Algebraic Equation Manipulation for Electric Circuits
Mathematics for Electronics
The electrical resistance of a conductor at any temperature may be calculated by the following equation:

Where,
R_{T} = Resistance of conductor at temperature T
R_{r} = Resistance of conductor at reference temperature T_{r}
α = Temperature coefficient of resistance at reference temperature T_{r}
Simplify this equation by means of factoring.
The equation for voltage gain (A_{V}) in a typical noninverting, singleended opamp circuit is as follows:

Where,
R_{1} is the feedback resistor (connecting the output to the inverting input)
R_{2} is the other resistor (connecting the inverting input to ground)
Suppose we wished to change the voltage gain in the following circuit from 5 to 6.8, but only had the freedom to alter the resistance of R_{2}:

Algebraically manipulate the gain equation to solve for R_{2}, then determine the necessary value of R_{2} in this circuit to give it a voltage gain of 6.8.
The equation for voltage gain (A_{V}) in a typical inverting, singleended opamp circuit is as follows:

Where,
R_{1} is the feedback resistor (connecting the output to the inverting input)
R_{2} is the other resistor (connecting the inverting input to voltage signal input terminal)
Suppose we wished to change the voltage gain in the following circuit from 3.5 to 4.9, but only had the freedom to alter the resistance of R_{2}:

Algebraically manipulate the gain equation to solve for R_{2}, then determine the necessary value of R_{2} in this circuit to give it a voltage gain of 4.9.
The following equations solve for the output voltage of various switching converter circuits (unloaded), given the switch duty cycle D and the input voltage:



Manipulate each of these equations to solve for duty cycle (D) in terms of the input voltage (V_{in}) and desired output voltage (V_{out}). Remember that duty cycle is always a quantity between 0 and 1, inclusive.
The formula for calculating total resistance of three seriesconnected resistors is as follows:

Algebraically manipulate this equation to solve for one of the series resistances (R_{1}) in terms of the other two series resistances (R_{2} and R_{3}) and the total resistance (R). In other words, write a formula that solves for R_{1} in terms of all the other variables.
The formula for calculating total resistance of three parallelconnected resistors is as follows:

Algebraically manipulate this equation to solve for one of the parallel resistances (R_{1}) in terms of the other two parallel resistances (R_{2} and R_{3}) and the total resistance (R). In other words, write a formula that solves for R_{1} in terms of all the other variables.
The decay of a variable over time in an RC or LR circuit follows this mathematical expression:

Where,
e = Euler’s constant ( ≈ 2.718281828)
t = Time, in seconds
τ = Time constant of circuit, in seconds
For example, if we were to evaluate this expression and arrive at a value of 0.398, we would know the variable in question has decayed from 100% to 39.8% over the period of time specified.
However, calculating the amount of time it takes for a decaying variable to reach a specified percentage is more difficult. We would have to manipulate the equation to solve for t, which is part of an exponent.
Show how the following equation could be algebraically manipulated to solve for t, where x is the number between 0 and 1 (inclusive) representing the percentage of original value for the variable in question:

Note: the “trick” here is how to isolate the exponent [(−t)/(τ)]. You will have to use the natural logarithm function!
Voltage and current gains, expressed in units of decibels, may be calculated as such:


Another way of writing this equation is like this:


What law of algebra allows us to simplify a logarithmic equation in this manner?
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