Algebraic Substitution for Electric Circuits

A bipolar junction transistor parameter similar to β is älpha,” symbolized by the Greek letter α. It is defined as the ratio between collector current and emitter current:

$$a = \frac{I_C}{I_E}$$

Apply algebraic substitution to this formula so that alpha is defined as a function of beta: α = f(β). In other words, substitute and manipulate this equation until you have alpha by itself on one side and no variable except beta on the other.

You may find the following equations helpful in your work:

$$\beta = \frac{I_C}{I_B} \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ I_E=I_C+I_B$$

The Q, or quality factor, of an inductor circuit is defined by the following equation, where X_s is the series inductive reactance and R_s is the series resistance:

$$Q = \frac{X_s}{R_s}$$

We also know that we may convert between series and parallel equivalent AC networks with the following conversion equations:

$$R_s R_p = Z^2 \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ X_sX_p = Z^2$$

Series and parallel LR networks, if truly equivalent, should share the same Q factor as well as sharing the same impedance. Develop an equation that solves for the Q factor of a parallel LR circuit.

The equation relating probability of continued performance for a component or a system versus time may be expressed as follows:

Where,

x = Probability (a number between 0 and 1, inclusive)

e = Euler’s constant ( ≈ 2.7182818)

t = Time of continuous operation

m = Mean Time Between Failure of the component or system

The unit of time for both t and m must be the same. That is, if t is measured in years, then m must also be expressed in years or else the equation will give very misleading answers.

Suppose, though, we were given m in years, and the operating time t in days. Substitute the relationship t_d = 365 t_y into the reliability equation so that we will have a new equation that can take t in days (t_d) and m in years, and still provide the correct answer.