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Introduction
The Concept
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1.0 Symbols Used
1.1 Earth as an Elec. Circuit
1.2 Series RLC Circuit
1.3 Complex Notation
1.4 Graphic Representation
1.5 EM Properties
1.6 Constitutive Relations
1.7 Complex EM Properties
1.8 EM Rock Properties
1.9 Coupling of E & H
Electromagnetic Spectrum
MT, TEM, & GPR Methods

1.2 Series RLC Circuit

Figure 1.2. Series RLC circuit.
A time-varying voltage, V(t) (assumed for simplicity to be sinusoidal), applied to a series RLC circuit, causes a sinusoidal current to flow given by

I(t) = Io sin wt (1.2.1)

where w = 2pf is angular frequency, f = 1/T is the harmonic frequency with period T. The instantaneous current through each circuit element is the same. This is referred to as being in-phase and since the current is in-phase in each circuit element, it is a good phase reference.

Series RLC Current-Voltage Relationships

Across the resistor Ohm's law says that there is a linear relationship between current and voltage, VR, i.e.,

VR = IR, (1.2.2)

where the resistance, R is the constant of linear proportionality.

Across the inductor, Faraday's law says that a voltage, VL, that is induced is proportional to the time rate of change of magnetic flux. This is expressed using the change in current flow, dI/dt as

VL = L dI/dt, (1.2.3)

where the inductance, L is the constant of linear proportionality.

Across the capacitor the voltage depends on the capacity to store electric charge. The ratio of the charge, q to the voltage (potential difference) across such a device defines the capacitance, C such that

VC = q/C. (1.2.4)

Applying Kirchhoff's voltage (second) law (the sum of the voltage drops in a closed loop equals the source voltage) yields

V = VR + VL + VC (1.2.5)

or
V = IR + L dI/dt + q/C. (1.2.6)

Using

I = Io sin wt (1.2.7)

and recalling that the definition of current is

I = dq/dt, (1.2.8)

we find after integration that

q = -(I0/w) cos wt. (1.2.9)

Whereupon, after substitution into equation 1.2.6 above yields

V = IoR sin wt + IowL cos wt - (Io/wC) cos wt. (1.2.10)


The quanities wL and 1/(wC) are the impedances of L and C which are called the inductive reactance, XL and capacitive reactance, XC , respectively. They clearly must have the units of ohms like R.

Ociloscope
Figure 1.3. A oscilloscope can measure current and voltage time variations.
Imagine placing a device that can record the time variations of the voltages with respect to the current variation through the circuit element. Such a device is an oscilloscope (Figure 1.3). Using the current for the phase reference one can now compare the temporal phase relations between I, VR , VL , and VC (Figure 1.4).


Figure 1.4. Temporal current and voltage relations across series RLC circuit elements.

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