RESPONSE CHARACTERISTICS OF A TRANSMISSION LINE BASED
TRANSFORMER
Akira Homma*
Division of Quantum Energy, Faculty of Engineering, Hokkaido University,North 13, West 8, Kita-ku, Sapporo 060-8628, Japan

Abstract
Response characteristics of a transmission line basedtransformer were studied to take into account signals thatinevitably appear in the space outside of the transmission
lines. A prototype of the transformer, using semi-rigid
co-axial cables, was constructed and tested to confirm theresults of the theory. Good agreement was obtained
between the theory and experiments in the 5 MHz to 3
GHz frequency range. A 3 dB bandwidth of thistransformer was also obtained from about 20 MHz to 1GHz.

1 INTRODUCTION
Transmission line based transformers (TLBT) havebeen widely used in fields requiring high-speed
high-power pulse techniques such as in linear accelerator
technology. However, only a little attention has been paid
to the frequency response characteristics. [1][2] In general,
the frequency response is one of the most importantcharacteristics when transformers are actually used. This
paper describes a theoretical approach to study frequency
response characteristics of a TLBT.
Based on the principles of operation TLBT can beclassified into two types. One, a transmission line
(normally co-axial cables) wound around a magnetic coresuch as ferrite or other magnetic material. A large
inductance can be made between input and output so thatthe transformer achieves high inductive isolation fortransient voltage changes between input and output.
However, even though an inversion of the signal ispossible, dc isolation of input and output is impossible.
The other type of TLBT uses one conductor of thetransmission line as the primary winding and the otherconductor as the secondary winding or vice versa. ATLBT of this type can invert the signals and isolation of
the dc potential of input and output is possible. [3][4] The
operational principle of this second type is similar to an
ordinary transformer. The TLBT considered in thisarticle belongs to the latter type.

impedance, Z/2. The inner and outer conductors of theco-axial lines correspond to the primary and secondarywindings of an ordinary transformer, which can isolate
the dc potential of the input and output. In the figure, thesolid arrows show the direction of the electric lines offorce for signals propagated inside the co-axial lines.
The input signal transmitted by the coaxial line withimpedance Z divides into lines 1 and 2 each withimpedances of Z/2 (at the junction S). From here thesignals travel in the lines and join at junction J where thesignal is reconstructed as the output and transmitted to theco-axial line with impedance Z. A small part of the signal
leaks from the junction of the co-axial lines at J. Thesignal is termed ‘the coil-mode signal’, and is illustrated
by the broken arrows in the figure. The coil-mode signal
travels in the space formed by the outer conductors of theco-axial lines 1 and 2, and this space is termed
the ’coil-mode line’. The signal is reflected at theshort-circuited end S, where the polarity is inverted andthe signal returns to junction J. Then a part of the signal isreflected and reenters the coil-mode line again, and theother part of the signal is transmitted as the output. Thus
the round trip of the coil-mode signal described above isrepeated. The coil-mode signal can be seen as anelectromagnetic field stored inside the space of thecoil-mode line.
The output voltage of the TLBT at any instant isdetermined by the sum of all the incident waves atjunction J. The following analysis assumes the coil-modeline to be a lossless ideal transmission line withimpedance αZ where the signal is transmitted by thefundamental electromagnetic mode with constantpropagation time. The coil-mode impedance wasdetermined with the characteristic impedance of a parallel
wire-line in air.
LTransmission Line 1 (Z/2)

Input (Z)Output (Z)JS
Coil Mode Signals

2 OPERATIONAL PRINCIPLE
As shown in Fig. 1, the transformer is composed oftwo co-axial lines, 1 and 2, with the same path length andTransmission Line 2 (Z/2)

Z=50Ω

Fig.1. The basis of the structure of the TLBT.
*homma@qe.eng.hokudai.ac.jp

3 RESPONSE FUNCTIONSThe unit impulse response function was determined
first. By considering multiple reflection of the voltage
wave in the coil-mode line, the unit impulse response ofthe TLBT is readily obtained from the total of the voltage
wave incidence in the output terminal. The detailed
procedure for deducing the response was reported
previously. [3][4] When an impulse signal is applied to theinput terminal at time t=0, the resulting impulse response
is given asn−1
2α 
2 ∞  2α −1 h(t ) =δ (t ) −∑ δ (t − nT ) (1)1+ 2α 1 + 2α n=1  1+ 2α where the T is the transit time for the coil-mode signal to
make a round trip on the coil-mode line. Here a stands
for the ratio of the coil-mode line impedance to the output(or input) impedance. The frequency response function
A(w) of the TLBT can be obtained by the Fouriertransform of Eq. (2) as follows

1+4 RESULTS
To confirm the theory, the TLBT shown in Fig. 1
was constructed and tested. Semi-rigid co-axial cables(Miyazaki Densen Kogyo 219TB-25) with impedance 25
W (Z/2) and outer diameter d=2.1 mm were used as theco-axial lines for the prototype TLBT. The TLBT had awire spacing D =30 mm and the line length L=120 mm.
Input and output of the TLBT maintain an impedance Z=50 W.
The gain was measured in the frequency range from5 MHz to 5 GHz by a Network Analyzer. The result waspresented in Fig. 3.

 2α −1 < 1 . (2) 1+ 2α 1A(ω) =The 3 dB bandwidth of the band pass region for theTLBT is determined by the fundamental coil-mode
frequency fc . As the ratio a increases from unity toinfinity, the bandwidth of the TLBT increases from zeroto 2fc . Although theoretically these pass bands repeat
indefinitely with increasing frequency, several effects nottaken into account in the theory presented here limits thehigh frequency response of the TLBT.

1  1+ cosωT 
4α 2  1− cosωT 

105coil mode line length=120 cm

The gain given by Eq. (2) has a maximum at thefrequency fc (wT=p) where the coil-mode line length is aquarter wavelength long. The fc is designated as thefundamental coil-mode frequency of the TLBT since thefrequency is determined by the time that is necessary forthe coil-mode signal to traverse the coil-mode line. The
gain is the minimum at even integer multiplies of fc . Eq.
(2) exhibits repeated symmetric band pass characteristics
centered at odd integer multiplies of fc .
To emphasize the properties of the gain
characteristics, the TLBT with the coil-mode line lengthof 150 mm (T =1 ns, fc =0.5 GHz) was calculated and isshown in Fig. 2.
10coil mode line length =150 cm
fc=0.5 GHz

52fc

4fc 6fc 8fc¦A(ω)¦ (dB)0α=12-5α=1

α=8α=4-10-15-20
5M

10M

100M

Frequency(Hz)1G

Fig. 2. Calculated response functions.

5G

Gain (dB)

4fc=2.09GHz
0-5-10

6fc=3.2GHz
2fc=1.05GHz-15-20
5M

10M

100M

1G

5GFrequency (Hz)Fig. 3. Response of the prototype TLBT.
The minimum gain at even integer multiplies of fc
were observed at frequencies of 2fc =1.05 GHz, 4fc =2.09
GHz, and 6fc =3.2 GHz. With averaged values of thesefrequencies, the coil-mode frequency was determined as fc=0.52 GHz (T = 0.96 nsec). At frequencies above 6fc , it
was difficult to identify the frequency of even integer
multiplies of fc because of a large increase in theattenuation at high frequencies that may be caused byelectromagnetic field leakage at the junctions of theco-axial lines or the coil-mode line.
At very low frequencies where the wavelength of theinput signal λ is much longer than the coil-mode line
length, the low cutoff frequency fLow-3dB can be written as

f Low − 3 dB =

1l

2πα T

λ1.

(3)

From the wire spacing and outer diameter of thecables, the coil-mode line impedance can be estimated as≅ 400 W (a= 8) and then fLow-3dB ≅21 MHz was obtainedby Eq. (3). With the result of the experiments shown inFig. 3, fLow-3dB can be estimated to be about ≅21 MHz
which agrees with the value estimated by the theory. If
good low frequency characteristics are required, anincrease in a with an increase in the distance between theco-axial lines or an increase in T lengthening thecoil-mode line will be effective. Another way to increasea is to place magnetic material around the space of thecoil-mode line. Then the transit time T will also increase
because of the decrease in propagation velocity of thecoil-mode signals.
The results also exhibit the 3 dB bandwidth of theTLBT as the practical operating range can be evaluatedfrom about 20 MHz to 1 GHz.
The measured fundamental coil-mode frequency fc=0.52 GHz was lower than the value of fc =0.625 GHz
expected as the coil-mode signal propagates with light
velocity in the 120 mm long coil-mode line. This implies
that the coil-mode signals may not propagate with thefundamental electromagnetic mode in the coil-mode line.
Therefore, an estimate of the transit time T would benecessary in the design of an actual TLBT when a precise
determination of the bandwidth is required.
The theory described here would be useful in thedesign of actual high frequency transformers and detailsof this will be reported elsewhere. [5]REFERENCES[1] J. O. Rossi and P. W. Smith, IEE Colloquium on
Pulsed Power ’96 (Ref. No. 1996 / 059), 19 / 1-3, 140(1996).
[2] A.N. Other, “A Very Interesting Paper”, EPAC’96,
Sitges, June 1996.
[3] A. Homma and H.Yamazaki, Bulletin of the Facultyof Engineering, Hokkaido University, No. 131, 1986,
p. 37.
[4] A.Homma, Rev. Sci. Instrum. 70, 232 (1999).
[5] A.Homma, Rev. Sci. Instrum. to be published.