Circuits

Electrical Circuits

• An electrical component is a
• An electrical element is a
• An electrical network is an interconnection of electrical components or a model of such an interconnection, consisting of electrical elements
• An electrical circuit is a network consisting of a closed loop, giving a return path for the current.
• A branch represents a single two-terminal element (such as a voltage source or a resistor)
• A node (or junction) represents a connection between two or more branches
  • Nodes that are connected by perfectly conducting wires are considered to be the same node
• A reference node
• A mesh
• A loop
• A short circuit is a circuit element with resistance approaching zero, so $I=\underset{R\to 0}{\lim }\frac{V}{R}=\infty$
• An open circuit is a circuit element with resistance approaching infinity, so $I=\underset{R\to \infty }{\lim }\frac{V}{R}=0$

Terminology

In some texts, a node is defined to be the junction between 3 or more elements. Another term for that is an essential node In this text, a node is defined to be the junction between 2 or more elements, by thisdefinition, some nodes may be redundant (i.e. not independent)

Electrical Elements

classification of elements:

• terminals number:
  • One-port elements (two terminals) - dioes, resistors, capacitors, inductors
  • Two-port elements (four terminals)
  • Multiport elements
• energy source:
  • passive elements do not have a source of energy
    • dioes, resistors, capacitors, inductors
  • active elements (or sources) have a source of energy
    • voltage sources, current sources
    • dependent sources
• linearity:
  • linear elements have a linear relationship between voltage and current
    • resistors, capacitors, inductors
  • nonlinear elements are elements in which the relation between voltage and current is a nonlinear function
    • dioes

Kirchhoff’s Circuit Laws

• Kirchhoff’s (circuit) laws (or Kirchhoff’s rules) are two equalities that deal with the current and potential difference.

Kirchhoff’s Current Law

• aka: first law, junction rule

• For any node in an electrical circuit, the algebraic sum of the currents flowing into and out of the node is zero. Mathematically $\sum _{i=1}^n{I}_i=0$

  • $I_i$ is the current flowing through the $i$-th branch
  • $n$ is the total number of branches with currents flowing towards or away from the node
  • Currents flowing into the node are considered positive, and currents flowing out of the node are considered negative (or vice versa, depending on the convention chosen)

• This law is based on the conservation of electric charge

Kirchhoff’s Voltage Law

• aka: second law, loop rule

• The voltage drop is the decrease in electric potential along the path of a current flowing in a circuit

• In one traversal of any closed loop, the sum of the voltage rises equals the sum of the voltage drops.

• given a circuit with a voltage source $V$ and resistors $R_1,R_2,\dots ,R_n$:

  • $I=\frac{V}{\sum R_i}$ is the current through the circuit
  • $V_0,V_1,V_2,\dots ,V_n$
  • $I\cdot R_i$ is the voltage drop across the $i$-th resistor
  • $V_i$ is equal to $V_{i-1}$ minus the voltage drop across $R_i$
    • $V_0=V$
    • $V_n=0$

• This law is based on the conservation of energy

Resistance & Conductance

• the electric current through a conductor between two points is directly proportional to the voltage across the two points
  • Ohm’s Law holds for ohmic materials (like most metals) but not for non-ohmic materials (like diodes, transistors, and other semiconductors)
  • The unit of resistance is the $\mathrm{ohm}$ ($\mathrm{\Omega }$) defined as $1\mathrm{\Omega }=1V/A$
  • The reciprocal of resistance is called the electrical conductance (in $\mathrm{S}$, siemens, which is $1\mathrm{S}=1{\mathrm{\Omega }}^{-1}$)
  • $V=IR$
    • $V$ is the voltage (in $\mathrm{V}$)
    • $I$ is the current (in $\mathrm{A}$)
    • $R$ is the resistance (in $\mathrm{\Omega }$)
• The (electrical) resistivity (or specific resistance) $\rho$ (in $\mathrm{\Omega }\cdot \mathrm{m}$) of a material is a measure of how strongly that material opposes the flow of electric current
  • $R=\rho \frac{L}{A}$ is the resistance of a conductor of length $L$ and cross-sectional area $A$ with resistivity $\rho$
  • $\rho =\frac{E}{J}$ where $E$ is the electric field (in $V/m$) and $J$ is the current density (in $A/{\mathrm{m}}^2$)
  • $\rho (T)={\rho }_0[1+\alpha (T-T_0)]$
  • The reciprocal of the resistivity, called the electrical conductivity (or specific conductance) is $\sigma =\frac{1}{\rho }$ (in $S/m$, siemens per meter, or $(\mathrm{\Omega }\cdot \mathrm{m}{)}^{-1}$)

Resistor

• The resistors could be simple resistors, or they could be lightbulbs, heating elements, or other resistive devices

Series Resistors

• When $n$ resistors are connected end to end along a single path they are said to be connected in series
  • $R_{\text{total}}=\sum _{i=1}^n{R}_i$ is the total resistance (or equivalent resistance) of the series
  • $V_i=V\frac{R_i}{R_{\text{total}}}=I{R}_i$ is the voltage across the $i$-th resistor
  • $I=\frac{V}{R_{\text{total}}}$ is the current through the circuit
  • $V=\sum _{i=1}^n{V}_i$ is the voltage across the voltage source, and it is equal to the sum of the voltage drops across each resistor
  • Any charge that passes through $R_1$ will pass through $R_2$ and so on, hence the same current flows through each resistor in series
  • When we add resistors in series:
    • ($\underset{n\to \infty }{\lim }I=0$) The current $I$ in the circuit decreases (more resistors to pass through)
    • ($\underset{n\to \infty }{\lim }{R}_{\text{total}}=\infty$) The total resistance $R_{\text{total}}$ increases
  • The power dissipated by the $i$-th resistor is $P_i=V_{i}I=I^2{R}_i$
    • When $R_i<R_j$ then $P_i<P_j$

Parallel Resistors

• We say that $n$ resistors are connected in parallel when the current from the source splits into $n$ paths
  • $I_i=\frac{V}{R_i}$ is the current through the $i$-th resistor
  • $I=\sum _{i=1}^n{I}_i=\frac{V}{R_{\text{total}}}$ is the total current through the circuit
  • $\frac{1}{R_{\text{total}}}=\frac{1}{R_1}+\frac{1}{R_2}+\dots +\frac{1}{R_n}$
  • All resistors in parallel have the same voltage drop $V$ across them
  • When we add resistors in parallel:
    • ($\underset{n\to \infty }{\lim }I=\infty$) The total current $I$ in the circuit increases (more paths for the current to flow)
    • ($\underset{n\to \infty }{\lim }{R}_{\text{total}}=0$) The total resistance $R_{\text{total}}$ decreases
  • The power dissipated by each resistor is $P_i=V{I}_i=\frac{V^2}{R_i}$
    • When $R_i<R_j$ then $P_i>P_j$
  • The total power dissipated from the source is $P=VI=\sum _{i=1}^n{P}_i=\sum _{i=1}^n\frac{V^2}{R_i}$

EXAMPLE

A parallel circuit with two resistors of $4\mathsf{\Omega }$ has a total resistance of $\frac{1}{R_{\text{eq}}}=\frac{1}{4}+\frac{1}{4}=\frac{1}{2}$, so $R_{\text{eq}}=2\mathsf{\Omega }$

NOTE

The total power dissipated from a source is greater in a parallel circuit than in a series circuit with the same resistors and voltage source

RC Circuits

• $\tau =RC$ (in $\mathrm{s}$) is the time constant (or relaxation time) of the RC circuit

Capacitor charging

• $q=Q_{\text{f}}(1-e^{-t/RC})$ is the charge on the capacitor in time $t$
  • $Q_{\text{f}}=C\mathcal{E}$
• $i=I_0{e}^{-t/RC}$ is the current in time $t$
  • $I_0=\frac{\mathcal{E}}{R}$
• $v_C=\mathcal{E}(1-e^{-t/RC})$ is the voltage across the capacitor in time $t$
• $v_R=\mathcal{E}-v_C$ is the voltage across the resistor in time $t$

Capacitor discharging

• $q=Q_0{e}^{-t/RC}$ is the charge on the capacitor in time $t$
  • $Q_0$ is the initial charge on the capacitor
• $i=I_0{e}^{-t/RC}$ is the current in time $t$
  • $I_0=\frac{Q_0}{RC}$
• $v_C=V_0{e}^{-t/RC}$ is the voltage across the capacitor in time $t$
  • $V_0$ is the initial voltage across the capacitor
• $v_R=v_C$ is the voltage across the resistor in time $t$

RL circuit

• $\tau =\frac{L}{R}$ (in $\mathrm{s}$) is the time constant (or relaxation time) of the RL circuit
• $i=\frac{\mathcal{E}}{R}(1-e^{-(R/L)t})$ is the current growth in time $t$
• $i=I_0{e}^{-(R/L)t}$ is the current decay in time $t$

RLC circuit

• $L\frac{d^{2}i(t)}{d{t}^2}+R\frac{di(t)}{dt}+\frac{1}{C}i(t)=\frac{d{v}_s(t)}{dt}$