Harmonic Generation…

2.2.1 Single-frequency Excitation

We will first describe the frequency spectrum at the output of the test circuit when it is excited with one sinusoidal source at a frequency 1. When the amplitude A1 of the input signal is small enough, then the output spectrum of the circuit only contain one frequency component above the noise floor, this is to say, the response corresponding to the circuit's linear behavior. This is a signal at the same frequency of the input signal, called the fundamental frequency. The amplitude of this signal change proportionally with the input amplitude.

When the input amplitude increased, the output spectrum contains signals at the frequencies 21 and 31, and these signals called the second and third harmonics, originate from second- and third-order nonlinear circuit behavior. Respectively, as we shall see below. Harmonics higher than the third, caused by higher-order behavior,

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come above the noise floor at even higher input amplitudes. It is seen that the amplitude of the nth harmonic increases with the nth power of the input amplitude: an increase of the input amplitude with 6dB yields an increase of the second harmonic at the output with 12dB, the third harmonic increases with 18dB and so on. At high input amplitudes, this is not true anymore. Then it is observed that the third-order nonlinear behavior also give rise to a component at the fundamental frequency which increases with the third power of the input amplitude. As a result, the fundamental response can increase faster than linear, which for an amplifier means that the gain slightly increase.

In this case one speak about gain expansion. On the other hand, if the increase is less than linear because the sign of the third-order contribution is opposite to the sign of the linear response, then a gain compression is observed in the output. Similarly, forth-order behavior gives a contribution to the second harmonic and so on. This situation is depicted in Figure 2.1. Signals caused by nonlinear behavior of order higher than five are not shown in this figure. Also it must be noted that a component at 0Hz is found at the output. This DC shift is caused by second-order, forth-order, or in general, even-order nonlinear behavior.



₁

0 ^ 

₁

 

₁

 

₁

 

₁

(2) (4)

(1) (3)

(5) (2)

(4) (3)

(5) (4) (5)

frequency

Figure 2.1 The differential harmonics at the output of an analog circuit excited by a sinusoidal signal at frequency 1. The numbers between brackets indicate the order of nonlinear behavior by which the signal is determined.

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The nonlinear behavior as discussed above in a qualitative way can be clarified mathematically with a simple example. Assume that the relationship between the input signal x(t) (which is either a current or a voltage) of a circuit and the output y(t) is given by the following relationship

y t( )K x t1

 

K2



x t

  

²K3



x t

  

³... (2.1) When the input-output relationship is given explicitly by an analytic relationship ^{y t}

 

^{ f}

  

^{x t}



(2.2) Then the coefficients K1,K2,K3,... can be identified with the coefficients of a Taylor series of f : second-order and third-order nonlinearity coefficients, respectively, in general, high-order nonlinearity coefficients.

Returning to our example, we assure that the input signal x(t) has the form x t

 

Acos



1t1



(2.6) Substituting this expression into equation (2.4) yields the output y(t) :

 

1



1 1



² 2



1 1



0Hz. Both signals are proportional to K2 and to the square of the input amplitude A.

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Therefore, these signals are denoted as second-order signals. And the third-order signals at the frequency 31 and at the fundamental frequency 1 are with the same reason. Assure K₃ has the same sign as K₁, in this case the third-order signal at the fundamental frequency as the same sign at the first-order signal. In other words, the amplitude of the fundamental signal has increased due to third-order behavior. This situation corresponds to expansion. If K1 and K3 have an opposite sign then we have compression.

2.2.1 Two-frequency Excitation

In the same way as in the previous section, the test circuit under consideration is now excited with two sinusoids A₁cos(1t) and A₂cos(2t), both applied at the same input port. When A₁ and A₂ are sufficiently low, the output spectrum contains two signals above the noise floor at the fundamental frequency 1 and 2 due to the circuit's linear behavior. Because in a linear circuit the superposition principle is valid, the two excitations don't produce any interfering signal. However, When A1 and A2

become larger, then, apart from the harmonics of 1 and 2, interfering signals grow above the noise floor at the frequency 1+2, ∣1-2∣, 21+2, ∣21-2∣,

1+22 and ∣-1+22∣. The signals at ∣1±2∣are caused by second-order nonlinear behavior and are called second-order intermodulation products. They increase with the first power of both A₁ and A₂. The other signals come from third-order behavior and are denoted as third-order intermodulation products. The signals at ∣1±2∣increase with the square of A1 and with the first power of A2, and so on.

For analog circuits such as amplifiers, the intermodulation products are usually unwanted. Therefore, they are denoted as intermodulation distortion. In communication circuits, these unwanted products are often denoted as spurious responses. Then consider again the test circuit with the input-output relationship given

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by equation (2.1). When the input signal x(t) consists of two signals of equal amplitude and with a different frequency 1 and 2:

x t

 

Acos

 

1t Acos

 

2t (2.8) Then the different responses are shown in Figure 2.2. It is assumed that the amplitude A is sufficiently low such that the circuit behaves in a weakly nonlinear way. The frequency w1 and w2 are have been given the value *10 MHz and *11 MHz. It is seen that harmonics of 1 and 2 are present at the output as well as intermodulation products. with the input-output relationship given by equation (2.1) to a combination of two sinusoidal signals with the same amplitude A. The frequency of the two input signals are 10 MHz and 11 MHz.

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2.3 Description of Nonlinearities in

在文檔中 具有加強轉換增益及雜訊相消機制的 V-band四倍頻器 (頁 16-21)