Distortion in RF Circuits

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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

(1)   \begin{equation*} v_o(t) = \alpha_o + \alpha_{\tiny 1} v_{i}(t) + \alpha_{\tiny 2} v_{i}^{\tiny 2}(t) + \ldots + \alpha_n v_{i}^n(t) \end{equation*}

If the input v_{i} contains only one frequency component, the output v_{o} contains desired fundamental frequency component and harmonics of the fundamental frequency which are generally undesirable components. Whereas if v_{i} 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.

For most of the circuits, the first three terms in Eq.(1) are sufficient to characterize the system with good amount of accuracy. Hence the output of the system governed by the Eq.(1), due to v_{i}(t) = A cos(\omega t) is given by,

(2)   \begin{eqnarray*} v_o(t) &=& \underbrace{ \alpha_o +\frac{1}{2}\alpha_{\tiny 2} A^{\tiny 2}}_{DC} + \underbrace{\left(\alpha_{\tiny 1} + \frac{3}{4}\alpha_{\tiny 3} A^{\tiny 2}\right)A cos(\omega t)}_{fundamental} \\ & & + \underbrace{\left(\frac{1}{2}\alpha_{\tiny 2}A\right)A\cos(2\omega t) + \left(\frac{1}{4} \alpha_{\tiny 3}A^{\tiny 2}\right) A cos(3\omega t) + \ldots}_{harmonics} \end{eqnarray*}

From Eq.(2), we can observe that output contains DC term, fundamenal and harmonic components even though the input has a single frequency component due to system’s nonlinearity. It also gives an idea how fundamental and harmonic gain terms are associated with signal amplitudes. This association of gain term with singal amplitude results in distortion. 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.

In general 2^{nd} and 3^{rd} order are significant contributors for total harmonic distortion. Hence they are of much interest compared to others.

From Eq.(2), 2^{nd} Order Harmonic Distortion(HD_{\tiny 2}) can be written as,

(3)   \begin{equation*} HD_{\tiny 2} = \frac{V_{o2}}{V_{o1}} = \frac{\frac{1}{2}\alpha_{\tiny 2} A}{\alpha_{\tiny 1} + \frac{3}{4}\alpha_{\tiny 3} A^{\tiny 2}} \end{equation*}

where V_{o1} and V_{o2} are the amplitudes of fundamental and 2^{nd} harmonic components in the output.

If {\alpha_{\tiny 1} A \gg \frac{3}{4}\alpha_{\tiny 3} A^{\tiny 3}},

(4)   \begin{equation*} HD_{\tiny 2} \approx \frac{\alpha_{\tiny 2}}{2\alpha_{\tiny 1}}A \end{equation*}

The condition under which the above approximation is valid is called low distortion condition and given by
A_{i,low} \ll \sqrt{\frac{4}{3}\frac{\alpha_{\tiny 1}}{\alpha_{\tiny 3}}}

Similarly 3^{rd} Order HarmonicDistortion(HD_{3}) can be written as,

(5)   \begin{equation*} HD_{\tiny 3} = \frac{V_{o3}}{V_{o1}} = \frac{\frac{1}{4}\alpha_{\tiny 3} A^{\tiny 3}}{\alpha_{\tiny 1} A + \frac{3}{4}\alpha_{\tiny 3} A^{\tiny 3}} \end{equation*}

It can also be approximated as

(6)   \begin{equation*} HD_{\tiny 3} \approx \frac{\alpha_{\tiny 3}}{4\alpha_{\tiny 1}}A^{\tiny 2} \end{equation*}

From Eq.(4) and Eq.(6), we can observe that HD_{2} \propto A and HD_{3} \propto A^{2}. In dB sense for every 1dB increase in input, HD_{2} increases 1dB and HD_{3} increases by 2dB.

Total Harmonics Distortion

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

(7)   \begin{equation*} THD = \frac{P_{\tiny 2}+P_{\tiny 3}+\ldots+P_{\tiny n}}{P_{\tiny 1}} = \frac{\sum_{i=\tiny 2}^n P_i}{P_{\tiny 1}} = \frac{\sqrt{V_{o\tiny 2}^{\tiny 2}+\ldots+V_{on}^{\tiny 2}}}{V_{o\tiny 1}} \end{equation*}

where P_{\tiny 2}, \ldots , P_{n} are power of 2nd to nth harmonics.

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, v_{i} = A_{\tiny 1} \cos(\varphi_{\tiny 1}) + A_{\tiny 2} \cos(\varphi_{\tiny 2})

where,
\varphi_{\tiny 1}=\omega_{\tiny 1}t+\phi_{\tiny 1} and
\varphi_{\tiny 2}=\omega_{\tiny 2}t+\phi_{\tiny 2}.

The output of a system represented by Eq.(1)., with the above input, is

(8)   \begin{eqnarray*} v_o &=& \underbrace{\left(\alpha_{\tiny 1}+ \alpha_{\tiny 3}\left(\frac{3}{4}A_{\tiny 1}^{\tiny 2} + \frac{3}{2}A_{\tiny 2}^{\tiny 2} \right) \right)A_{\tiny 1}\cos{\varphi_{\tiny 1}} + \left(\alpha_{\tiny 1}+ \alpha_{\tiny 3}\left(\frac{3}{4}A_{\tiny 2}^{\tiny 2} + \frac{3}{2}A_{\tiny 1}^{\tiny 2}\right) \right) A_{\tiny 2} \cos{\varphi_{\tiny 2}}}_{fundamental} \\ & &+\underbrace{\frac{1}{2}\alpha_{\tiny 2} \left(A_{\tiny 1}^{\tiny 2}\cos{2\varphi_{\tiny 1}}+ A_{\tiny 2}^{\tiny 2} \cos{2\varphi_{\tiny 2}} \right)}_{2^{nd}~~harmonic }\\ & &+\underbrace{\frac{1}{4}\alpha_{\tiny 3} \left(A_{\tiny 1}^{\tiny 3} \cos(3\varphi_{\tiny 1}) + A_{\tiny 2}^{\tiny 3} \cos(3\varphi_{\tiny 2})\right)}_{3^{rd}harmonic}\\ & &+\underbrace{\frac{1}{2}\alpha_{\tiny 2}A_{\tiny 1}A_{\tiny 2}\left(\cos(\varphi_{\tiny 1}-\varphi_{\tiny 2}) + \cos(\varphi_{\tiny 1}+\varphi_{\tiny 2}) \right)}_{2^{nd}~~order~~intermods}\\ & &+\underbrace{\frac{3}{4}\alpha_{\tiny 3}\left(A_{\tiny 1}A_{\tiny 2}^{\tiny 2}\cos(\varphi_{\tiny 1}-2\varphi_{\tiny 2}) + A_{\tiny 1}^{\tiny 2}A_{\tiny 2}\cos(2\varphi_{\tiny 1}-\varphi_{\tiny 2})+ A_{\tiny 1}A_{\tiny 2}^{\tiny 2}\cos(\varphi_{\tiny 1}+2\varphi_{\tiny 2}) + A_{\tiny 1}^{\tiny 2}A_{\tiny 2}\cos(2\varphi_{\tiny 1}+\varphi_{\tiny 2})\right)}_{3^{rd}~~order~~intermods}\end{eqnarray*}

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.

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, the output due to the fundamental component can be written as,

(9)   \begin{equation*} \alpha_{\tiny 1}A\left( 1 + \frac{3}{4}\frac{\alpha_{\tiny 3}}{\alpha_{\tiny 1}}A^{\tiny 2}\right) \end{equation*}

The gain of fundamental component deviates from small signal gain due to the term \alpha_{\tiny 3}. Gain expands if \alpha_{\tiny 3} > 0 , or compresses if \alpha_{\tiny 3} < 0. For most practical devices \alpha_{\tiny 3} < 0 and gain compresses as the amplitude of the input signal increases. The point at which the output power or gain compresses by 1dB is called 1dB compression point(P_{{\tiny 1}dB}).

From the definition,

(10)   \begin{eqnarray*} P_{{\tiny 1}dB} &=& P_{o(ideal)} - 1dB \\ 20\log\left(1 + \frac{3}{4}\frac{\alpha_{\tiny 3}}{\alpha_{\tiny 1}} A_{\tiny -1dB}^{\tiny 2}\right) &=& -1 \\ A_{\tiny -1dB} &=& \sqrt{0.145 \|\frac{\alpha_{\tiny 3}}{\alpha_{\tiny 1}}\|} \end{eqnarray*}

The gain compression is due to odd order nonlinearities, current and/or voltage limiting. When the gain compression is caused by the odd-order nonlinearities in the transfer functions of the devices in the circuit, the gain decreases more linearly with increase in input signal power. Whereas with current or voltage limiting the gain drops abruptly.

3rd Order Intercept Point

IP3 is a figure-of-merit for the linearity of a two port network. The point at which the extrapolated curves of third order distortion products (\frac{3}{4}\alpha_{\tiny 3}A^{\tiny 3}) equal the desired linear, uncompressed output power (\alpha_{\tiny 1}A) is called Third Order Intercept Point(IP3). If it is referred to input it is called Third Order Input Intercept Point(IIP3).

(11)   \begin{equation*} A_{\tiny IIP3} = \sqrt{\frac{4}{3}\frac{\alpha_{\tiny 1}}{\alpha_{\tiny 3}}} \end{equation*}

From Eq(10), 1dB compression point and IP3 are related as

(12)   \begin{equation*} A_{\tiny -1dB} = \sqrt{0.109 \frac{4}{3}\|\frac{\alpha_{\tiny 3}}{\alpha_{\tiny 1}}\|} = \sqrt{0.109} A_{\tiny IIP3} \end{equation*}

in dB sense,

(13)   \begin{equation*} A_{\tiny -1dB} = A_{\tiny IIP3} - 9.64dB \end{equation*}

IP3 determines the amount of IMD produced in the system when subjected to high level interference.

Cross Modulation

Consider a weak signal A_{\tiny 1}\cos{\varphi_{\tiny 1}} and a strong signal A_{\tiny 2}\cos{\varphi_{\tiny 2}}, whose amplitude is modulated as A_{\tiny 2} = A_{\tiny m}(1+m\cos\varphi_{\tiny m}), passing through the system defined by Eq.(1).

Gain of the fundamental component at frequency \varphi_{\tiny 1} from the Eq.1.4., is given as

(14)   \begin{equation*} \alpha_{\tiny 1} + \alpha_{\tiny 3}\left(\frac{3}{4}A_{\tiny 1}^{\tiny 2} + \frac{3}{2}A_{\tiny 2}^{\tiny 2}\right) \end{equation*}

Therefore the fundamental gain due to A_{\tiny 2}=A_{\tiny m}(1+m\cos\varphi_{\tiny 2}) is given by,

(15)   \begin{equation*} \alpha_{1} + \alpha_{3}\left({3 \over 4}A_{1}^{2}+{3 \over 2}A_{m}^{2}\left[1 + \frac{m^{\tiny 2}}{2} + 2m\cos\varphi_{\tiny m} + \frac{m^{\tiny 2}}{2}\cos(2\varphi_{\tiny m}) \right]\right) \end{equation*}

If A_{1} \ll A_{m}, then the second term in the gain expression becomes dominant and is modulated by the strong signal(A_{\tiny 2}). It result in 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. The fundamental gain is a function of strong signal modulation index, modulation amplitude and modulation frequency.

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