Balanced Slope Demodulator EEC 112
The circuit below is a balanced FM slope demodulator.
ω 01 Ideal Ideal ω 02
+ v o(t)−
+ +
i i(t) C1 L1 R1 R0 C0 v o1 v o2 C0 R 0 R2 L2 C2 i i(t)
− −
It is the same as the circuit in Fig. 12.4-3 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 as ω 0 and the
resonant frequencies as ω C1 and ω C2 . These notes define the carrier frequency as ω C and
the resonant frequencies as ω 01 and ω 02 , in keeping 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 δ ω above ω C , and the other is tuned to δ ω below ω C . Let
ω 01 be the resonant frequency of the R1 L1 C1 circuit, and assume that ω 01 > ω C . Also, let
ω 02 be the resonant frequency of the R2 L2 C2 circuit. Then δ ω = ω 01 − ω C = ω C − ω 02 .
The balanced slope demodulator has v o(t) = 0 for 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 com-
pared 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 + ∆ω f (t)) for
ω in the filter transfer functions, where ∆ω is the peak frequency deviation of the FM sig-
nal. Here, the filters are the tuned circuits R1 L1 C1 and R2 L2 C2 . The book derives the fol-
lowing inequality, which shows when the quasi-static approximation is valid:
2
∆ω ω m ∆ωω m ∆ωω m ∆ω 1 ∆ω
= = = << 1 (11.3-17)
α α α2 α 2 ∆ω β α
where β = ∆ω /ω m is the modulation index, and ω m is the modulation frequency. Also,
α1 = ω 01 /(2Q1) and α 2 = ω 02 /(2Q2), where Q1 and Q2 are the quality factors of the two
tuned circuits. The derivation of (11.3-17) is not repeated here, but the book points out
that this equation shows the quasi-static approximation is valid when ∆ω << α and/or
ω m << α . Since α = ω 0 /2Q is half the bandwidth of a tuned circuit, the quasi-static con-
dition is that the modulation frequency and/or the peak frequency deviation are much less
than half the bandwidth of each of the tuned circuits.
-2-
This result can be understood intuitively by remembering the plot of the instanta-
neous frequency versus time for the case of co-sinusoidal frequency modulation shown
below and in class.
ω i(t) ωm
ω C + ∆ω
ωC
ω C − ∆ω
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
t
i i(t) = I1 cos ω C t + ∆ω
∫ f (θ )dθ
(12.4-10)
Also, the output is
v o(t) = v o1(t) − v o2(t)
= I1|Z11[ jω C + j∆ω f (t)]|1 − I1|Z11[ jω C + j∆ω f (t)]|2
= I1 {|Z11[ jω C + j∆ω f (t)]|1 − |Z11[ jω C + j∆ω f (t)]|2 }
= I1|Z T [ jω C + j∆ω f (t)]| (12.4-11)
where |Z T ( jω )| = |Z11( jω )|1 − |Z11( jω )|2 and Z11(s)1 and Z11(s)2 are the input impedances of
the parallel R1 L1 C1 and R2 L2 C2 circuits, respectively.
Since |Z T ( jω )| = |Z11( jω )|1 − |Z11( jω )|2 ,
|Z T ( jω )|′ = |Z11( jω )|′1 − |Z11( jω )|′2
1 − |Z11( jω )|′′
|Z T ( jω )|′′ = |Z11( jω )|′′ 2
1 − |Z11( jω )|′′′
|Z T ( jω )|′′′ = |Z11( jω )|′′′ 2
etc. Therefore, a Taylor-series analysis of the balanced slope demodulator involves find-
ing 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 |Z11( jω )| and its first five
derivatives. More details than shown in the book are presented below.
-3-
The input impedance of one parallel tuned circuit was derived in class and is
R R
Z11( jω ) = ≈
ω ω0 ω − ω0
1 + jQ − 1+ j
ω0 ω α
Therefore, the magnitude of this impedance is
R
|Z11( jω )| = 0.5
2
ω − ω0
1 +
α
The first five derivatives of |Z11( jω )| are as follows:
R ω − ω0
α α
|Z11( jω )|′ = − 1.5
2
ω − ω0
1 +
α
2
R ω − ω0
2 − 1
α2 α
|Z11( jω )|′′ = 2.5
2
ω − ω0
1 +
α
3
−3R ω − ω 0 ω − ω 0
2 − 3
α3 α α
|Z11( jω )|′′′ =
2 3.5
ω − ω0
1 +
α
4 2
3R ω − ω 0 ω − ω0
8 − 24 + 3
α4 α α
|Z11( jω )|′′′′ = 4.5
2
ω − ω0
1 +
α
5 3
15R ω − ω 0 ω − ω0 ω − ω 0
−8 + 40 − 15
α5 α α α
|Z11( jω )|′′′′′ =
2 5.5
ω − ω0
1 +
α
-4-
Evaluating these terms at the carrier frequency ω = ω C gives
R
|Z11( jω C )| = 0.5
2
ωC − ω0
1 +
α
R ωC − ω0
α α
|Z11( jω C )|′ = − 1.5
2
ωC − ω0
1 +
α
2
R ωC − ω0
2 − 1
α2 α
|Z11( jω C )|′′ =
2 2.5
ωC − ω0
1 +
α
3
−3R ω C − ω 0 ω C − ω 0
2 − 3
α3 α α
|Z11( jω C )|′′′ =
2 3.5
ωC − ω0
1 +
α
4 2
3R ω C − ω 0 ωC − ω0
8
− 24 + 3
α
4 α α
|Z11( jω C )|′′′′ =
2 4.5
ωC − ω0
1 +
α
5 3
15R ω C − ω 0 ωC − ω0 ω − ω 0
−8 + 40 − 15 C
α
5 α α α
|Z11( jω C )|′′′′′ =
2 5.5
ωC − ω0
1 +
α
-5-
Now extend the above results to the balanced slope demodulator, which uses two
tuned circuits with resonant frequencies ω 01 and ω 02 . Consider |Z T ( jω C )|, which is
|Z T ( jω C )| = |Z11( jω C )|1 − |Z11( jω C )|2
R1 R2
|Z T ( jω C )| = −
2 0.5 2 0.5
ω C − ω 01 ω C − ω 02
1 + 1 +
α1 α2
Assume that R1 = R2 = R and α1 = α 2 = α . Since ω 01 − ω C = ω C − ω 02 = δ ω ,
R R
|Z T ( jω C )| = − =0
2 0.5 2 0.5
−δ ω δ ω
1 + 1 +
α α
because (−δ ω )2 = (δ ω )2 . This result is important because it means that a properly
designed balanced slope demodulator produces zero output at the carrier frequency. As a
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 R1 ≠ R2 for example) allows some AM to appear
at the output.
Similarly,
1 − |Z11( jω C )|′′
|Z T ( jω C )|′′ = |Z11( jω C )|′′ 2
2 2
R1 ω C − ω 01 R2 ω C − ω 02
2 − 1 2 − 1
α1 2 α1
α 22 α 2
=0
|Z T ( jω C )|′′ = 2.5
− 2.5
2 2
ω C − ω 01 ω C − ω 02
1 + 1 +
α1 α2
when R1 = R2 = R, α1 = α 2 = α , and ω 01 − ω C = ω C − ω 02 = δ ω because
2 2
(−δ ω ) = (δ ω ) . 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 | is constant. 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 .
-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:
(ω − ω C )3
|Z T ( jω )| = |Z T ( jω C )|′(ω − ω C ) + |Z T ( jω C )|′′′
3!
(ω − ω C )5 ...
+ |Z T ( jω C )|′′′′′
+ (12.4-12)
5!
In this equation, the third-order term can be set to zero by design by choosing the reso-
nant frequencies of the two tuned circuits to be symmetrical around the carrier frequency
with a carefully chosen frequency difference (δ ω = ω 01 − ω C = ω C − ω 02 ). In the third-
order term,
1 − |Z11( jω C )|′′′
|Z T ( jω C )|′′′ = |Z11( jω C )|′′′ 2
3
3R1 ω C − ω 01 ω − ω 01 3R ω C − ω 02 3 ω C − ω 02
− 2
−3 C − 32 2 −3
α1 α1 α1 α 2
− α2 α
3
= 2
2 3.5 2 3.5
ω C − ω 01 ω C − ω 02
1 + 1 +
α1 α2
2 2
3R1 ω C − ω 01 ω C − ω 01 3R2 ω C − ω 02 ω C − ω 02
− 3 2 − 3 − 3 2 − 3
α1 α1 α1
− α2
α2 α2
=
2 3.5 2 3.5
ω C − ω 01 ω − ω 02
1 + 1+ C
α1 α2
As mentioned above, let R1 = R2 = R, α1 = α 2 = α , ω C − ω 01 = − δ ω , and
ω C − ω 02 = δ ω . Then
2 2
3R −δ ω −δ ω 3R δ ω δ ω
− 3 2 − 3 − 3 2 − 3
α α α − α
α α
|Z T ( jω C )|′′′ = 3.5 3.5
2 2
−δ ω δ ω
1 + 1 +
α α
2
6R δ ω δ ω
2 − 3
α3 α α
|Z T ( jω C )|′′′ = 3.5
2
δ ω
1 +
α
Therefore, |Z T ( jω C )|′′′ = 0 when 2(δ ω /α )2 = 3 or
√
3
δω = α (12.4-14)
2
-7-
With this frequency difference, the coefficients in the first- and fifth-order terms in
(12.4-12) can be calculated as follows. In the first-order term,
R1 ω C − ω 01 R2 ω C − ω 02
α1 α1 α2 α2
|Z T ( jω C )|′ = |Z11( jω C )|1′ − |Z11( jω C )|2′ = − 1.5
+ 1.5
2 2
ω C − ω 01 ω C − ω 02
1 + 1 +
α1 α2
2R 3
R −δ ω R δ ω 2R δ ω
α
√
2
√
α α α α α α 4R 3
=− + = = =
2 1.5 2 1.5 2 1.5 √
2. 5(2. 5) 5α 5
−δ ω δ ω δ ω
1 + 1 + 1 +
α α α
-8-
In the fifth-order term,
|Z T ( jω C )|′′′′′ = |Z11( jω C )|1′′′′′ − |Z11( jω C )|2′′′′′
5 3
15R1 ω C − ω 01 ω C − ω 01 ω C − ω 01
−8 + 40 − 15
α15 α1 α1 α 1
=
2 5.5
ω C − ω 01
1 +
α1
5 3
15R2 ω C − ω 02 ω C − ω 02 ω C − ω 02
−8 + 40 − 15
α 25 α 2 α2 α 2
−
2 5.5
ω C − ω 02
1 +
α2
5 3 5 3
15R −δ ω −δ ω −δ ω 15R δ ω δ ω δ ω
−8 + 40 − 15 −8 + 40 − 15
α5 α α α α5 α α α
= −
2 5.5 2 5.5
−δ ω δ ω
1 + 1+
α
α
5 3 5 3
15R δ ω δ ω δ ω 15R δ ω δ ω δ ω
8
− 40 + 15 8 − 40 + 15
α α
5 α α α α
5 α α
= +
2 5.5 2 5.5
δ ω δ ω
1 + 1+
α
α
5 3
30R 3 3
√
√
√
5 3 3
30R δ ω δ ω δ ω 8 − 40 + 15
8 − 40 + 15
α α
5 α α α5 2 2 2
= =
2 5.5 5.5
δ ω 3
1 + 1 +
α 2
√
30R 3 9 3
√
8 − 40 + 15 30R 3
(−27)25
√
α5 2 4 2
= = α5 5 = − 96R 54 3
√ (2. 5) [(2. 5) 5] (5) 5 α 5 625 5
-9-
Substituting |Z T ( jω C )|′ and |Z T ( jω C )|′′′′′ into (12.4-12) gives
√
5
4R 3 ω − ω C 96R 54 ω − ω C
|Z T ( jω )| = − + ...
5 5 α 5! 625 α
√
5
4R 3 ω − ωC 54 ω − ω C
= − + ... (12.4-15)
5 5 α 625 α
Substituting this equation into (12.4-11) with ω = ω C + ∆ω f (t) gives
v o(t) = I1|Z T [ jω C + j∆ω f (t)]|
√
5
4I R 3 ∆ω f (t) 54 ∆ω f (t)
= 1 − + ... (12.4-16)
5 5 α 625 α
The ratio of the magnitudes of the fifth-order term to the first-order term in (12.4-16) is
4
5th order term 54 ∆ω
= f (t)
1st order term 625 α
If this ratio is less than 0.01, then the output of the balanced slope demodulator is approx-
imately proportional to f (t). Assuming that f (t) ≤ 1,
4
54 ∆ω
f (t) ≤ 0. 01
625 α
when
∆ω
≤ 0. 583
α
However, when ∆ω /α is this large, the quasi-static approximation is strained. From
(11.3-17), the quasi-static approximation is valid when
2
∆ω ω m ∆ωω m ∆ωω m ∆ω 1 ∆ω
= = = << 1
α α α2 α 2 ∆ω β α
The book assumes that this condition is satisfied when << 1 in the above inequality means
∆ω
no more than 5%. Substituting = 0. 583 gives
α
1
(0. 583)2 ≤ 0. 05
β
This condition is satisfied for β = ∆ω /ω m ≥ 6. 8. To allow β ≥ 5, the book states that
∆ω /α = 0. 5 is usually chosen in practice.
