# Causal Cosine and Sine Response

Chapter

## Abstract

We are all familiar with the sine/cosine system response, especially looking at it from steady state points of view. But the causal sine/cosine response is another story! What the steady state sine/cosine response is missing is the transient part of the solution. But this is not a problem for the causal sine/cosine response; not only do they reproduce the transient solution, they also produce the steady state solution for a net result of capturing the whole solution! Again we start with the transfer function of the system, multiply by either $$\omega _0/(s^2 + \omega _0^2)$$ (for sine) or $$s/(s^2+\omega _0^2)$$ (for cosine, and where ω0 is the angular frequency of sine/cosine input), then do inverse transform to get the solution in the time domain. Typically multiplying the transfer function by the Laplace transform of the sine/cosine will reproduce the transfer function but with a spike around ω0, and this is discussed in the chapter. Once the response is known in the frequency domain we can find the inverse transform analytically or by numerical integration. We illustrate the causal sine/cosine response on various RLC circuits and study along the way some of the interesting observations about these circuits. Another advantage of knowing the causal sine/cosine response is that we can use them to figure the response to any other causal periodic function simply by using the Fourier series! Keep in mind here we would be using the causal Fourier series, since our basis functions are causal to start with.