Info on RF Circuits and Systems

RF Introduction

For a memoryless two port active network the input-output characteristic is generally a nonlinear function, that can be approximated by a polynomial over some signal range as

Polynomial representation of Nonlinear Equation

(1.1)

If the input Vin contains only one frequency component, the output Vo contains desired fundamental frequency component and harmonics of the fundamental frequency which are generally undesirable components. Whereas if Vin contains more that one frequency component, it will result in output components which are mathematical combinations of the frequency of the input signals called inter-modulation products, will be described in the later sections.

The output of the system governed by the Eq.1.1., due to V_in<(t) = A cos(wt) is given by,

(1.2)

From Eq.1.2., we can observe that oputput contains DC term, fundamenal and harmonic componennts even though the input has a single frequency component. It also gives an idea how fundamental and harmonic gain terms are associated with signal amplitudes. The more excursions in the signal amplitude, the more distortion in the output.

Harmonic Distortion

The ratio of the amplitude of the

n

^th harmonic to its fundamental amplitude is called

n

^th order Harmonic Distortion, or it is the ratio of

n

^th harmonic power to the fundamental component power.

From Eq.1.2., $3$ ^rd Order HarmonicDistortion(HD₃) can be written as,

(1.3)

where P1 and P3 are power of the fundamental and 3^rd order harmonics.

Total Harmonic Distortion

The ratio of sum of the harmonics power to the fundamental power is called Total Harmonic distortion(THD).

(1.4)

where P₂, P₃, ..., P_n are power of 2^nd to n^th harmonics.

1dB Compression Point (P1dB)

It is a measure of linearity. The point at which the actual gain is reduced by 1dB from the small signal linear gain is called 1dB compression point. Beyond P1dB, the output power remains almost constant even as input power increases.
From Eq.1.2., the output due to the fundamental component can be written as,

(1.5)

The gain of fundamental component deviates from small signal gain due to the term

a

₃. Gain increases if

a

₃>0 , or compresses if

a

₃<0. For most practical devices

a

₃<0 and gain compresses as the amplitude of the input signal increases. This point at which the output power or gain compresses by 1dB is called 1dB compression point.
From the definition,

(1.6)

3^rd Order Intercept Point

IP3 is a figure-of-merit for the linearity of a two port network. The point at which the third order distortion products equal the desired linear, uncompressed output power is called Third Order Output Intercept Point(OIP3). If it is referred to input it is called Third Order Input Intercept Point(IIP3). IP3 determines the amount of IMD produced in the system when subjected to high level interference.
Equivalent system input intercept point is given by,

(1.7)

Inter-Modulation Distortion (IMD)

If the input containing two or more frequency components are mixed together, it will result in the output frequency components that are mathematical combination of the frequency components in the input. These output frequency components are called inter-modulation products.

Let,

Vin(t)=A1cos(f

₁t) + A2cos(f₂t)
where,

f1 = ω

₁ + Φ₁ and

f2 = ω

₂ + Φ₂ and

The output of a system represented by Eq.1.1., with the above input contains,

Second order terms :

(1.8)

(1.9)

From the above Equations we can observe that second order terms does not contribute to fundamental gain but it will be there from third order terms.

Cross Modulation

If a weak signal and a strong signal pass through the nonlinear system described by Eq.1.1., then there will be transfer of amplitude modulation of the strong signal to the amplitude of the weak signal. This transfer of modulation from strong signal to weak signal is called cross modulation.
Consider a weak signal A₁cos(ω₁t) and a strong signal A₂cos(ω₂t) passing through the system defined by Eq.1.1. Let's say A₂ = A_m[1+m.cos(ω_mt)]
The fundamental gain term from the Eq.1.9., is given as

(1.10)

Substituting A₂ in gain term we get,

(1.11)

From the Eq.1.11., we can observe that fundamental gain is a function of strong signal modulation index and modulation frequency and amplitude.

RF Circuits & Systems