Balanced Slope Demodulator EEC 11 The circuit below isabalanced FM slope demodulator. ω 01 i i (t) C 1 L 1 1 Ideal +v o (t) 0 C 0 v o1 v o + + C 0 Ideal 0 ω 0 L C i i (t) It is the same as the circuit in Fig. 1.4- in the Clarke and Hess book except that the variables used to define the FM carrier frequency and the resonant frequencies of the two tuned circuits have been changed. The book defines the carrier frequency asω 0 and the resonant frequencies as ω C1 and ω C. These notes define the carrier frequency asω C and the resonant frequencies as ω 01 and ω 0,inkeeping with the notation used in class this term. The balanced slope demodulator consists of two single-ended slope demodulators. One is tuned to a frequency δω above ω C,and the other is tuned to δω below ω C. Let ω 01 be the resonant frequency ofthe 1 L 1 C 1 circuit, and assume that ω 01 > ω C. Also, let ω 0 be the resonant frequency ofthe L C circuit. Then δω ω 01 ω C ω C ω 0. The balanced slope demodulator has v o (t) 0for ω i ω C,which means it is insensitive to inputs with amplitude modulation (or envelope variations) at ω C. Also, the balanced slope demodulator has less nonlinearity than a single-ended demodulator. The analysis assumes that the envelope detectors do not load the tuned circuits and that the quasi-static approximation is valid. The quasi-static approximation is that f (t), a signal whose amplitude is 1 and is proportional to the modulation, changes slowly compared to the duration of the impulse responses of the filters that operate on the FM signal. When the quasi-static approximation is valid, the frequency-domain outputs of the filters can be found by substituting the instantaneous FM frequency (ω i (t)ω C + ωf(t)) for ω in the filter transfer functions, where ω is the peak frequency deviation of the FM signal. Here, the filters are the tuned circuits 1 L 1 C 1 and L C. The book derives the following inequality, which shows when the quasi-static approximation is valid: ω ω m ωω m ωω m ω ω 1 ω β << 1 (11.-17) where β ω /ω m is the modulation index, and ω m is the modulation frequency. Also, 1 ω 01 /(Q 1 )and ω 0 /(Q ), where Q 1 and Q are the quality factors of the two tuned circuits. The derivation of (11.-17) is not repeated here, but the book points out that this equation shows the quasi-static approximation is valid when ω << and/or ω m <<. Since ω 0 /Q is half the bandwidth of a tuned circuit, the quasi-static condition is that the modulation frequency and/or the peak frequency deviation are much less than half the bandwidth of each of the tuned circuits.
-- This result can be understood intuitively by remembering the plot of the instantaneous frequency versus time for the case of co-sinusoidal frequency modulation shown below and in class. ω i (t) ω C + ω ω C ω C ω ω m t In this case, f (t) cos ω m t. When this modulation signal changes slowly, the maximum slope in the above plot is small. Since the slope of this cosine wave is proportional to both its frequency ω m and its peak amplitude ω,the condition that ω m and/or ω be small for the quasi-static approximation to be valid is reasonable. The book assumes that input currents in the balanced slope demodulator are Also, the output is v o (t) v o1 (t)v o (t) t i i (t) I 1 cos ω C t+ ω f(θ)dθ (1.4-10) I 1 Z 11 [ jω C +j ωf(t)] 1 I 1 Z 11 [ jω C +j ωf(t)] I 1 { Z 11 [ jω C +j ωf(t)] 1 Z 11 [ jω C +j ωf(t)] } I 1 Z T [jω C +j ωf(t)] (1.4-11) where Z T ( jω) Z 11 ( jω) 1 Z 11 ( jω) and Z 11 (s) 1 and Z 11 (s) are the input impedances of the parallel 1 L 1 C 1 and L C circuits, respectively. Since Z T ( jω) Z 11 ( jω) 1 Z 11 ( jω), Z T ( jω ) Z 11 ( jω) 1 Z 11 ( jω) Z T ( jω ) Z 11 ( jω) 1 Z 11 ( jω) Z T ( jω) Z 11 ( jω) 1 Z 11 ( jω) etc. Therefore, a Taylor-series analysis of the balanced slope demodulator involves finding the magnitude of Z T ( jω) and its derivatives. This derivation starts by finding the magnitude of the input impedance of one parallel tuned circuit Z 11 ( jω) and its first five derivatives. More details than shown in the book are presented below.
-- The input impedance of one parallel tuned circuit was derived inclass and is Z 11 ( jω) 1 +jq ω ω 0 1 +j ωω 0 ω 0 ω Therefore, the magnitude of this impedance is Z 11 ( jω) 1 + ω ω 0 The first five derivatives of Z 11 ( jω) are as follows: Z 11 ( jω) 1 + ω ω 0 ω ω 0 0. 1. Z 11 ( jω) ω ω 0 1 + ω ω 0 1. Z 11 ( jω) ω ω 0 1 + ω ω 0 ω ω 0. Z 11 ( jω) 4 8 ω ω 0 4 4 1 + ω ω 0 ω ω 0 4. + Z 11 ( jω) 1 8 ω ω 0 + 40 ω ω 0 1 + ω ω 0. 1 ω ω 0
-4- Evaluating these terms at the carrier frequency ω ω C gives Z 11 ( jω C ) 1 + ω C ω 0 Z 11 ( jω C ) 1 + ω C ω 0 ω C ω 0 0. 1. Z 11 ( jω C ) ω C ω 0 1 + ω C ω 0 1. Z 11 ( jω C ) ω C ω 0 1 + ω C ω 0 ω C ω 0. Z 11 ( jω C ) 4 8 ω C ω 0 4 4 1 + ω C ω 0 ω C ω 0 4. + Z 11 ( jω C ) 1 8 ω C ω 0 + 40 ω C ω 0 1 + ω C ω 0. 1 ω C ω 0
-- Now extend the above results to the balanced slope demodulator, which uses two tuned circuits with resonant frequencies ω 01 and ω 0.Consider Z T ( jω C ),which is Z T ( jω C ) Z 11 ( jω C ) 1 Z 11 ( jω C ) 1 Z T ( jω C ) 1 + ω C ω 01 0. 1 1 + ω C ω 0 Assume that 1 and 1.Since ω 01 ω C ω C ω 0 δω, Z T (jω C ) 1 + δω 0. 1 + δω 0. 0 because (δω) (δω). This result is important because it means that a properly designed balanced slope demodulator produces zero output at the carrier frequency. Asa result, amplitude modulation (AM) on the carrier does not appear at the output under ideal conditions. In practice, however, mismatch between the two halves of the balanced slope demodulator (possibly caused by 1 for example) allows some AM to appear at the output. Similarly, Z T ( jω C ) Z 11 ( jω C ) 1 Z 11 ( jω C ) Z T ( jω C ) 1 1 ω C ω 01 1 1 + ω C ω 01 1 1. ω C ω 0 1 0 1 + ω C ω 0. when 1, 1, and ω 01 ω C ω C ω 0 δω because (δω) (δω). Furthermore, Z T ( jω C ) and all other even-order terms in the Taylor series for Z T ( jω) turn out to be zero under the same assumptions for the same reason. This result could have been anticipated based on the following observation. Z T ( jω C ) is the difference in magnitudes of the input impedances of two parallel tuned circuits. Each of these input impedances can be expressed as a Taylor series with both even- and odd-order terms. In each of these Taylor series, changing the frequency ω from greater than ω C to less than ω C does not change the amplitude of the even-order terms as long as the magnitude of the frequency difference ω ω C isconstant. Therefore, the even-order terms in the Taylor series for one input impedance are identical to those terms in the Taylor series for the other input impedance, and the difference in these impedances has zero amplitude in all even-order terms, causing Z T ( jω C ) to have odd symmetry around ω C. 0.
-6- As a result, a properly designed balanced slope demodulator not only is insensitive to AM, but also is more linear than the single-ended slope demodulator. Thus, a Taylor series expansion of Z T ( jω) can be simplified to show only odd-order terms: Z T ( jω) Z T ( jω C ) (ω ω C ) + Z T ( jω C ) (ω ω C) + Z T ( jω C ) (ω ω C) +... (1.4-1)! In this equation, the third-order term can be set to zero by design by choosing the resonant frequencies of the two tuned circuits to be symmetrical around the carrier frequency with a carefully chosen frequency difference (δω ω 01 ω C ω C ω 0 ). In the thirdorder term, Z T ( jω C ) Z 11 ( jω C ) 1 Z 11 ( jω C )! 1 1 ω C ω 01 1 1 + ω C ω 01 1 ω C ω 01 1. ω C ω 0 1 + ω C ω 0 ω C ω 0. 1 1 ω C ω 01 1 ω C ω 01 1 1 + ω C ω 01 1. ω C ω 0 ω C ω 0 1 + ω C ω 0 As mentioned above, let 1, 1, ω C ω 01 δω, and ω C ω 0 δω.then δω δω Z T ( jω C ) 1 + δω Z T ( jω C ) 6 δω δω 1 + δω. Therefore, Z T ( jω C ) 0when (δω/) or. δω δω 1 + δω. δω (1.4-14).
-7- With this frequency difference, the coefficients in the first- and fifth-order terms in (1.4-1) can be calculated as follows. In the first-order term, 1 ω C ω 01 Z T ( jω C ) Z 11 ( jω C ) 1 Z 11 ( jω C ) 1 1 1 + ω C ω 01 1 1. + ω C ω 0 1 + ω C ω 0 1. δω 1 + δω 1. + δω 1 + δω 1. δω 1 + δω 1.. (.) 4
In the fifth-order term, Z T ( jω C ) Z 11 ( jω C ) 1 Z 11 ( jω C ) -8-1 1 1 8 ω C ω 01 1 + 40 ω C ω 01 1 1 + ω C ω 01 1. 1 ω C ω 01 1 1 8 ω C ω 0 + 40 ω C ω 0 1 + ω C ω 0. 1 ω C ω 0 1 8 δω + 40 δω 1 + δω 1 δω. 1 8 δω + 40 δω 1 + δω. 1 δω 1 8 δω 0 8 δω 40 δω 1 + δω 40 δω 1 + δω +1 δω. + +1 δω. 1 8 δω 0 8 40 δω 1 + δω. 40 1 + +1 δω. +1 0 8 9 4 40 +1 (.)[(.) ] 0 (7) 96 () 4 6
-9- Substituting Z T ( jω C ) and Z T ( jω C ) into (1.4-1) gives Z T ( jω) 4 4 ω ω C 96! ω ω C 4 6 4 6 ω ω C ω ω C Substituting this equation into (1.4-11) with ω ω C + ω f (t) gives v o (t) I 1 Z T [jω C +j ωf(t)] 4I 1 ω f (t) 4 6 ω f (t) +... +... (1.4-1) +... (1.4-16) The ratio of the magnitudes of the fifth-order term to the first-order term in (1.4-16) is thorder term 1st order term 4 6 ω f (t) If this ratio is less than 0.01, then the output of the balanced slope demodulator is approximately proportional to f (t). Assuming that f (t) 1, when 4 ω 6 ω f (t) 4 0. 8 0. 01 However, when ω / is this large, the quasi-static approximation is strained. From (11.-17), the quasi-static approximation is valid when ω ω m ωω m ωω m 4 ω ω 1 ω β The book assumes that this condition is satisfied when << 1 in the above inequality means no more than %. Substituting ω 0. 8 gives 1 β (0. 8) 0. 0 This condition is satisfied for β ω /ω m 6. 8. Toallow β, the book states that ω / 0. is usually chosen in practice. << 1