ) = C ⋅ ∫ t 0 t V d V W ( t ) = C 2 [ V ( t ) 2 − V ( t 0 ) 2 ] W(t) = \int_{t_0}^{t}\ P(t)\ dt \newline
W(t) = \int_{t_0}^{t}\ CV \frac{dV}{dt}\ dt \newline
W(t) = C \cdot\ \int_{t_0}^{t}\ V\ dV \newline
W(t) = \frac{C}{2} [V(t)^2 - V(t_0)^2] W ( t ) = ∫ t 0 t P ( t ) d t W ( t ) = ∫ t 0 t C V d t d V d t W ( t ) = C ⋅ ∫ t 0 t V d V W ( t ) = 2 C [ V ( t ) 2 − V ( t 0 ) 2 ] If V = 0 V = 0 V = 0 t 0 t_0 t 0
W ( t ) = C ⋅ v ( t ) 2 2 ∥ q = C V W ( t ) = v ( t ) q ( t ) 2 ∥ C = q V W ( t ) = q ( t ) 2 2 C ∥ V = q C W(t) = \frac{C \cdot\ v(t)^2}{2} \quad \| \quad q = CV \newline
W(t) = \frac{v(t)\ q(t)}{2} \quad \| \quad C = \frac{q}{V} \newline
W(t) = \frac{q(t)^2}{2C} \quad \| \quad V = \frac{q}{C} W ( t ) = 2 C ⋅ v ( t ) 2 ∥ q = C V W ( t ) = 2 v ( t ) q ( t ) ∥ C = V q W ( t ) = 2 C q ( t ) 2 ∥ V = C q V ( t ) = 1 C ∫ t 0 t i ( t ) d t + V ( t 0 ) V(t) = \frac{1}{C} \int_{t_0}^{t}\ i(t)\ dt + V(t_0) V ( t ) = C 1 ∫ t 0 t i ( t ) d t + V ( t 0 ) Capacitors in series and parallel:
Parallel:
C e q = C 1 + C 2 + … + C N C_{eq} = C_1 + C_2 + \ldots\ + C_N C e q = C 1 + C 2 + … + C N Series:
C e q = 1 C 1 + 1 C 2 + … + 1 C N C_{eq} = \dfrac{1}{C_1} + \dfrac{1}{C_2} + \ldots\ + \dfrac{1}{C_N} C e q = C 1 1 + C 2 1 + … + C N 1
Capacitors are open circuits to DC voltage (If V V V I = 0 I = 0 I = 0
The voltage on a capacitor cannot jump (Change instantaneously, since then we would have infinite current).
Capacitors store energy (I ⋅ V > 0 I \cdot\ V > 0 I ⋅ V > 0 I ⋅ V < 0 I \cdot\ V < 0 I ⋅ V < 0
Inductors:
An inductor is a component in an electrical circuit that utilizes electromagnetic induction to resist changes in current flow by generating a voltage that opposes the change.
V ( t ) = L d I d t V(t) = L \dfrac{dI}{dt} V ( t ) = L d t d I I ( t ) = 1 L ∫ t 0 t V ( t ) d t + I ( t 0 ) I(t) = \frac{1}{L} \int_{t_0}^{t}\ V(t) dt + I(t_0) I ( t ) = L 1 ∫ t 0 t V ( t ) d t + I ( t 0 ) Power & Energy in Inductors
P ( t ) = I ( t ) V ( t ) = I ( L d I d t ) = d W d t P(t) = I(t) V(t) = I(L\ \frac{dI}{dt}) = \frac{dW}{dt} P ( t ) = I ( t ) V ( t ) = I ( L d t d I ) = d t d W W = L I 2 2 W = \frac{LI^2}{2} W = 2 L I 2 Inductors in series and parallel:
Series:
L e q = L 1 + L 2 + … L N L_{eq} = L_1 + L_2 + \ldots\ L_N L e q = L 1 + L 2 + … L N Parallel:
L e q = 1 L 1 + 1 L 2 + … 1 L N L_{eq} = \dfrac{1}{L_1} + \dfrac{1}{L_2} + \ldots\ \dfrac{1}{L_N} L e q = L 1 1 + L 2 1 + … L N 1
Inductors are short circuits to DC voltages (If I I I V = 0 V = 0 V = 0
The current through an inductor cannot jump (change instantaneously, otherwise we would have infinite voltage).
Inductors store energy (I ⋅ V > 0 I \cdot\ V > 0 I ⋅ V > 0 I ⋅ V < 0 I \cdot\ V < 0 I ⋅ V < 0
Time-Varying Circuits:
V C ( t ) = V i e − t τ , where τ is: τ = R C V_{C}(t) = V_{i}\ e^{\frac{-t}{\tau}} \quad \text{, where $\tau$ is:} \newline
\tau = RC V C ( t ) = V i e τ − t , where τ is: τ = R C I ( t ) = I 0 e − R t L τ = L R I(t) = I_{0}\ e^{\frac{-Rt}{L}} \newline
\tau = \frac{L}{R} I ( t ) = I 0 e L − R t τ = R L Electrical and Magnetic Fields
Charge:
e − = 1.602 ⋅ 10 − 19 C e^{-} = 1.602 \cdot\ 10^{-19} C e − = 1.602 ⋅ 1 0 − 19 C Coulomb’s Law:
F 12 ⃗ = k e q 1 q 2 0 r 2 r 12 ^ \mathbf{\vec{F_{12}}} = k_{e}\ \frac{q_1 q_20}{r^2} \hat{r_{12}} F 12 = k e r 2 q 1 q 2 0 r 12 ^ ε 0 = 10 − 9 36 π ≈ 8.841 ⋅ 10 − 12 [ F m ] ( Farads per meter ) \varepsilon_{0} = \frac{10^{-9}}{36\pi} \approx 8.841 \cdot\ 10^{-12}\ \left[\dfrac{F}{m}\right]\ (\text{Farads per meter}) ε 0 = 36 π 1 0 − 9 ≈ 8.841 ⋅ 1 0 − 12 [ m F ] ( Farads per meter ) k e = 1 4 π ε 0 ≈ 9 ⋅ 10 9 [ N m 2 C 2 ] k_e = \frac{1}{4\pi\varepsilon_{0}} \approx 9 \cdot\ 10^{9}\ \left[\dfrac{Nm^2}{C^2}\right] k e = 4 π ε 0 1 ≈ 9 ⋅ 1 0 9 [ C 2 N m 2 ] E ⃗ = k e q r 2 r ^ \mathbf{\vec{E}} = k_{e}\ \frac{q}{r^2}\hat{r} E = k e r 2 q r ^ F E ⃗ = q E ⃗ \mathbf{\vec{F_{E}}} = q\ \mathbf{\vec{E}} F E = q E Dipoles:
p ⃗ = q d ⃗ \mathbf{\vec{p}} = q\mathbf{\vec{d}} p = q d Placing a dipole in an electrical field:
τ ⃗ = p ⃗ × E ⃗ τ = p ⋅ E s i n ( θ ) \mathbf{\vec{\tau}} = \mathbf{\vec{p}} \times \mathbf{\vec{E}} \newline
\tau = p \cdot\ E\ sin(\theta) τ = p × E τ = p ⋅ E s in ( θ ) Electrical Flux 1D:
Φ = ∑ i = 1 N E ⃗ ⋅ n ^ \Phi = \sum_{i = 1}^{N}\ \mathbf{\vec{E}} \cdot\ \hat{n} Φ = i = 1 ∑ N E ⋅ n ^ Φ = ∫ L 1 L 2 E l ⃗ ⋅ n ^ ⋅ d l \Phi = \int_{L_{1}}^{L_{2}}\ \mathbf{\vec{E_{l}}} \cdot\ \hat{n} \cdot\ dl Φ = ∫ L 1 L 2 E l ⋅ n ^ ⋅ d l Electrical Flux 2D:
Φ = ∬ E ⃗ ⋅ n ^ ⋅ d A ⃗ = ∬ E ⃗ ⋅ d A ⃗ c o s ( θ ) \Phi = \iint \mathbf{\vec{E}} \cdot\ \hat{n} \cdot\ d\mathbf{\vec{A}} = \iint \mathbf{\vec{E}} \cdot\ d\mathbf{\vec{A}}\ cos(\theta) Φ = ∬ E ⋅ n ^ ⋅ d A = ∬ E ⋅ d A cos ( θ ) Electrical Flux closed contour:
Φ = ∯ E ⃗ ⋅ d A ⃗ \Phi = \oiint\ \mathbf{\vec{E}} \cdot\ d\mathbf{\vec{A}} Φ = ∬ E ⋅ d A Gauss’s Law:
Φ E = ∯ E ⃗ ⋅ d A ⃗ = q ε 0 \Phi_{E} = \oiint \mathbf{\vec{E}} \cdot\ d\mathbf{\vec{A}} = \frac{q}{\varepsilon_{0}} Φ E = ∬ E ⋅ d A = ε 0 q Work to move charges:
W = k e q 1 q 2 R W = \dfrac{k_e q_1 q_2}{R} W = R k e q 1 q 2 Work in an electrical field:
W = − q E 0 d W = -qE_0d W = − q E 0 d Cheat sheet:
W = − ∫ F ⃗ ⋅ d r W = - \int \mathbf{\vec{F}} \cdot\ dr W = − ∫ F ⋅ d r E ⃗ = F ⃗ Q \mathbf{\vec{E}} = \dfrac{\mathbf{\vec{F}}}{Q} E = Q F Δ V = W Q \Delta V = \dfrac{W}{Q} Δ V = Q W Δ V = − ∫ E ⃗ ⋅ d r \Delta V = - \int \mathbf{\vec{E}} \cdot\ dr Δ V = − ∫ E ⋅ d r Capacitors:
q = σ A q = \sigma A q = σ A E = q ε 0 A E = \dfrac{q}{\varepsilon_0 A} E = ε 0 A q Energy stored in a capacitor:
W = 1 C Q 2 2 W = \dfrac{1}{C}\ \dfrac{Q^2}{2} W = C 1 2 Q 2 Magnetic Fields:
F ⃗ B = q v ⃗ × B ⃗ = ∣ q ∣ v B s i n ( θ ) \mathbf{\vec{F}_B} = q\mathbf{\vec{v}} \times \mathbf{\vec{B}} = |q|vB\ sin(\theta) F B = q v × B = ∣ q ∣ v B s in ( θ ) B ⃗ [ 1 T = 1 N A ⋅ m ] \mathbf{\vec{B}} \left[1\ T = 1 \dfrac{N}{A \cdot\ m}\right] B [ 1 T = 1 A ⋅ m N ] Biot-Savarts law:
d B ⃗ = μ 0 4 π I d s ⃗ × r ⃗ r 2 d\mathbf{\vec{B}} = \dfrac{\mu_0}{4\pi} \dfrac{I\ d\mathbf{\vec{s}} \times \mathbf{\vec{r}}}{r^2} d B = 4 π μ 0 r 2 I d s × r μ 0 = 4 π ⋅ 10 − 7 ≈ 1.2566 ⋅ 10 − 6 \mu_0 = 4\pi \cdot\ 10^{-7}\ \approx 1.2566 \cdot\ 10^{-6} μ 0 = 4 π ⋅ 1 0 − 7 ≈ 1.2566 ⋅ 1 0 − 6 B ⃗ = μ 0 I 4 π ∫ d s ⃗ × r ⃗ r 2 \mathbf{\vec{B}} = \dfrac{\mu_0 I}{4\pi} \int \dfrac{d\mathbf{\vec{s}} \times \mathbf{\vec{r}}}{r^2} B = 4 π μ 0 I ∫ r 2 d s × r B = μ 0 I 2 π r B = \dfrac{\mu_0 I}{2\pi r} B = 2 π r μ 0 I B = μ 0 N I l = μ 0 n l B = \dfrac{\mu_0 NI}{l} = \mu_0 nl B = l μ 0 N I = μ 0 n l Ampere’s Law:
∮ B ⃗ ⋅ d s ⃗ = μ 0 I e n c \oint \mathbf{\vec{B}} \cdot\ d\mathbf{\vec{s}} = \mu_0 I_{enc} ∮ B ⋅ d s = μ 0 I e n c Lorentz Force:
F ⃗ E = q E ⃗ \mathbf{\vec{F}_E} = q\mathbf{\vec{E}} F E = q E F ⃗ B = q v ⃗ × B ⃗ \mathbf{\vec{F}_B} = q\mathbf{\vec{v}} \times \mathbf{\vec{B}} F B = q v × B F ⃗ = F ⃗ E + F ⃗ B = q ( E ⃗ + v ⃗ × B ⃗ ) \mathbf{\vec{F} = \mathbf{\vec{F}_E} + \mathbf{\vec{F}_B} = q(\mathbf{\vec{E}} + \mathbf{\vec{v}} \times \mathbf{\vec{B}})} F = F E + F B = q ( E + v × B ) Magnetic Flux:
Φ B = ∫ ∫ B ⃗ ⋅ d A ⃗ = B A c o s ( θ ) \Phi_B = \int \int \mathbf{\vec{B}} \cdot\ d\mathbf{\vec{A}} = BA\ cos(\theta) Φ B = ∫∫ B ⋅ d A = B A cos ( θ ) Lenz Law:
The induced current produces a magnetic field, which oppose the change in magnetic flux that induces such currents.
Maxwell’s Equations:
Gauss’s Law for electrostatics :
Φ E = ∯ E ⃗ ⋅ d A ⃗ = q μ 0 \Phi_E = \oiint \mathbf{\vec{E}} \cdot\ d\mathbf{\vec{A}} = \dfrac{q}{\mu_0} Φ E = ∬ E ⋅ d A = μ 0 q Gauss’s Law for magnetism :
Φ B = ∯ B ⃗ ⋅ d A ⃗ = 0 \Phi_B = \oiint \mathbf{\vec{B}} \cdot\ d\mathbf{\vec{A}} = 0 Φ B = ∬ B ⋅ d A = 0 Faraday’s Law :
ε = ∮ E ⃗ ⋅ d s ⃗ = − d Φ B d t \varepsilon = \oint \mathbf{\vec{E}} \cdot d\mathbf{\vec{s}} = -\dfrac{d \Phi_B}{dt} ε = ∮ E ⋅ d s = − d t d Φ B Ampere-Maxwell Law :
ε = ∮ B ⃗ ⋅ d s ⃗ = μ 0 ( I + I d ) = μ 0 ( I + ε 0 d Φ E d t ) \varepsilon = \oint \mathbf{\vec{B}} \cdot d\mathbf{\vec{s}} = \mu_0(I + I_d) = \mu_0\left(I + \varepsilon_0 \dfrac{d \Phi_E}{dt}\right) ε = ∮ B ⋅ d s = μ 0 ( I + I d ) = μ 0 ( I + ε 0 d t d Φ E ) If q = 0 and I = 0:
Gauss’s Law for electrostatics :
Φ E = ∯ E ⃗ ⋅ d A ⃗ = 0 \Phi_E = \oiint \mathbf{\vec{E}} \cdot\ d\mathbf{\vec{A}} = 0 Φ E = ∬ E ⋅ d A = 0 Gauss’s Law for magnetism :
Φ B = ∯ B ⃗ ⋅ d A ⃗ = 0 \Phi_B = \oiint \mathbf{\vec{B}} \cdot\ d\mathbf{\vec{A}} = 0 Φ B = ∬ B ⋅ d A = 0 Faraday’s Law :
ε = ∮ E ⃗ ⋅ d s ⃗ = − d Φ B d t \varepsilon = \oint \mathbf{\vec{E}} \cdot d\mathbf{\vec{s}} = -\dfrac{d \Phi_B}{dt} ε = ∮ E ⋅ d s = − d t d Φ B Ampere-Maxwell Law :
ε = ∮ B ⃗ ⋅ d s ⃗ = μ 0 ( I + I d ) = μ 0 ε 0 d Φ E d t \varepsilon = \oint \mathbf{\vec{B}} \cdot d\mathbf{\vec{s}} = \mu_0(I + I_d) = \mu_0 \varepsilon_0 \dfrac{d \Phi_E}{dt} ε = ∮ B ⋅ d s = μ 0 ( I + I d ) = μ 0 ε 0 d t d Φ E Electromagnetic Waves:
E B = ω k = c \dfrac{E}{B} = \dfrac{\omega}{k} = c B E = k ω = c EMFs in circuits:
Electrical field opposes change in voltage
C = ε 0 A d U = 1 2 C ∣ V 2 ∣ I ( t ) = C d V d t C = \varepsilon_0 \dfrac{A}{d} \newline
U = \dfrac{1}{2} C|V^2| \newline
I(t) = C \dfrac{dV}{dt} C = ε 0 d A U = 2 1 C ∣ V 2 ∣ I ( t ) = C d t d V
Magnetic field opposes change in the current
L = μ 0 N 2 A l U = 1 2 L I 2 V ( t ) = L d I d t L = \mu_0 N^2 \dfrac{A}{l} \newline
U = \dfrac{1}{2} LI^2 \newline
V(t) = L \dfrac{dI}{dt} L = μ 0 N 2 l A U = 2 1 L I 2 V ( t ) = L d t d I AC
General:
Note: Can be cos \cos cos
V ( t ) = V m sin ( ω t + θ ) V(t) = V_m \sin(\omega t + \theta) V ( t ) = V m sin ( ω t + θ ) V m − Amplitude [ V ] ω − Angular frequency [ r a d s ] θ − Phase Shift [ ∘ or r a d ] T − Period [ s ] f − Frequency [ H z ] V_m - \text{Amplitude}\ [V] \newline
\omega - \text{Angular frequency}\ \left[\dfrac{rad}{s}\right] \newline
\theta - \text{Phase Shift} \left[^\circ\ \text{or}\ rad \right] \newline
T - \text{Period}\ [s] \newline
f - \text{Frequency}\ [Hz] \newline V m − Amplitude [ V ] ω − Angular frequency [ s r a d ] θ − Phase Shift [ ∘ or r a d ] T − Period [ s ] f − Frequency [ H z ] T = 2 π ω = 1 f ω = 2 π f T = \dfrac{2\pi}{\omega} = \dfrac{1}{f} \newline
\omega = 2\pi f T = ω 2 π = f 1 ω = 2 π f Root Mean Square (RMS):
V R M S = 1 T ∫ 0 T V 2 ( t ) d t = V m 2 I R M S = 1 T ∫ 0 T I 2 ( t ) d t = I m 2 V_{RMS} = \sqrt{\dfrac{1}{T} \int_0^T V^2(t) dt} = \dfrac{V_m}{\sqrt{2}} \newline
I_{RMS} = \sqrt{\dfrac{1}{T} \int_0^T I^2(t) dt} = \dfrac{I_m}{\sqrt{2}} V R M S = T 1 ∫ 0 T V 2 ( t ) d t = 2 V m I R M S = T 1 ∫ 0 T I 2 ( t ) d t = 2 I m P A v g = ( V R M S ) 2 R = ( V m 2 ) 2 R = V m 2 2 R P_{Avg} = \dfrac{(V_{RMS})^2}{R} = \dfrac{\left(\dfrac{V_m}{\sqrt{2}}\right)^2}{R} = \dfrac{V_m^2}{2R} \newline P A v g = R ( V R M S ) 2 = R ( 2 V m ) 2 = 2 R V m 2 P A v g = ( I R M S ) 2 R = ( I m 2 ) 2 R = I m 2 R 2 P_{Avg} = (I_{RMS})^2 R = \left(\dfrac{I_m}{\sqrt{2}}\right)^2 R = \dfrac{I_m^2 R}{2} P A v g = ( I R M S ) 2 R = ( 2 I m ) 2 R = 2 I m 2 R Trigonometry:
r a d = d e g ⋅ π 180 rad = deg \cdot\ \dfrac{\pi}{180} r a d = d e g ⋅ 180 π s i n ( ω t ) = c o s ( ω t − 90 ∘ ) c o s ( ω t ) = s i n ( ω t + 90 ∘ ) sin(\omega t) = cos(\omega t - 90^\circ) \newline
cos(\omega t) = sin(\omega t + 90^\circ) s in ( ω t ) = cos ( ω t − 9 0 ∘ ) cos ( ω t ) = s in ( ω t + 9 0 ∘ ) Rules for comparing the phase of two wave functions:
Both must be written in either sine or cosine.
Both must be written with positive amplitude.
Each have the same constant frequency.
Phasors:
Rectangular form:
z = x + j y z = ∣ z ∣ ( c o s ( φ ) + j s i n ( φ ) ) z = x + jy \newline
z = |z|(cos(\varphi) + j sin(\varphi)) z = x + j y z = ∣ z ∣ ( cos ( φ ) + j s in ( φ )) Polar form:
z = ∣ z ∣ ∠ θ ∘ z = ∣ z ∣ e j θ ∘ z = |z|\angle{\theta^\circ} \newline
z = |z|e^{j \theta^\circ} z = ∣ z ∣∠ θ ∘ z = ∣ z ∣ e j θ ∘ Polar to rectangular form:
z = r ∠ θ ∘ x = r c o s ( θ ∘ ) y = r c o s ( θ ∘ ) z = x + j y z = r\angle{\theta^\circ} \newline
x = r cos(\theta^\circ) \newline
y = r cos(\theta^\circ) \newline
z = x + j y z = r ∠ θ ∘ x = r cos ( θ ∘ ) y = r cos ( θ ∘ ) z = x + j y Rectangular to polar form:
z = x + j y r = x 2 + y 2 θ = arctan ( y x ) z = r ∠ θ ∘ z = x + jy \newline
r = \sqrt{x^2 + y^2} \newline
\theta = \arctan{\left(\dfrac{y}{x}\right)} \newline
z = r\angle{\theta^\circ} \newline z = x + j y r = x 2 + y 2 θ = arctan ( x y ) z = r ∠ θ ∘ Operations:
Multiplication:
a = ( b ∠ c ∘ ) ⋅ ( d ∠ e ∘ ) a = ( b ⋅ d ∠ c ∘ + e ∘ ) a = (b\angle{c^\circ}) \cdot\ (d\angle{e^\circ}) \newline
a = (b \cdot\ d \angle{c^\circ + e^\circ}) a = ( b ∠ c ∘ ) ⋅ ( d ∠ e ∘ ) a = ( b ⋅ d ∠ c ∘ + e ∘ ) Division:
a = ( b ∠ c ∘ ) ( d ∠ e ∘ ) a = ( b d ∠ c ∘ − e ∘ ) a = \dfrac{(b\angle{c^\circ})}{(d\angle{e^\circ})} \newline
a = (\dfrac{b}{d} \angle{c^\circ - e^\circ}) a = ( d ∠ e ∘ ) ( b ∠ c ∘ ) a = ( d b ∠ c ∘ − e ∘ ) Phasor relations:
V = R I V = RI \newline V = R I V = j ω L I V = j\omega LI \newline V = j ω L I V = 1 j ω C I V = \dfrac{1}{j\omega C} I V = j ω C 1 I Impedance:
Z = V I Z = \dfrac{V}{I} Z = I V This means that:
Z R = R Z L = j ω L Z C = 1 j ω C = − j 1 ω C Z_{R} = R \newline
Z_{L} = j\omega L \newline
Z_{C} = \dfrac{1}{j\omega C} = -j \dfrac{1}{\omega C} Z R = R Z L = j ω L Z C = j ω C 1 = − j ω C 1 Power in AC - general:
V ( t ) = V m c o s ( ω t + θ V ) I ( t ) = I m c o s ( ω t + ϕ I ) V(t) = V_m cos(\omega t + \theta_V) \newline
I(t) = I_m cos(\omega t + \phi_I) V ( t ) = V m cos ( ω t + θ V ) I ( t ) = I m cos ( ω t + ϕ I ) P ( t ) = V m I m c o s ( ω t + θ V ) c o s ( ω t + ϕ I ) P(t) = V_m I_m cos(\omega t + \theta_V) cos(\omega t + \phi_I) P ( t ) = V m I m cos ( ω t + θ V ) cos ( ω t + ϕ I ) P ( t ) = 1 2 V m I m c o s ( θ V − ϕ I ) + 1 2 V m I m c o s ( 2 ω t + θ V + ϕ I ) P(t) = \dfrac{1}{2} V_m I_m cos(\theta_V - \phi_I) + \dfrac{1}{2} V_m I_m cos(2 \omega t + \theta_V + \phi_I) P ( t ) = 2 1 V m I m cos ( θ V − ϕ I ) + 2 1 V m I m cos ( 2 ω t + θ V + ϕ I ) P a v g = V m I m 2 c o s ( θ V − ϕ I ) P_{avg} = \dfrac{V_m I_m}{2} cos(\theta_V - \phi_I) P a v g = 2 V m I m cos ( θ V − ϕ I ) P a v g = V r m s I r m s c o s ( θ V − ϕ I ) P_{avg} = V_{rms} I_{rms} cos(\theta_V - \phi_I) P a v g = V r m s I r m s cos ( θ V − ϕ I ) Different Types:
Real Power: P = V r m s I r m s c o s ( θ V − ϕ I ) [ W ] P = V_{rms} I_{rms}\ cos(\theta_V - \phi_I)\ [W] P = V r m s I r m s cos ( θ V − ϕ I ) [ W ]
Reactive Power: Q = V r m s I r m s s i n ( θ V − ϕ I ) [ V A R ] Q = V_{rms} I_{rms} sin(\theta_V - \phi_I) [VAR] Q = V r m s I r m s s in ( θ V − ϕ I ) [ V A R ]
Complex Power: S = P + j Q S = P + jQ S = P + j Q S = V r m s I r m s ∠ θ V − ϕ I [ V A ] S = V_{rms} I_{rms} \angle \theta_V - \phi_I [VA] S = V r m s I r m s ∠ θ V − ϕ I [ V A ]
Apparent Power: ∣ S ∣ = V r m s I r m s [ V A ] |S| = V_{rms} I_{rms} [VA] ∣ S ∣ = V r m s I r m s [ V A ]