[[The Bare Essentials of Electrical Engineering.pdf#page=45&offset=99,702|The Bare Essentials of Electrical Engineering, Fundamental Laws]] # Resistance - Ohm's Law: $v=iR$, $v(t)=i(t)R$ - If we let $R\to \infty\Omega$, can treat the resistor as the point at which an open circuit is created - If we let $R=0\Omega$, can treat the resistor as if it doesn't exit - Resistivity: natural material property that quantifies its ability to impede current flow through it - Resistance can be calculated using $R=\rho \frac{l}{A}$ where $\rho$ is material resistivity - Resistivity is a material property while resistance is a property of an object - Conductivity: a material's ability to conduct electricity, the inverse of resistivity -> measured as $\sigma=\frac{1}{\rho}$ while conductance is measured as $G=\frac{1}{R}$ - Ohm's Law can be written in terms of conductance as $v(t)=\frac{i(t)}{G}$ ## Resistive Power $p=vi=v\left( \frac{v}{R} \right)=\frac{v^2}{R}\text{ or }p=i^2R$ - Equation above is described using the term "resistive losses" - Can also find the power in terms of current & conductance: $p=vi=\left( \frac{i}{G} \right)i=\frac{i^2}{G}$ # Kirchhoff's Laws - Node: junction or point where $\geq2$ circuit elements are connected - Essential node is one in which $3\geq$ elements are jointed together. Non essential nodes only connect $2$ circuit element ## Kirchhoff's Current Law (KCL) - KCL states that the sum of all current leaving a node is equal to zero, $\sum_{n=1}^N i_{n}(t)=0A$ - In the mathematical statement above, every current leaving a node is taken as positive while every current entering the node is considered negative - When three branches meet at a single point, the current entering them is *evenly* split across each - Multiple interpretations / statements can define KCL - The sum of all currents entering a node is equal to zero, in which case every current leaving a node would be taken as negative, while every current entering a node would be considered positive - The sum of all currents entering a node should equal the sum of all currents leaving the node - **Every element directly touching that node contributes a current term.** ### Resistors in parallel - Resistors in parallel share the same nodes and necessarily have the same voltage drop across them - Checking if the voltage drop across two or more resistors is the same is the only safe way to determine whether they are connected in parallel or not. - Current on the other hand will be divided among the resistors - Can rewrite a series of resistors in parallel as one *equivalent* resistance $\frac{1}{R_{eq}}=\sum^N _{n=1} \frac{1}{R_{n}}$ - Can be made more visually appealing by replacing resistance with conductance $G_{eq}=\sum_{n=1}^N G_{n}$ ### Current division - To find current through parallel connected resistors, use $i_{n}(t)=\left( \frac{\frac{1}{R_{n}}}{\frac{1}{R_{1}}+\frac{1}{R_{2}}+\dots +\frac{1}{R_{n}}+\dots +\frac{1}{R_{N}}} \right)i(t)$ -> can be rewritten in terms of $G$ instead of $R$ also - Current division between two resistors in parallel can be written as $i_{1}=\left( \frac{R_{2}}{R_{1}+R_{2}} \right)i(t)$ and $i_{2}=\left( \frac{R_{1}}{R_{1}+R_{2}} \right)i(t)$ - See [[The Bare Essentials of Electrical Engineering.pdf#page=63|Example 2.6.7]] / [[The Bare Essentials of Electrical Engineering.pdf#page=64|Example 2.6.8]] and for application ## Kirchhoff's Voltage Law (KVL) - ![[Pasted image 20260205181819.png|right]]KVL is applied to closed loops, which is any path where you start and end at the same point regardless of whether there are wires or devices connecting all points of the path - In Figure 2.28 (*see right*), the path (a,b,c,d->a) makes the most obvious closed loop; the path (d,c,f,e->d) makes another closed loop; and the path (a,b,c,f,e,d->a) makes a third closed loop. Despite the fact that the second and third loops are not *physically* closed does not stop us from considering them as closed paths. - **Kirchhoff's Voltage Law** states that the sum of all voltages around a closed path is zero, $\sum^N_{n=1}v_{n}(t)=0V$ - To be consistent in every loop while assigning polarity to voltages, consider the following guidelines (see [[The Bare Essentials of Electrical Engineering.pdf#page=66|Figure 2.29]]) 1. Start from lowest left corner of each loop 2. Move clockwise around a loop 3. Use the first voltage sign encountered. - When applying KVL systematically to solve a circuit, we don't need to find all possible loops - only those needed to solve for the ones that contain our unknowns w/ the fewest number of equations (see [[The Bare Essentials of Electrical Engineering.pdf#page=66|Example 2.7.1]] / [[The Bare Essentials of Electrical Engineering.pdf#page=67|Example 2.7.3]]) - In parallel, voltage is the same across every branch ### Resistors in Series - As a reminder, the same current *must* flow through all components connected in series - The equivalent resistance for series in resistors is given by $R_{eq}=\sum^N_{n=1}R_{n}=\frac{v}{i}$ ### Voltage Division - For resistors connected in series, we can find the voltage with the equation below (after the derivation) $v_{2}=iR_{2}=\frac{v}{R_{eq}}R_{2}=\left( \frac{R_{2}}{R_{1}+R_{2}+\dots R_{N-1}+R_{n}} \right)v\to\text{in general, }v_{n}(t)=\left( \frac{R_{n}}{R_{1}+R_{2}+\dots +R_{n}+\dots +R_{N-1}+R_{N}} \right)v(t)$ - Above equation is called voltage division. Total voltage is distributed among the resistors according to their values - amount of voltage drop across each resistor dependso n its value with respect to other resistors - See [[The Bare Essentials of Electrical Engineering.pdf#page=72|Example 2.7.6]] for application of voltage division ## Finding the Equivalent Resistance of a Circuit ### Circuits w/ *no* Sources - A circuit with no sources can easily be reduced to an equivalent resistance if it contains only series and parallel combinations of resistor - A circuit will sometimes contain a delta or a wye connection, which if you know the formula to transform it, the solution can keep moving forward - If not familiar w/ delta or wye connection, $i-v$ test method is used (discussed later) - See [[The Bare Essentials of Electrical Engineering.pdf#page=79|Example 2.9.3]] ### Circuits w/ sources #### Independent Sources Only - Finding $R_{eq}$ in a circuit that contains independent sources requires turning off all independent sources first - Remember that voltage sources are replaced by short circuits & current sources are replaced with open circuits - See [[The Bare Essentials of Electrical Engineering.pdf#page=81|Example 2.9.4]] #### Independent *and* Dependent Sources - To find $R_{eq}$ for circuit w/ independent & dependent sources: dependent sources remain active, so the *only* way to find $R_{eq}$ is to perform the $i-v$ test and apply the definition, meaning: 1. Apply a test voltage (or current) source at the required port 2. Calculate the resulting current (or voltage) 3. Apply the definition $R_{eq}=\frac{v}{i}$ - Refer to [[The Bare Essentials of Electrical Engineering.pdf#page=82|Figure 2.5.2]] for basic process. Need to add an ind. test source at the terminal where we wish to calculate $R_{eq}$ - can be left as voltage/current source. Value of test source can be left as $V_{test},I_{test}$ (choosing value of 1V or 1A may make calculations easier) - You initially create a test source, either $I_{test}$ or $V_{test}$, then the one that you didn't choose appears at the same location, just with an unknown value - See [[The Bare Essentials of Electrical Engineering.pdf#page=83|Example 2.9.5]] for full analysis # Finding $R_{eq}$ of a Circuit 1. Identify terminals where $R_{eq}$ is seen 2. Deactivate independent sources inside the network - Independent voltage source -> short circuit - Independent current source -> open circuit 3. Apply test source at the terminals - Either set a test voltage $V_{t}$ (often 1$V$) across the terminals, *or* - Set a test current $I_{t}$ (often 1$A$) into the terminals 4. Define the terminal current/voltage consistently - If you apply $V_{t}$, define $I_{t}$ as the current entering the network at the positive terminal - If you apply $I_{t}$, define $V_{t}$ as the voltage at the positive terminal relative to the negative terminal 5. Write KCL/KVL to relate $I_{t}$ and $V_{t}$ 1. Use node voltage (KCL) if the circuit is mostly parallel/junctions 2. Use loop equations (KVL) if it is mostly series loops 6. Solve for the ratio: $R_{eq}=\frac{V_{t}}{I_{t}}$