# ECE 140: Linear Circuits
## Voltage, current, and resistance
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.
$$i=\frac{dq}{dt}\approx \frac{\Delta q}{\Delta t}$$
### Power
Power represents the rate of doing work, in unit watts ($\pu W$, \pu{J/s})
$$P=\frac{dw}{dt}$$
It is also directly related to voltage and current:
$$P=vi$$
Much like relative velocity, power is directional and relative, with a positive sign indicating the direction of conventional current.
$$P_{CB}=-P_{BC}$$
In a closed system, conservation of energy applies:
$$\sum P_\text{in}=\sum P_\text{out}$$
The **ground** is the "absolute zero" voltage with a maximum potential difference. It is also known as the "reference voltage".
### Independent energy sources
!!! definition
- 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.
(Source: Wikimedia Commons)
A **generic current source** provides a known amperage between its two terminals that is defined by the source. The resultant voltage can be calculated.
(Source: Wikimedia Commons)
!!! tip
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.
$$
v=kv_0\ |\ ki_0 \\
i=kv_0\ |\ ki_0
$$
(Source: Wikimedia Commons)
(Source: Wikimedia Commons)
### Applications
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.
### Resistance
A **resistor** *always absorbs power*, so must be oriented such that current goes into the positive sign.
According to Ohm's law, the voltage, current, and resistance are related:
$$v=iR$$
The **conductance** of a resistor is the inverse of its resistance, and is expressed in siemens ($\pu{S}$)
$$G=\frac 1 R = \frac I V$$
Therefore, power can be expressed by manipulating the equations:
$$
\begin{align*}
P &= IR^2 \\
&= V^2G \\
&= \frac{V^2}{R}
\end{align*}
$$
## Kirchhoff's laws
!!! definition
- A **node** is any point in the circuit to which 3+ elements are *directly* connected (i.e., all junctions).
- A **supernode** is any connected group in the circuit to which 3+ elements are *directly* connected.
- A **loop** is any closed path of elements.
Kirchhoff's **current law** states that the sum of all current entering a node must be zero, where positive indicates current entrance.
$$\sum i_\text{entering node}=0$$
Kirchoff's **voltage law** states that the sum of all voltage in a **closed loop** must be zero.
$$\sum v_\text{loop}=0$$
### Nodal analysis
Nodal analysis uses the voltages at the **nodes** instead of elements to calculate things in a three-step process:
1. Determine a reference node with $v=0$ and stick a ground out of that node.
2. Use KCL and Ohm's law on non-reference nodes to get their currents in terms of the reference node.
3. Solve the system of equations with the formula below.
On either side of a resistor, the current flowing that entire segment can be determined via the following formula:
$$i=\frac{v_\text{higher}-v_\text{lower}}{R}$$