Please see [SL Physics 1#Electric potential](/g11/sph3u7#electric-potential) for more information on voltage.
Please see [SL Physics 1#5.2 - Heating effect of electric currents](/g11/sph3u7/#52-heating-effect-of-electric-currents) for more information on current.
Please see [SL Physics 1#Resistance](/g11/sph3u7/#resistance) for more information on resistance.
**Electric charge** $Q$ quantises the charge of electrons and positive ions, and is expressed in coulombs (**C**).
Objects with charge generate electric fields, thus granting potential energy that is released upon proximity to another charge.
!!! warning
Voltage and current are capitalised in **direct current only** ($V$, $I$). In general use, their lowercase forms should be used instead ($v, $i$).
**Voltage** is related to the change in energy ($dw$) over the change in charge ($dq$), or alternatively through Ohm's law:
$$i=\frac{dw}{dq}=\frac{i}{R}$$
**Current** represents the rate of flow of charge in amps (**A**). Conventional current moves opposite electron flow because old scientists couldn't figure it out properly.
- A **ground** is the reference point that all **potential differences are relative to**.
A **generic voltage source** provides a known potential difference between its two terminals that is defined by the source. The resultant current can be calculated.
A **generic current source** provides a known amperage between its two terminals that is defined by the source. The resultant voltage can be calculated.
A current in the **positive direction** indicates that the source is releasing power (is a source). Otherwise, it is consuming power (is a load).
### Dependent energy sources
A **dependent <​T: voltage | current> source** has a **T** dependent on the voltage or current elsewhere in the circuit. $k$ is a function that is likely but not guaranteed to be linear.
A **cathode ray tube** produces an electron beam of variable intensity depending on the input signal. Electrons are deflected by the screen to produce imagery.
- A **supermesh** is a combination of multiple meshes that share a common current source.
Mesh / loop analysis is used to determine unknown currents, using KVL instead of KCL to create a system of equations.
1. Assign mesh currents to each loop.
2. Use KVL and Ohm's law to get voltages in terms of mesh currents.
3. Solve the system of equations.
It may be easier to delete the branch of the current source in supermeshes, treating the region as one mesh with multiple mesh currents.
## Linearity
Circuits are linear if and only if their voltages, resistances, and currents can be expressed in terms of linear transformations of one another. They contain only linear loads, linear dependent sources, and independent souces.
$$\text{output}\propto\text{input}+C$$
!!! example
Halving voltage must halve current (or at least halve it relative to a base current / voltage).
In linear circuits, the superposition principle states that the voltage/current through an element is equal to the sum of the voltages/currents from each independent source alone.
$$
v=\sum v_x \\
i=\sum i_x
$$
To do so, each unused independent source should be replaced with a short circuit (voltage) or an open circuit (current).
In linear circuits, a voltage source in series with a resistor can be replaced by a current source in parallel to that resistor (or vice versa), so long as Ohm's law is followed for the replacement:
$$v_1=i_2R$$
The arrow of the current source must point in the positive direction of the voltage source. This can also be used with dependent sources.
Any part of a circuit including an independent source can be replaced with exactly one voltage source and a resistor in series. Two circuits are **Thevenin equivalent** if their $\lambda$ are equal in $V=\lambda I$.
A **summing amplifier** splits an inverting amplifier's input into multiple voltage sources in series with resistances, all parallelised into the opamp:
Capacitors are open circuits in DC that store energy in electric fields. Capacitance is measured in **farads** ($\pu{1 F = 1 C/V}$).
Where $A$ is the cross-section area of the wire, $\epsilon$ is the permittivity of the dielectric, and $d$ is the distance between plates:
$$C=\frac{\epsilon A}{d}$$
Capacitors charge only when power is positive ($VI>0$).
For linear capacitors:
$$i=C\frac{dv}{dt}$$
$$v(t)=\frac{1}{C}\int^t_{t_0}i(t)dt+v(t_0)$$
The energy in a capacitor can be interconverted.
$$U=\frac 1 2 CV^2$$
Capacitor rules are the opposite of resistor rules.
- In parallel: $C_{eq} = C_1 + C_2 + ...$
- In series: $\frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + ...$
## Inductors
Inductors store energy in their magnetic field. Inductance is measured in **henrys** ($\pu{1 H = 1 V\cdot S/A}$). An ideal inductor has zero resistance and capacitance
Where $L$ is the inductance (opposition of charge flow):
$$V=L\frac{di}{dt}$$
Inductor rules are the same as resistor rules.
### Selenoids
Selenoids have an inductance based on their cross sectional area $A$, number of coils $N$, length $\ell$, and core permeability $\mu$:
$$L=\frac{N^2\mu A}{\ell}$$
Where $i(t_0)$ is the total current for $-\infty<t<t_0$
$$i=\frac 1 L\int^t_{t_0}v(t)dt + i(t_0)$$
Much like capacitors, inductors have energy now based on current.
The **scalar** quantity of **impedance** represents the opposition to electron flow, measured in ohms.
$$Z=\frac{1}{j\omega C}=j\omega L$$
It is effectively generalised resistance. Where $X$ is a positive value representing **reactance** such that $+jX$ implies inductance while $-jX$ implies capacitance:
$$Z=\frac{\bold V}{\bold I}=R\pm jX$$
**Admittance** is the inverse of impedance with units Siemens/mhos with factors **conductance** and **susceptance**:
$$Y=G+jB$$
Arranging equations yields
$$
G=\frac{R}{R^2+X^2} \\
B=-\frac{X}{R^2+X^2}
$$
### Steady state analysis
**Kirchoff's laws** only hold for phasor forms.
1. Convert to phasor forms
2. Solve phasor forms
3. Convert back to time domain
Superposition must be summed at the end only, although individual components can first be solved.
1. Convert to phasor forms
2. Solve each individual current/voltage that make KCL/KVL
When applying source transformations, different equivalent circuits for **each frequency** must be calculated individually — reducing it to one equivalent circuit is not possible.
The same rules for maximum power transfer apply with resistance, but with $Z_L$ as the **complex conjugate** of $Z_{Th}$. The maximum power has a shortcut formula:
$$P_{max}=\frac{|V_{Th}^2}{8R_{Th}}$$
The **effective value** of a sinusoid is its DC equivalent. It is the root mean square.
$$X_{rms}=\sqrt{\frac 1 T\int^T_0x^2dt}$$
The **apparent power** $S$ is the seemingly true power.
$$S=V_{rms}I_{rms}$$
The **power factor (pf)** is the required factor to take the apparent power into real power.
$$pf=\frac P S = \cos(\theta_v-\theta_i)$$
The **power factor angle** $\theta_v-\theta_i$ is the angle of local impedance between voltage and current.
The two components of complex power are actual power $P=I^2_{rms}R$ and reactive power $Q=I^2_{rms}X$, the latter with units VAR (volt-ampere reactive).
