Circuit Symbols and Conventions

Circuits model electrical systems

• Voltage is work done per unit charge
• Potential difference- difference in electrical potential between two points in an electric field
• A force used to move charge between two points in space

$$V=\frac{W}{q}=\frac{dW}{dq}$$

• Moving charges produce an electric current
• Moving charges can do electrical work the same way moving objects do mechanical work

$$I=\frac{dq}{dt}$$

• Electrical energy is the capacity to do electrical work
• Electrical power is the rate at which work is done

$$P=IV=\frac{dq}{dt}\cdot \frac{dW}{dq}=\frac{dW}{dt}$$

Resistance

• Resistance is the opposition to the flow of current
• Ohm's Law:

$$R=\frac{V}{I}$$

• Resistance is also proportional to the Resistivity of the material
  • $l$ and $A$ are the length and area of the conductor, respectively.

$$R=\frac{\rho \cdot l}{A}$$

Sources and Nodes

Everything in a circuit can be modelled as either a source, or a node.

Voltage Sources

• DC and AC voltage sources
• DC source has positive and negative terminals
• Ideal voltage source has 0 internal resistance (infinite conductance)
• Supplies constant voltage regardless of load
  • This is an assumption, is not the case in reality

Current Sources

• Ideal current source has infinite resistance (0 conductance)
• Supplies constant current regardless of load
  • Also an assumption
  • In reality, will have some internal resistance and therefore a maximum power limit

Dependant sources

• Diamond-shaped
• Sources depend on values in other parts of the circuit
• Model real sources more accurately

Nodes

All passive elements: generate no electrical power.

• Resistors provide resistance/impedance in Ohms ($\Omega$)
• Inductors provide inductance in Henries ($L$)
• Capacitors provide capacitance in Farads ($F$)

The voltage rise across an impedance conducting current is in opposition to the flow of current in the impedance.

Basic Conventions

Electrical current always flows from high to low potential.

• If the direction of the current in a circuit is such that it leaves the positive terminal of a voltage source and enters the negative terminal, then the voltage is designated as negative
• If the direction of the current is such that it leaves the negative and enters the positive, then the voltage is positive
  • The sign of the loop current is the terminal that it flows into

The power absorbed/produced by a source is $P=IV$.

• A voltage source is absorbing power if it is supplying a negative current
• A voltage source is producing power if it is supplying a positive current

The power dissapated in a resistor is $P=I^{2}R$.

Resistors in series and parallel

Resistors in series: $R_t=R_1+R_2$

Resistors in parallel: $\frac{1}{R_t}=\frac{1}{R_1}+\frac{1}{R_2}$

Resistors dissipate electrical power, so there is a drop in voltage accross them, in the direction of current flow. Therefore, the voltage rise is in opposition to the direction of current

Voltage dividers

Using two resistors to divide a voltage

In the general case:

$$V_{out}=V_{in}\times \frac{Z_2}{Z_1+Z_2}$$

Current Dividers

Similar deal to voltage divider

$$I_{R1}=I_T\times \frac{R_2}{R_1+R_2}$$ $$I_{R2}=I_T\times \frac{R_1}{R_1+R_2}$$