7.6 KiB
ECE 140: Linear Circuits
Voltage, current, and resistance
Please see SL Physics 1#Electric potential for more information on voltage.
Please see SL Physics 1#5.2 - Heating effect of electric currents for more information on current.
Please see SL Physics 1#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 (\(\pu W\), )
\[P=\frac{dw}{dt}\]
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:
- Determine a reference node with \(v=0\) and stick a ground out of that node.
- Use KCL and Ohm’s law on non-reference nodes to get their currents in terms of the reference node.
- 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:
\[i=\frac{v_\text{higher}-v_\text{lower}}{R}\]
Mesh analysis
!!! definition - A mesh is a loop with no inner loops. - 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.
- Assign mesh currents to each loop.
- Use KVL and Ohm’s law to get voltages in terms of mesh currents.
- 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).
Superposition
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).
Source transformation
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.
Thevenin’s theorem
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 \(V=\lambda I\).
If there are no dependent sources, all independent sources should be removed to determine the resistance across points \(AB\):
\[R_{Th}=R_{AB}\]
Otherwise, \(V_{AB}\) and \(I_{AB}\) should be found by repeating these steps:
- Cut off the load (open if finding voltage, short if finding current)
- If dependent sources depend on elements inside the load branch, zero them
- Use analysis to determine the desired quantity
Across the load:
\[ I_L=\frac{V_{Th}}{R_{Th}+R_L} \\ V_L=R_LI_L = \frac{R_L}{R_{Th}+R_L}V_{Th} \]
!!! warning A negative resistance \(R_{L}\) indicates that the load supplies power.
Maximum power transfer
To maximise the power transferred from the circuit to the load, \(R_L\) should be equal to \(R_{Th}\).
\[P_L=v_Li_L\]