- #1

mrjeffy321

Science Advisor

- 875

- 1

A circuit was constructed with an inductor, capacitor, and resistor all wired in series. A sinusoidal input signal was put through the circuit and the output voltage was measured across the resistor as a function of the input angular frequency (ω).

My task is to find the transfer function of this circuit….to find the ratio of the output voltage to the input voltage.

The inductor and capacitor has frequency dependent impedances, but the resistor’s impedance is constant for all frequencies.

Since all the circuit components are wires in series, the total impedance of the circuit is simply the sum of each component’s impedance.

Impedance…

of resistor = R,

of capacitor = -j / ωC,

of inductor = jωL,

The transfer function cab be found as,

R / (R + jωL + -j / ωC)

By multiply the denominator by its complex conjugate in order to remove the imaginary parts from the bottom of the fraction, we get,

R [R - j(ωL – 1/(ωC))] / (R^2 + (ωL – 1/(ωC))^2

If we say that X = R and Y = (ωL – 1/(ωC)), then we can re-write this as,

(X^2 + jXY) / (X^2 + Y^2)

And we can divide everything by X to get,

(X + jY) / (X + Y^2 / X)

Then to get the magnitude of the output voltage, we need to find the magnitude of the numerator,

sqrt (X^2 + Y^2) / (X + Y^2 / X)

But when I graph this transfer function on the same graph with my actual data points (on a logarithmic scale), I do not get anything which comes even close.

My theoretical values from the above function give me a downward sloping line, but my data points (and knowledge of what

Where did I go wrong?

I was told I am aiming to get a transfer function which looks like,

(X – jY) / (1 + Y^2)

For some values of X and Y.

EDIT:

If I keep playing with the equation, I can simplify it down to:

|H(ω)| = R / sqrt (R^2 + (ωL – 1/ωC)^2)

But this just gives me the same graph I had before, which looks nothing like my data points or how I know the graph should behave.

My task is to find the transfer function of this circuit….to find the ratio of the output voltage to the input voltage.

The inductor and capacitor has frequency dependent impedances, but the resistor’s impedance is constant for all frequencies.

Since all the circuit components are wires in series, the total impedance of the circuit is simply the sum of each component’s impedance.

Impedance…

of resistor = R,

of capacitor = -j / ωC,

of inductor = jωL,

The transfer function cab be found as,

R / (R + jωL + -j / ωC)

By multiply the denominator by its complex conjugate in order to remove the imaginary parts from the bottom of the fraction, we get,

R [R - j(ωL – 1/(ωC))] / (R^2 + (ωL – 1/(ωC))^2

If we say that X = R and Y = (ωL – 1/(ωC)), then we can re-write this as,

(X^2 + jXY) / (X^2 + Y^2)

And we can divide everything by X to get,

(X + jY) / (X + Y^2 / X)

Then to get the magnitude of the output voltage, we need to find the magnitude of the numerator,

sqrt (X^2 + Y^2) / (X + Y^2 / X)

But when I graph this transfer function on the same graph with my actual data points (on a logarithmic scale), I do not get anything which comes even close.

My theoretical values from the above function give me a downward sloping line, but my data points (and knowledge of what

*should*happen) show an increasing function with some peak value when then decreases back down towards zero again. Since the impedances of the capacitor and inductor are frequency dependent, there will be some frequency (the resonance frequency) where the two should cancel each other out and the only impedance at this frequency should be that of the resistor (yields maximum power). But this is not happening in my theoretical function.Where did I go wrong?

I was told I am aiming to get a transfer function which looks like,

(X – jY) / (1 + Y^2)

For some values of X and Y.

EDIT:

If I keep playing with the equation, I can simplify it down to:

|H(ω)| = R / sqrt (R^2 + (ωL – 1/ωC)^2)

But this just gives me the same graph I had before, which looks nothing like my data points or how I know the graph should behave.

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