Time Domain and Frequency Domain

1. Time Domain and Frequency Domain • A signal can often be written as a sum of frequencies: • Voltage in a transmission line: • Electric field propagating through a dielectric medium • Why is sinusoidal behavior so common in practice? • Most systems that we care about are time-invariant (… or are “nearly” time-invariant) • All systems that we care about are linear at small amplitudes • The time evolution in any system with these two properties is a sum (perhaps infinite) of complex exponentials!

2. Time Domain and Frequency Domain • Some familiar examples: • A simple RLC circuit: • We may now find algebraically and from that i(t) • A transmission line: L

3. Time Domain and Frequency Domain • Some familiar examples: • A transmission line: Our partial differential equation isnow just an ordinary differential equation But there is an added advantage…

4. Time Domain and Frequency Domain In general: all depend on w This behavior is reasonably simple in the frequency domain, …but in the time domain we get messy convolutions!

5. Maxwell’s Equations: Frequency Domain Name of Law Time Domain Phasor Domain Gauss’s Law Faraday’s Law Gauss’s Law of Magnetics Ampere’s Law • We have used D = eE and B = mH • In the frequency domain, we can easily generalize

6. Time Domain and Frequency Domain • Some familiar examples: • Plane wave propagation • In a source free region: • In a spatially invariant medium: • where • and propagation in any direction is allowed. • In a waveguide or transmission line, this freedom is constrained! • The wave only propagates in one direction

7. Time Domain and Frequency Domain • Physical requirements for the frequency domain to be useful • Linearity • Time invariance • Is that all we use when we write

8. Time Domain and Frequency Domain • Physical requirements for the frequency domain to be useful • Linearity • Time invariance • Is that all we use when we write • Other requirements: • Spatial locality • Causality • Isotropy (Note that this is not the same as homogeneity!)

9. Time Domain and Frequency Domain Without isotropy so that Without causality This change looks small and innocent… but it is very profound!

10. Time Domain and Frequency Domain Without spatial locality Without time invariance Again, a small change with profound effects! Without linearity

11. Time Domain and Frequency Domain • Without linearity • Anisotropy leads to second-order nonlinearities • Nonlinearity is bad in communication transmission lines, including free-space transmission, optical fibers,… • Nonlinearity is needed in switches, oscillators, and storage elements

12. Time Domain and Frequency Domain • A final note on causality and time invariance • for passive electromagnetic systems • PASSIVE = SYSTEM WITH NO ENERGY SUPPLY • NOTE: • Many causal functions do not have causal inverses • Hence, the physical requirement constrains the dielectric function. • So, when is it true? What dielectric functions are “allowed” • In general, the math problem is difficult, but discretization yields useful insights.

13. Time Domain and Frequency Domain Discretization: The discretized equation is an upper triangular matrix equation. We may now take advantage of what we know from linear algebra! The use of discretization will be an oft-repeated theme! It is the essence of most computational methods!! 0

14. Time Domain and Frequency Domain • What linear algebra tells us: • The inverse of an upper triangular matrix is upper triangular • The inverse of a diagonally-invariant matrix is diagonally invariant • The matrix will always have an inverse as long as in the limit as and the interval • Example:

15. Time Domain and Frequency Domain Example (continued): We conclude

16. Time Domain and Frequency Domain Example (continued): In the limit e0 = 0:

17. Time Domain and Frequency Domain Example (continued): The discretization: For singular functions In matrix form: 0 0 0 0

18. Time Domain and Frequency Domain Example (continued): MATLAB Code: Dielectric.m