Electrical Systems

Similar to mechanical systems, models of electrical systems can be constructed. Similar deal to ES191.

Variables

• Current $I(t)$ in amps (A)
• Voltage $e(t)$ in volts (V) -- not v for voltage, e is used in systems
• Power in watts $P=I(t)\cdot e(t)$

Elements

Capacitors

• Store electrical energy in a reversible form
• Capacitance $C$ measured in Farads (L)

Elemental equation:

$$I(t)=C\frac{d}{dt}{e}_{12}(t)$$

Energy stored: $$W=\frac{1}{2}C{e}^2$$

Inductors

• Store magnetic energy in a reversible form
• Inductance $L$ measured in Henries (H)

Elemental equation: $$e_{12}(t)=L\frac{d}{dt}I(t)$$

Energy Stored: $$W=\frac{1}{2}L{I}^2$$

Resistors

• Dissapates energy
  • Non-reversible
• Resistance $R$ measured in Ohms ($\Omega$)

Elemental Equation (Ohm's law): $$e_{12}(t)=I(t)\cdot R$$

Voltage Source

• Provides an input of energy to the system.
• Input voltage $e_i(t)$

Kirchhoff's Laws

• Describe how elements interconnect and transfer energy between them
• KVL - voltages around a closed loop sum to zero
• KCL - currents about a node sum to zero

Example

Form a differential equation to model the following electrical system/circuit:

Elements:

• Resistor: $e_r=IR$
• Capacitor: $I=C\frac{d}{dt}{e}_c$
• Inductor: $e_L=L\frac{d}{dt}I$

KVL - the voltages round the loop sum to zero:

$$e_i-e_r-e_l-e_o=0$$ $$e_i-IR-L\frac{dI}{dt}-e_o=0$$

Using the capacitor equation, and the fact that $e_o=e_c$:

$$e_i-RC\frac{d}{dt}{e}_o-LC\frac{d^2}{d{t}^2}{e}_o-e_o=0$$ $$LC\frac{d^2}{d{t}^2}{e}_o(t)+RC\frac{d}{dt}{e}_o(t)+e_o(t)=e_i(t)$$